Machine-learning Guided Search for Phonon-mediated Superconductivity in Boron and Carbon Compounds

Niraj K. Nepal¹ [email protected],[email protected] Lin-Lin Wang^1,2 [email protected] [1] Ames National Laboratory, Ames, Iowa 50011, USA [2] Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA

Abstract

We present a workflow that iteratively combines ab-initio calculations with a machine-learning (ML) guided search for superconducting compounds with both dynamical stability and instability from imaginary phonon modes, the latter of which have been largely overlooked in previous studies. Electron-phonon coupling (EPC) properties and critical temperature (T_c) of 417 boron, carbon, and borocarbide compounds have been calculated with density functional perturbation theory (DFPT) and isotropic Eliashberg approximation. Our study addresses T_c convergence of Brillouin zone sampling with an ansatz test, stabilizing imaginary phonon modes for significant EPC contributions and comparing performance of two ML models especially when including compounds of dynamical instability. We predict a few promising superconducting compounds with formation energy just above the ground state convex hull, such as Ca₅B₃N₆ (35 K), TaNbC₂ (28.4 K), Nb₃B₃C (16.4 K), Y₂B₃C₂ (4.0 K), Pd₃CaB (7.0 K), MoRuB₂ (15.6 K), RuVB₂ (15.0 K), RuSc₃C₄ (6.6 K) among others.

I Introduction

The pursuit of high-temperature superconductivity (SC) is a challenging and active research area. The recent discovery of superconducting temperature (T_c) near 200 K in H₃S at the high pressure of 150 GPa[1] has re-energized the field to focus on phonon-mediated SC. However, at ambient pressure, increasing T_c even just higher than the 40 K of MgB₂ [2, 3, 11, 5, 6] has been difficult [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. While the recent report of a T_c=32 K for MoB₂ under the pressure of 110 GPa is a promising development [20], the large temperature and pressure gaps between MgB₂ and H₃S still remain, which motivates intensive search for new phonon-mediated SC compounds. Despite the challenges faced in experiments, theoretical studies continue to provide valuable insights into potential SC materials and their properties, such as SC in FeB₄[21, 22] was predicted and verified. Density functional perturbation theory (DFPT) [23, 24] is one of the most robust ab-initio methods [10] to compute the electron-phonon coupling (EPC) matrices over the full Brillouin zone (BZ). One can then employ either isotropic Eliashberg approximation or Green function-based anisotropic Migdal-Eliashberg equations to compute T_c [26, 27, 28, 1, 30, 31, 32, 33]. Therefore, computational exploration of compounds containing atoms slightly heavier than H, such as B and C[34, 35, 36, 37, 38, 39], with strong EPC is a promising venue to search for high-temperature SC at ambient pressure and also fill the large materials gap between MgB₂ and H₃S.

Machine learning (ML) and Artificial intelligence (AI) are increasingly taking important roles for predicting materials properties including SC. Previous studies performing ML predictions based on random forest [40], regression [41, 42, 43, 44], classification [41], natural language processing (NLP) [45], and deep learning [46] models have trained on experimental data, mostly from SuperCon database [36]. Comparing to the more expensive and time consuming experiments to explore many new compounds to generate SC data for ML, the high throughput (HT) ab-initio first-principles approaches are valuable tools to obtain the data that can be trained to predict potential SC compounds. Several recent studies have utilized ab-initio computed data for training ML model and predicting SC. One approach involves performing BCS-inspired screening of materials to identify potential candidates based on certain key properties such as the Debye temperature and the density of states at the Fermi level (N(E_F)) [9]. A study similar to that described in Ref. [9] has been conducted on a vast range of materials, but restricting the size of the compounds to eight or fewer atoms, as reported in Ref. [49]. In a recent study [50], a ML approach was used to predict the maximum T_c and corresponding pressure of binary metal hydrides. The input layer consisted of atomic properties of the heavier metallic atom, while the output layer had two nodes representing T_c and pressure. The ab-initio data utilized in this study has been collected from literature. Recently, ML-driven search with experimental feedback was also performed to discover a novel superconductor in Zr-In-Ni systems [51].

Notably in these previous HT and ML studies, compounds of dynamical instability with imaginary phonon modes have been largely discarded. However, as shown by our recent EPC study [52] on Y₂C₃ with experimentally known T_c=18K[18], imaginary phonon of C dimer wobbly motion once stabilized can carry significant EPC contributions, which explains well the observed sizable T_c. In recent model Hamiltonian studies, phonon softening and anharmonicity have also been found to enhance T_c [53, 54]. Here we present a workflow that iteratively combines ab-initio calculations with an ML-guided search across the dataset of compounds with both dynamical stability and instability from imaginary phonon modes by focusing on boron/carbon/borocarbide (B/C/B+C) compounds. Ab-initio calculations were performed to compute the EPC strength ( $\lambda$ ), the logarithmic average phonon frequency ( $\omega_{log}$ ) and T_c of 417 compounds employing DFPT and isotropic Eliashberg approximation. Two major issues arise during DFPT calculations: choosing appropriate BZ sampling (k and textbfq-mesh) for convergence and the problem of calculated dynamic instability. To address the convergence problem, we developed an ansatz test to check the convergence of EPC properties, particularly the T_c. For dynamically unstable compounds, we employed large electronic smearing, lattice distortion, and pressure to stabilize them. We then calculated their EPC properties, which were included in building the ML models. We evaluated ML models, specifically the crystal graph convolutional neural network (CGCNN) and the atomistic line graph neural network (ALIGNN), trained utilizing ab initio computed data to predict SC properties. Among the two models, especially when including the dynamically unstable compounds, ALIGNN consistently outperforms the CGCNN in predicting EPC properties. We predict a few promising SC compounds with formation energy just above the ground state convex hull. For dynamically stable systems, we predict TaNbC₂ (28.4 K), Nb₃B₃C (16.4 K), Y₂B₃C₂ (4.0 K) among others. For systems with dynamical instability and imaginary phonon, we predict Ca₅B₃N₆ with a T_c as high as 35-42.4 K, besides Pd₃CaB (7.0 K), and a few Ru compounds of MoRuB₂ (15.6 K), RuVB₂ (15.0 K), and RuSc₃C₄ (6.6 K).

II Results

II.1 Machine learning guided workflow

A ML-guided search workflow is employed in this study. It can be divided into three parts: data extraction, DFPT calculations, and training ML models. We will discuss each part in the following three sections (A, B and C) before presenting the main results in the last three sections (D, E and F). As illustrated in Fig. 1 for obtaining the crystal structures of all the known B/C/B+C compounds, we utilized the Materials Project (MP) database [55, 56, 57], which offers a diverse range of compounds, including experimentally synthesized and theoretically predicted ones. We selected those B and C compounds in the MP database that meet certain criteria: being metallic with negative formation energy, excluding oxides, C₆₀ and Lanthanides except for La. Approximately 1500 compounds fall within this category, out of which 400 exhibit magnetic moments, and these magnetic cases are not considered in the present study. Consequently, our focus narrows down to around 1100 nonmagnetic compounds, forming the pool for investigating phonon-mediated superconductivity. To manage computational cost, we set a further criterion that considers only systems with primitive cells containing 40 atoms or less and composed of up to four different elements ( $N_{type}\leq 4$ ). This refinement narrows our selection to approximately 700 compounds. These 700 include 121 compounds with known T_c from experimental measurement as in SuperCon database (113 being dynamically stable and 8 dynamically unstable) and the other 579 compounds of unknown T_c. We will first discuss the 113 compounds with known T_c and also dynamical stability (no imaginary phonon modes), while the remaining 8 compounds with dynamical instability will be discussed later.

Refer to caption — Figure 1: Machine learning workflow. The numbers are the counts of compounds involved in each step. See main text for the detailed summary.

Figure 2 provides a statistical description of the 113 B/C/B+C compounds with known T_c and dynamic stability, which are 53 superconductors (SC) and 60 non-superconductors (NSC). We reviewed and corrected any inaccuracies or discrepancies found in the SuperCon database through an extensive literature review [58, 15, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 2, 87, 88, 89, 44, 91, 92, 93, 94, 95, 85, 96, 97, 98, 99, 100, 42, 102, 103, 104, 105, 106, 107, 41, 21, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 20, 127, 128, 129, 130, 37, 132, 133, 134, 135, 136, 19, 137, 138, 139, 79, 140, 141, 142, 46].

Figure 2 (a)-(c) illustrate the distribution of different elements in these systems, allocation of B/C/B+C compounds and thermodynamic stability, respectively. The thermodynamic stability of these compounds, indicated by the energy above the ground state convex hull ( $\Delta E_{h}$ ), are obtained from the MP database[55]. Approximately 71% of the compounds are on the ground state convex hull with $\Delta E_{h}\sim 0$ . Around 18% of the cases have an $\Delta E_{h}$ within 0.05 eV/atom, while the remaining 11% have $\Delta E_{h}$ larger than 0.05 eV/atom. Figure 2(d) shows the distributions of these compounds based on their space groups (SGs). In Fig. 2(a) and (d), each bar is partitioned into two segments with the red segment representing the number of SC, while blue segment for NSC. From Fig. 2(a), many known SC compounds are associated with transition metals (TM) such as Y, La, Ni, Rh, Mo, Nb and others. Figure 2(e)-(m) depict the crystal structures of the representative SC compounds in the top 6 SGs among the experimentally known SC. In these structures, B and C atoms form various structural motifs: honeycomb lattice of B in MgB₂ (Fig. 2(e)); monomers in NbC (Fig. 2(f)), MgCNi₃ (Fig. 2(h)) and Mo₂GaC (Fig. 2(i)); dimers in YC₂ (Fig. 2(k)); graphene sheets in SrC₆ (Fig. 2(j)); chains of lighter elements in Mo₂BC (Fig. 2(g)) and LaPt₂B₂C (Fig. 2(m)); octahedral cage structures in YB₆ (Fig. 2(l)). In terms of the lattice types of these known SC compounds, highly symmetric structures with hexagonal, tetragonal and cubic SGs have the most compounds, as shown in Fig. 2(d). Similarly, statistical description for the SC compounds with predicted T_c that have not been measured, akin to Fig. 2, is illustrated in Supplementary Materials (SM) Fig.S1.

II.2 Overview of DFPT calculations

After the crystal structures of B and C compounds have been collected, the next step as shown in Fig.1 is to do HT calculations with DFPT on EPC properties for compounds with both known and unknown T_c using our recently developed high-throughput electronic structure package (HTESP) [147]. We performed DFPT calculations and computed EPC properties using the isotropic Eliashberg approximation. The accuracy of the EPC data is crucial for building reliable ML models. We have encountered two major challenges in the HT calculations with DFPT. One is the convergence with respect to BZ sampling and the other is dynamic instability. The first obstacle involved determining appropriate BZ samplings (k- and q- meshes) to compute the EPC properties, as these calculations become computationally expensive with dense meshes. To reduce the computational cost, we initiated an efficient screening by using a k-point mesh that accurately describes the ground-state structures and energetics. EPC quantities are then interpolated to fine k-mesh only twice the size of the coarse k-mesh. But we noticed that calculations with such grid combination can lead to inaccurate predictions, with discrepancies as high as approximately 10% of the total compounds with known T_c, giving NSC for known SC and vice versa. To address this problem, we developed an ansatz test to assess the convergence of T_c with respect to the k-point mesh. This approach leverages the decaying behavior of T_c with respect to Gaussian broadening width ( $\sigma$ ), which is used in the double-delta integration. Initially, we acquired results using the DFPT method with the k-point mesh size from the MP database. Subsequently, we assessed the convergence of these results based on the convergence ansatz test. To ensure convergence, we repeated calculations with denser k-mesh for the cases where results did not pass this test. We applied this technique to the 113 dynamically stable compounds, and obtained reasonable accuracy for the calculated T_c with a mean absolute error (MAE) of 2.21 K compared to experimental data. Further details regarding the convergence with respect to the k- and q-point meshes are discussed in the Method section and SM. In addition to the 113 dynamically stable compounds with known T_c, we also computed the EPC properties of 268 compounds of unknown T_c with dynamical stability, resulting from the ML-guided search as summarized in Fig.1.

The second obstacle encountered in EPC calculations was the presence of imaginary phonon modes and dynamical instability in almost 146 compounds. Among them, the imaginary phonon modes in 36 compounds show large EPC contribution. Stabilizing these imaginary modes is crucial for calculating EPC properties in such systems. These imaginary modes can be stabilized through lattice distortion, pressure, and electronic smearing, with the later method being particularly effective in HT screening[52]. A comprehensive analysis of dynamical instability and its implications for superconductivity will be presented in the “Imaginary phonon modes and superconductivity” section. In addition to these instances, there are cases up to 173 compounds, where EPC calculations are not complete due to either numerical problems or too large size of unit cells. For now, we will set these cases aside and will revisit them in the future.

II.3 Training and testing ML models

Besides the data extraction and DFPT calculations, training and testing ML models also play crucial roles in the ML-guided search workflow in Fig.1. We utilized two different ML models: CGCNN)[148] and ALIGNN)[149]. CGCNN maps 3D crystal structures to 2D graphs by using chemical element information and neighbor bonding distance to encode them into local chemical environments through convolution operations and updating the node features based on these descriptors. The updated node features are then aggregated to represent the entire crystal, which is connected to the output via a neural network. ALIGNN includes extra local chemical information, such as bonding angles, in addition to the crystal graphs in CGCNN with another auxiliary graph of bonding distances and angles. The parameters, including weights and biases, of the neural network connections are learned through training with available DFPT data.

As shown in Fig. 1, we trained the initial ML models in Run 1 using a dataset of 250 dynamically stable compounds with 109 SC (calculated T_c $>$ 1 K) and 141 NSC (calculated T_c $<$ 1 K) including those 113 from SuperCon database that were already measured in experiments. This dataset encompasses 45 distinct SGs. For the purpose of evaluation, a separate collection of 58 stable compounds (18 SC and 40 NSC), belonging to 27 unique SGs (6 of which are not part of the training 45), was reserved exclusively for independent testing and was not involved in the ML training process. Moving from Run1 to Run2, we use the ML model from Run1 to predict T_c for the remaining dataset of B and C compounds. We then sort the predictions and select the top candidates of both SC and NSC for more DFPT calculations to close the ML-guided loop. In Run2, the original 250 stable systems were augmented with an additional 73 dynamically stable systems to refine the ML models. In the concluding stage of Run 3, we incorporated the results derived from the compounds with dynamic instability. Notably, this stage included results from 2 new SGs. Among the 36 results in this category, we incorporated 28 into the training set and added 8 compounds into the independent test set. In the overall count of 417 compounds with converged EPC properties, 181 were classified as SCs, whereas 236 were categorized as NSCs. The progress from Run 1 to Run 3 constitutes a loop, as illustrated in Fig. 1. In summary, our methodology has a series of iterations involving training and testing ML models with increasingly comprehensive datasets from dynamically stable to unstable compounds.

II.4 Comparison of ML models

Next, we will present the main results of the ML models and guided search, highlight the notable compounds, including cases of dynamical stability and instability. Figures 3(a-d) depict the ML-predicted vs. DFPT-calculated $\lambda$ , $\omega_{log}$ , T_c, and T ${}^{\prime}_{c}$ using the dynamically stable systems in Run 1, respectively. Here, T_c represents the critical temperature computed using DFPT-computed $\lambda$ and $\omega_{log}$ or predicted directly from ML models, while T ${}^{\prime}_{c}$ is calculated from the ML-predicted $\lambda^{ML}$ and $\omega^{ML}_{log}$ using Eq. 1 in a postprocessing manner.

T^{\prime}_{c}=\frac{\omega^{ML}_{log}}{1.2}exp\left[-\frac{1.04(1+\lambda^{ML})}{\lambda^{ML}-\mu^{*}_{c}(1+0.62\lambda^{ML})}\right],

(1)

Here, It is important to note that the data first used do not include dynamically unstable cases. For Run 1, we trained ML models using 250 compounds, divided into training, validation, and testing sets in a ratio of 0.8:0.1:0.1. The mean absolute errors (MAEs) for the 10% test set in each training iteration are documented in SM Table-S6. An additional independent test set of 58 compounds was used to assess the predictability of the ML models and their MAE and predictions are plotted in Fig. 3. The training process consisted of 3000 epochs using default settings provided by the ML packages, which have been thoroughly investigated in the original work [148, 149]. In both CGCNN and ALIGNN, training and validation procedures are employed. Checkpoints are established at regular intervals to store crucial parameters such as model weights and architecture. The ML packages operate automatically to retain and update the model exhibiting the best performance, determined by the lowest validation error. This iterative process also acts to mitigate overfitting concerns. The performance of the models was evaluated by computing the MAE between the predicted and target quantities for the independent test set. The MAEs for CGCNN-predicted $\lambda$ and $\omega_{log}$ stand at 0.23 and 105 K, respectively, while the corresponding numbers for ALIGNN are slightly higher at 0.28 and 113 K (Figs. 3(a) and (b)). In terms of predicting T_c, the CGCNN and ALIGNN models yield MAEs of 2.4 K and 3.8 K, respectively. Despite the relatively small magnitude of MAEs, the ML outcomes exhibit distinct clustering patterns, as shown in Fig. 3(c). Specifically, for the CGCNN model, the results tend to cluster closely along the “DFPT” axis (red arrow in Fig. 3(c)), whereas for ALIGNN, the clustering is pronounced along the “ML prediction” axis (blue arrow in Fig. 3(c)). An alternative approach, rather than directly training and predicting T_c, is to utilize ML-predicted values, specifically $\lambda^{ML}$ and $\omega^{ML}_{log}$ , to estimate T ${}^{\prime}_{c}$ using Eq. 1. This modification not only enhances predictive performance for both models but also slightly ameliorates the issue of clustering [Fig. 3(d)]. A comparable approach has been employed in a recent study [150], wherein $\lambda$ and $\omega_{log}$ can be directly acquired from first principles calculations.

Then we use the predicted T ${}^{\prime}_{c}$ to rank the remaining compounds to pick the ones with high and low T ${}^{\prime}_{c}$ for additional DFPT calculations. In this next stage, we utilized ML-guided search to expand the dataset to 323 systems characterized by 54 distinct SGs, the ML models underwent further training. Subsequently, the improved ML models were subjected to the same independent testing, employing the same set of 58 system test cases. The outcomes of the Run 2 are presented in Figs. 3(e)-(h). The results demonstrate a notable improvement in addressing the issue of prediction clustering with the expansion of the training dataset, all while maintaining reasonable accuracy. ALIGNN improves the prediction of $\lambda$ and $\omega_{log}$ with MAEs of 0.24 and 93K, respectively, compared to CGCNN’s MAEs of 0.26 and 97K. However, T ${}^{\prime}_{c}$ calculated from ML-predicted $\lambda^{ML}$ and $\omega^{ML}_{log}$ shows a significant improvement for ALIGNN, with an MAE of 2.7K. Notably, $\omega_{log}$ is more accurately predicted than $\lambda$ and then T_c, because the former is directly related to the overall bonding strength and cohesive energy, while the later depends on the details of the Fermi surface and the EPC matrix elements.

As mentioned above, from the ML-guided search, we find quite some number of compounds with imaginary phonon and dynamical instability. In all the previous high throughput phonon-mediated SC studies, these dynamically unstable compounds are simply discarded, but as shown by our recent study [52] on Y₂C₃, some imaginary phonon modes once stabilized can carry a large EPC and give rise to a large T_c. Thus, specifically here we also include dynamically unstable compounds in ML. As far as we know, this is the first ML trained with both dynamically stable and unstable compounds for phonon-mediated SC. We carried out a distinct training and testing phase for ML models, incorporating dynamically unstable cases after stabilization, denoted as Run 3. These results predominantly included nonzero T_c. Among these, 28 results were added to the training dataset, while the remaining 8 were added for independent testing. In total, the independent test set now consists of the original 58 dynamically stable and additional 8 stabilized compounds with imaginary phonon modes. Both the training and test datasets were balanced across various SGs. The outcomes of Run 3 are illustrated in Fig. 4(a)-(d). In Run 3, the ALIGNN model consistently outperforms the CGCNN model across different superconducting properties. The ALIGNN model achieves MAEs of 0.27 for $\lambda$ , 86 K for $\omega_{log}$ , 4.3 K for T_c, and 3.5 K for T ${}^{\prime}_{c}$ . In contrast, the CGCNN model records MAEs of 0.34, 104 K, 4.5 K, and 5.3 K for the respective properties. The reason is that ALIGNN includes the bonding angles as part of training parameters, which better describe the dynamically unstable compounds, because imaginary phonon modes often involves lower-energy bond rotation with changing angle than higher-energy bond stretching vibration. Based on the information presented in Figs. 3 and 4, it is apparent that the ML model performs better in learning the parameters $\omega_{\text{log}}$ than $\lambda$ and then T_c. As a result, it is justifiable to utilize Eq. 1 to estimate the ML-predicted critical temperature T ${}^{\prime}_{c}$ rather than relying solely on the directly predicted T_c values.

II.5 Dynamically stable systems

Table 1: DFPT calculated EPC results for experimentally unknown systems with T_c

>

10 K; SG is spacegroup, and

\Delta E_{h}

is taken from materials project. [55]

Compound	SG	$\Delta E_{h}$ (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
B₂CN	R3m	0.34	1.66	578	60.7
B₂CN	P3m1	0.35	1.11	567	34.9
Mo₇B₂₄	P-6m2	0.15	1.28	386	29.6
TaNbC₂	R-3m	0.02	1.41	326	28.4
TcB	P6₃/mmc	0.26	1.32	333	26.5
TaC	P-6m2	0.41	1.54	235	22.8
ZrBC	P6₃/mmc	0.45	1.12	321	20.0
Ta₂CN	I4₁/amd	0.13	1.94	162	19.8
Ta₂CN	P4/mmm	0.15	2.70	127	19.5
B₂CN	P-4m2	0.32	0.80	644	18.8
TaB₂	P6/mmm	0.0	1.22	254	18.1
NbFeB	P-6m2	0.42	1.64	173	18.0
Nb₃B₃C	Cmcm	0.02	1.25	229	16.4
V₂CN	R-3m	0.12	1.00	323	16.2
NbC	P6₃/mmc	0.15	0.84	455	15.3
ZrMoB₄	P6/mmm	0.09	1.30	193	15.1
NbVCN	R3m	0.16	0.99	274	13.4
Ta₄C₃	Pm-3m	0.13	1.46	138	12.6
NbVC₂	R-3m	0.11	0.83	365	11.9
Nb₂CN	R-3m	0.08	0.88	310	11.7
Nb₄B₃C₂	Cmcm	0.05	1.01	219	11.3
TaVC₂	R-3m	0.08	0.80	343	10.3

In Table 1, we present the results of our ML-guided search for dynamically stable compounds with DFPT-calculated T_c exceeding 10 K, while the complete list of EPC properties of the 381 dynamically stable compounds are presented in SM from Table S7 to S13. It is worth noting that the compounds with high T_c tend to be thermodynamically metastable, as indicated by their formation energy significantly above the convex hull. B₂CN in different phases, with $\Delta E_{h}$ up to 0.35 eV/atom, has been predicted to exhibit SC, consistent with earlier theoretical findings [151]. TaC and NbC, both have sizable T_c in the metastable hexagonal structure, while their more stable cubic structures in $Fm-3m$ have already been observed with SC [152, 153]. In Table 1, the first compound close to the ground state convex hull ( $\Delta E_{h}$ $\sim$ 0.02 eV/atom) is TaNbC₂, whose crystal structure and EPC properties are plotted in Fig. 5(a)-(c). It crystallizes in the trigonal structure of $R-3m$ . The calculated EPC properties are $\lambda=1.4$ , $\omega_{\log}=326$ K, and $T_{c}=28.4$ K, with the majority of the contribution coming from phonons within the 3-6 THz range (Fig. 5(b) and (c)), as well as a significant contribution from the 16-20 THz range of C-dominated modes. Other compounds close to the ground state convex hull with $\Delta E_{h}$ $\leq$ 0.02 is TaB₂. The presence of SC in TaB₂ remains a subject of debate as discussed in the previous studies [85].

Moreover, we predict other ternary superconductors, such as Nb₃B₃C ( $\Delta E_{h}$ of 0.018 eV/atom [55]), with T_c of 16.4 K (Figs. 5 (d), (f), (h) and (j)). Comparison between the calculated and experimental T_c are also plotted (blue circles) for the Nb-B-C superconductors with known T_c. These ternary metallic borocarbides were all experimentally synthesized [154], however, some of their SC (red circles) have not been reported yet, with Nb₃B₃C has a predicted T_c as high as 16.4 K. The crystal structure of Nb₃B₃C exhibits a novel layer-like arrangement as in Fig. 5(f), where B atoms form strips of honeycomb lattice, while C being monomers. The $\lambda$ -projected phonon dispersion (Fig. 5(h)) analysis indicates that the SC is attributed to the presence of low-frequency soft phonon modes in the vicinity of the $Z$ and $T$ points of the BZ. For ternary Nb-B-C systems, EPC calculations with isotropic approximation tend to overestimate T_c, while for Y-B-C systems, EPC calculations have also notable underestimations as plotted in Fig. 5(e). Therefore, we also show the EPC properties of Y₂B₃C₂ with a predicted T_c of 4 K, but having $\Delta E_{h}$ $\sim$ 0[55], along with other experimentally measured Y-B-C systems in Fig. 5(e). Unlike Nb₃B₃C, the crystal structure of Y₂B₃C₂ shows a layer of mixed B and C network sandwiching the Y layer (Fig. 5(g)). The $\lambda$ -projected phonon dispersion shows that the EPC properties are mostly contributed by phonon within the 4-10 THz energy range around the $\Gamma$ and $Z$ points (Figs. 5 (i) and (k)). Other compounds with calculated T_c $<$ 10 K and their EPC properties are listed in SM.

II.6 Imaginary phonon modes and superconductivity

In this section, we address the issue of dynamical instability observed in the DFPT-computed phonon dispersion (146 out of 700 compounds). Out of these 146 dynamical unstable compounds, 34 exhibit large EPC in their imaginary phonon modes. Figure 6 (a) and (b) provide the breakdown of these 34 compounds based on their constituent elements, while (c) and (d) show the distribution of systems in terms of formation energy and SG, respectively. As expected, a considerable portion of these compounds with imaginary phonon have formation energies above the convex hull. Superconductivity in Sc₂C₃, shown in Fig. 6 (d), are very likely because SC has been observed in the same structure with larger cations of the same group as in Y₂C₃ [155] and La₂C₃ [156], which are closer to the convex hull. We have categorized these compounds into two groups based on whether the imaginary phonon modes occur at the $\Gamma$ -point or elsewhere. The instabilities at $\Gamma$ -point are represented by Sc₂C₃, Ta₂B, and La₃InB (Fig.6(e-g)) which constitutes 11 cases, whereas the other 23 compounds including MoB₂ represents the case of dynamic instability outside of $\Gamma$ -point, as illustrated in Figs. 6(h) for MoB₂. The unstable phonon modes possess a large mode-resolved $\lambda$ in the vicinity of instability, $\lambda_{\textbf{q}\nu}=$ $\frac{\gamma_{\textbf{q}\nu}}{\pi N(E_{F})\omega^{2}_{\textbf{q}\nu}}$ , where $\gamma_{\textbf{q}\nu}$ is the change in phonon linewidth due to EPC (See Fig. 6). The systems with dynamical instability at $\Gamma$ -point can be stabilized by obtaining a low-symmetry ground state structure through lattice distortion along the direction of the imaginary eigenmodes at $\Gamma$ and performing a full ionic relaxation. The analysis presented here is expanded from our earlier work on Y₂C₃ [52]. The application of smearing to stabilize imaginary phonon modes has been explored in other systems, including $\beta$ -phase Ni_xAl_(1-x) [157], NiTi [158] and EuAl₄[159]. In these cases, instead of optical modes, the instability arises from acoustic modes. It was proposed that the dynamic instability observed in these materials is the result of strong EPC between nested electronic states near the Fermi level [157, 158, 159]. Previously as exmplified with Y₂C₃ [52] these imaginary phonon modes can significantly contribute to $\lambda$ after stabilization. Although these cases represent only a small fraction of the compounds, disregarding them would exclude potential SC compounds with a sizable T_c.

Performing EPC calculations on low-symmetry structures can be computationally expensive, especially when the system contains a large number of atoms like Sc₂C₃. Hence, it is recommended to first try stabilization using pressure and smearing. The former option of stabilizing through pressure can give interesting pressure-dependent properties, whereas the latter with increasing electronic smearing can be beneficial for HT computations. To stabilize the dynamically unstable systems, we applied pressure (ranging from 5 to 60 GPa) or used a larger electronic smearing (0.05, 0.06, 0.08, 0.1 Ry). In each case, we fully relaxed the structures. We computed the EPC properties for the stabilized systems around $\Gamma$ -point and presented the results in Table 2 together with their SG and $\Delta E_{h}$ . Most of these compounds crystallize in high-symmetry structures such as cubic, hexagonal and tetragonal lattices. The relative ground state energy of relaxed low-symmetry structures compared to the original high-symmetry ones are listed as Distorted GS. We utilize pressure and the electronic smearing to stabilize the imaginary phonons in Y₂C₃, La₂C₃ and Sc₂C₃. However, for Ta₂B and La₃InB systems, pressure and smearing were insufficient for stabilization, so lattice distortion was employed. The calculated T_c using DFPT for the stabilized systems agrees well with experimental data. For example, Al₂Mo₃C exhibits an instability at $\Gamma$ akin to that observed in Sc₂C₃. After stabilization with electronic smearing, the calculated $T_{c}$ is 12.05 K, which is comparable to the experimental $T_{c}$ of 9.2 K [160]. We predict the EPC properties for Sc₂C₃, YBC, and MoB₄ with a sizable T_c of 27.9 K, 10.15 K and 7.58 K under ambient or moderate pressure. These compounds have formation energy higher (0.11, 0.42, and 0.26 eV/atom respectively) than the ground state convex hull, indicating they are metastable.

Table 2: Stabilization of imaginary phonon modes (around

\Gamma

) utilizing Distortion (D), Pressure (P), and Smearing (S). The values of pressure and smearing are presented as P-value (GPa) and S-value (Ry) respectively. Crystal is distorted along the direction represented by eigenmode at

\Gamma

and relaxed to obtain low-symmetry ground-state structures. Distorted GS is the difference between ground-state total energy of an undistorted and distorted structures at equilibrium. SG is spacegroup and

\Delta E_{h}

represents the energy above the ground state hull.

Compound	SG	$\Delta E_{h}$ (eV/atom)[55]	Distorted GS (meV/atom)	EPC ( $\lambda$ )	$\omega_{log}$ (K)	$T_{c}$ (K)	T ${}^{Expt}_{c}$ (K)
Y₂C₃	I-43d	0.04	0.9	1.94 (P-10)	175.23 (P-10)	21.27 (P-10)	18 [155]
				1.14 (S-0.1)	227.97 (S-0.1)	14.50 (S-0.1) [52]
La₂C₃	I-43d	0.0	2.2	1.15 (S-1)	219.4 (S-1)	14.3 (S-1)	13.4 [156]
Sc₂C₃	I-43d	0.11	3.6	1.435 (P-30)	303.18 (P-30)	27.90 (P-30)	-
				1.99 (S-0.1)	205.6 (S-0.1)	25.5 (S-0.1)	-
Al₂Mo₃C	P4_132	0.05	0.6	1.19 (S-0.1)	174.95 (S-0.1)	12.05 (S-0.1)	9.2 [160]
YBC	Cmmm	0.42	150	0.73 (P-20)	454.42 (P-20)	10.15 (P-20)	-
W₂B	I4/mcm	0.0	0.6	0.81 (D)	215.68 (D)	6.61 (D)	3.10 [15]
Mo₂B	I4/mcm	0.03	3.6	0.79 (D)	284.65 (D)	8.12 (D)	4.74 [15]
Ta₂B	I4/mcm	0.03	8.9	0.44 (D)	236.84 (D)	0.34 (D)	3.12 [15]
MoB₄	P6/mmm	0.26	0.7	0.69 (D)	418.42 (D)	7.58 (D)	-
La₃InC	Pm-3m	0.0	0.6	1.04 (D)	108.63 (D)	5.844 (D)	2.6 [161]
La₃In B	Pm-3m	0.20	2.3	1.18 (D)	83.03 (D)	5.60 (D)	10 [161]

Figure 6(j)-(m) displays the $\lambda$ -projected phonon dispersion for four different stabilized compounds: Sc₂C₃ (P-30), Ta₂B (D), La₃InB (D), and MoB₂ (S-0.1), utilizing pressure of 30 GPa, distortion (D), distortion (D), and smearing of 0.1 Ry, respectively. By comparing these plots with Figs. 6(e)-(h), we can see that the soft optical phonon modes are stabilized and contribute significantly to $\lambda$ near the $\Gamma$ -point, represented by green open circles. Despite the slight lifting of phonon band degeneracy caused by distortion, the large contribution to EPC remains, which give rise to SC. The discovery of such systems is interesting as it presents opportunities for stabilization through pressure and alloying, leading to potentially high T_c metastable compounds that may be synthesizable in experiment.

For the compounds with imaginary phonon modes away from the $\Gamma$ point, i.e. MoB₂-type, we also stabilized these dynamically unstable compounds with larger electronic smearing of 0.1 Ry and tabulated the results in Table. 3. For example, compounds like MoB₂ in the MgB₂ structure have shown experimentally measured T_c of 32 K under high pressure around 110 GPa [20]. Another notable example in Table 3 is Ca₅B₃N₆ with $\Delta E_{h}$ of 0.03 eV/atom, which exhibits dynamical instability at the $H$ point and displayed a significant mode-resolved $\lambda$ near its unstable phonon modes, as depicted in Fig. 7 (d)-(e). After stabilizing the system with electronic smearing of 0.06 Ry, Ca₅B₃N₆ exhibits a $\lambda$ of 1.5. The $\omega_{log}$ is found to be 372 K. Moreover, when computing $\lambda$ (with broadening parameter $\sigma$ = 0.01 Ry), the T_c is determined to be 35 K with Coulomb potential $\mu^{*}_{c}$ =0.16 and 42.4 K with $\mu^{*}_{c}$ =0.10, much larger than that of MgB₂ (16-20 K) computed from isotropic approximation. It should be noted that Ca₅B₃N₆ has been synthesized in cubic structure and Im3^′m (229) SG with partial occupancy in the 8c site of Ca [162]. The crystal structure of this boronitride (Fig. 7(a)-(c)) has a cage-like structure, similar to XB₃C₃ borocarbides (X $=$ Ca,Ba,Sr,Y,La). For the stoichiometric Ca₅B₃N₆, Fig. 7(f) and (g) present the isotropic Eliashberg spectral function and electronic band structure, respectively. It shows an electron-doped band structure that connects to strong EPC as also found in other predicted SC compounds and studies. As listed in Table III, interestingly some of the trigonal compounds of NbMoC₂ are in the same structure as the stable TaNbC₂ in Table 1. This shows that in this particular structure, with the substitution of neighboring group of early TMs, although bringing dynamical instability, the phonons once stablized can still provide a large EPC contribution for a sizable T_c. As also listed in Table III, other compounds with sizable predicted T_c after stabilization that also near GS hull are some notable ternary Ru compounds, MoRuB₂ at 15.6 K, RuVB₂ at 15.0 K, Pd₃CaB at 7.0 K and RuSc₃C₄ at 6.6 K.

Table 3: DFPT calculated EPC results for MoB₂-type instability (outside of

\Gamma

), stabilized with electronic smearing of 0.10 Ry; SG is spacegroup, and

\Delta E_{h}

\Delta E_{h}

taken from MP database. Experimental results for some high-pressure T_c are also reported.

Compound	SG	$\Delta E_{h}$ (eV/atom) [55]	$\lambda$	$\omega_{log}$ (K)	T_c (K)	T ${}^{Expt}_{c}$ (K)
Ca₅B₃N₆	Im3^′m	0.03	1.5	372	35.0 ¹¹1Stabilized with electronic smearing of 0.06 Ry
MoB₂	P6/mmm	0.156	2.12	209	27.3	32[20]
NbMoC₂	R-3m	0.172	1.72	215	23.5
TaMo₂C₃	P-3m1	0.197	2.01	171	21.4
TaMoC₂	R-3m	0.15	1.69	189	20.3
WVC₂	R-3m	0.225	1.31	249	19.8
TaTiWC₃	P3m1	0.119	1.42	213	18.9
ScC	P6₃/mmc	0.6	0.95	410	18.5
TaWC₂	R-3m	0.211	2.04	129	16.3
MoRuB₂	Pmc2₁	0.09	1.29	202	15.6
RuC	P-6m2	0.649	1.02	288	15.1
RuVB₂	Pmc2₁	0.073	1.09	255	15.0
Ni₃AlC	Pm-3m	0.166	1.86	116	13.6
Nb₄C₃	Pm-3m	0.167	0.99	267	13.2
Ta₄C₃	Pm-3m	0.134	0.99	203	10.1
RhC	F-43m	0.561	0.91	228	9.3
MoC	F-43m	0.585	0.97	191	9.0
Pd₃CaB	Pm-3m	0.0	1.23	96	7.0
RuSc₃C₄	C2/m	0.0	0.72	321	6.6
Ta₂S₂C	P-3m1	0.007	0.69	268	5.1
Mo₃ZrB₂C₂	Amm2	0.046	0.64	282	4.0
HfC	P-6m2	0.763	1.06	56	3.2
Sc₄C₃	I-43d	0.0	0.14	486	0.0

III Discussion

In this work, we have employed the machine learning (ML) models that utilize the data generated from ab-initio calculations using the DFPT and the isotropic Eliashberg approximation to iteratively guide the search for new phonon-mediated superconductors among B and C compounds. Our study also focuses on addressing the challenges encountered during DFPT calculations, such as convergence of Brillouin zone (BZ) sampling and the problem of calculated dynamic instability. To address the convergence issue, we developed an ansatz test to verify the convergence of superconductivity (SC) critical temperature $T_{c}$ . This test uses the variation of $T_{c}$ with respect to Gaussian broadening to compute the double delta summation. For dynamically unstable compounds, we applied large electronic smearing, lattice distortion, and pressure to stabilize imaginary phonon modes. We then calculated their EPC properties and incorporated these into the ML models. Between the two ML models, ALIGNN consistently outperforms CGCNN in predicting EPC properties especially after including the stabilized compounds with imaginary phonon and dynamical instability. Our ML-guided search demonstrates promising predictability for T_c values. For example, we predict SC in compounds with calculated dynamically stability such as TaNbC₂ (28.4 K), Nb₃B₃C (16.4 K), Y₂B₃C₂ (4.0 K), among others. In addition to studying dynamically stable compounds, we also focus on compounds with calculated dynamic instability, an area that, to our knowledge, has not been systematically explored before. We predicted SC in compounds showing dynamic instability such as Ca₅B₃N₆ (35 K), Pd₃CaB (7.0 K), some ternary Ru compounds, MoRuB₂ (15.6 K), RuVB₂ (15.0 K), Pd₃CaB (7.0 K) and RuSc₃C₄ (6.6 K) with $\Delta E_{h}$ mostly below 0.1 eV/atom. With further refinement and larger dataset, our workflow can be improved in accurately predicting more SC compounds. In this regard, identifying metastable compounds with calculated dynamic instability, where soft phonon exhibit significant EPC contribution, plays a crucial role.

IV METHODS

IV.1 Convergence with respect to Brillouin zone sampling (k-point mesh)

To identify cases of convergence failure (i.e., incorrect prediction of SC/NSC) related to Brillouin zone sampling, we analyzed the variation of T_c with respect to Gaussian broadening ( $\sigma$ ) in the double delta integration and developed a simple ansatz based on the converged results of MgB₂, as described in the ”Convergence ansatz” section of the Supplemental Material (SM). This ansatz involves extracting T_c, similar to MgB₂, and estimating the decay parameter (A) in the exponential variation of T_c with $\sigma$ ,

T_{c}=\exp(-A\sigma^{1/3}+B)

(2)

where A is the variable that quantifies the rate of exponential decay, and B is the constant associated with Debye temperature. Unconverged results show larger values of A, which decrease with denser k-meshes. With larger k- or q- grids, the T_c vs $\sigma$ curve becomes less steep. Our convergence analysis of MgB₂ suggests that A ${}_{MgB_{2}}$ = 12 $-$ 13 can be used as a threshold for this study: calculations are considered as unconverged for A $>$ A ${}_{MgB_{2}}$ , requiring a denser k-mesh, while those with A $<$ A ${}_{MgB_{2}}$ can be regarded as converged for accurate prediction.

IV.2 Computational details

Data extraction from the MP database, as well as input preparation for ground state calculations, calculation submission, result extraction and input preparation for ML studies, and plotting, were performed using the high-throughput electronic structure package (HTESP) [147]. A script, ‘fitting_elph_smearing.py‘, is included in the HTESP package to compute the decay parameter from $T_{c}$ vs. broadening $\sigma$ data. Ground-state DFT and EPC calculations were performed using the QE code [163, 164]. Ultrasoft or norm-conserving pseudopotentials (PP) from the efficiency standard solid-state pseudopotentials (SSSP) dataset [165] were employed, with the replacement of projector-augmented wave (PAW) PPs by GBRV ultrasoft norm-conserving high-throughput PPs [166]. The exchange-correlation energy was approximated using the Perdew-Burke-Ehrnzerhof (PBE) generalized-gradient approximation (GGA) [167]. The Brillouin zone was sampled using a k-point mesh from the MP database to compute the ground-state charge density for EPC calculations. The q-point mesh required for EPC calculations was obtained by halving the k-point mesh ( $\textbf{q}=\textbf{k}/2$ ), with odd k-points changed to even by adding 1 to it. The structures were fully relaxed using Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization [168] until the total energy and forces for ionic minimization converged within 10^-5 Ry and 10^-4 Ry/Bohr, respectively. The self-consistent electronic energy and charge density were minimized with a convergence threshold of 10^-12 Ry. Similarly, the SCF convergence for phonon calculations is achieved with an energy cutoff of $10^{-14}$ . Starting from the default $\alpha_{\text{mix}}$ value of 0.7, it is reduced to 0.3 if the EPC calculation does not converge. A fine k-point grid, twice the size of the k-point mesh used for charge density convergence, was used for interpolating EPC matrix elements to compute double-delta integration and the $\lambda$ . Gaussian smearing of 0.02 Ry was applied for charge-density optimization. To compute $\lambda$ for various $\sigma$ values, we computed double-delta integration using 10 broadening ( $\sigma$ ) values ranging from 0.005 to 0.05 Ry for the k-mesh. For the q-mesh integration, a fixed smearing of 0.5 meV was employed. The reported results were obtained for $\sigma=0.01$ Ry with a Coulomb potential $\mu^{*}_{c}$ of 0.16.

V Data availability

Data computed and utilized in this study are tabulated in STs ST7 $-$ ST13 in the SM, Tables. 2 and 3. Raw data is available from the corresponding authors upon reasonable request.

VI Acknowledgements

We thank Dr. Paul C. Canfield for the funding support, initiating the idea of searching for new phonon-mediated superconductors among boron and carbon compounds, and the helpful discussion throughout the project. This work was supported by Ames National Laboratory LDRD and U.S. Department of Energy, Office of Basic Energy Science, Division of Materials Sciences and Engineering. Ames National Laboratory is operated for the U.S. Department of Energy by Iowa State University under Contract No. DE-AC02-07CH11358.

VII Author contributions

L.-L.W. conceived and supervised the work. N.K.N. and L.-L.W. designed and performed the high throughput calculations with the machine learning guided approach. N.K.N. developed the ansatz to test the convergence of Tc calculation. All authors discussed the results and contributed to the final manuscript.

VIII Competing interests

The authors declare no competing interests.

IX References

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Supplementary Materials for “Machine-learning Guided Search for Phonon-mediated Superconductivity in Boron and Carbon Compounds”

Statistics for compounds with unknown T_c

In Fig. S1, we present the descriptions of 268 dynamically stable compounds for which the experimental T_c is not known. Figure S1 (a)-(c) illustrate the distribution of different elements in these systems, the allocation of B/C/B+C compounds, and the deviation of their formation energy ( $\Delta E_{h}$ ) from stable structures, respectively. Most unknown compounds are associated with transition metals (TM) such as Nb, Ta, Mo, V, Y, and others. Approximately 80% of the structures are close to the ground state convex hull with $\Delta E_{h}\leq 0.05$ eV/atom, while the remaining 20% have $\Delta E_{h}$ greater than 0.05 eV/atom. Panel S1(d) showcases the distributions of these compounds based on their space groups. In Fig. S1(a) and (d), each bar is partitioned into segments, with the segments colored in red representing the number of SC, while the segments in blue denote the NSC. The overall description is similar to that of known compounds, as shown in Fig. 2 of the main text.

Theory of superconductivity: Isotropic Approximation

We have adopted the DFPT calculation with isotropic Eliashberg approximation to compute SC properties, which provides a harmony between accuracy and efficiency. The critical temperature can be calculated using the Allen-Dynes formula [1],

T_{c}=\frac{\omega_{log}}{1.2}exp\left[-\frac{1.04(1+\lambda)}{\lambda-\mu^{*}_{c}(1+0.62\lambda)}\right],

(3)

where, $\lambda=2\int_{0}^{\infty}\frac{d\omega}{\omega}\alpha^{2}F(\omega)$ is the EPC strength constant, $\omega_{log}=exp\left[\frac{2}{\lambda}\int_{0}^{\infty}\frac{d\omega}{\omega}\alpha^{2}F(\omega)log\omega\right]$ , $\alpha^{2}F(\omega)$ is frequency ( $\omega$ ) resolved Eliashberg spectral function, and $\mu^{*}_{c}$ is the Coulomb potential. The spectral function $\alpha^{2}F(\omega)$ is defined as,

\alpha^{2}F(\omega)=\frac{1}{2}\sum_{\nu}\int_{BZ}\frac{d\textbf{q}}{\Omega_{BZ}}\omega_{\textbf{q}\nu}\lambda_{\textbf{q}\nu}\delta(\omega-\omega_{\textbf{q}\nu}).

(4)

Here, $\omega_{BZ}$ is the volume over the Brillouin zone $\int_{BZ}\frac{d\textbf{q}}{\Omega_{BZ}}\rightarrow\frac{1}{N_{\textbf{q}}}\sum_{\textbf{q}}$ , $\omega_{\textbf{q}\nu}$ is the mode ( $\nu$ ) resolved phonon frequency, and $\lambda_{\textbf{q}\nu}$ is the mode resolved EPC strength constant,

\lambda_{\textbf{q}\nu}=\frac{1}{N(\epsilon_{F})\omega_{\textbf{q}\nu}}\sum_{mn}\int_{BZ}\frac{d\textbf{k}}{\Omega_{BZ}}|g_{mn,\nu}(\textbf{k},\textbf{q})|^{2}\delta(\epsilon_{n\textbf{k}}-\epsilon_{F})\delta(\epsilon_{m\textbf{k+q}}-\epsilon_{F}).

(5)

$N(\epsilon_{F})$ is the density of states at the Fermi level $\epsilon_{F}$ , and $g_{mn,\nu}(\textbf{k},\textbf{q})$ is the EPC matrix element which quantifies the scattering process between Kohn-Sham states mk+q and nk. In Ref. [2], an effective approach to approximate double delta integration is explored, specifically tailored for situations where it is permissible to disregard the dependence on the momentum vector (q). This method finds utility in scenarios such as the study of extensive molecular systems like alkali fullerides, where momentum dependence can be safely overlooked. The net EPC strength constant is computed as

\lambda=\sum_{q\nu}\lambda_{q\nu}

(6)

For computational feasibility, these Dirac deltas can be approximated by Gaussian functions with a broadening parameter $\sigma$ [3], and EPC strength is given by Eqs.5 and 6 can be redefined as

\lambda\approx\frac{1}{N(\epsilon_{F})N_{\textbf{q}}N_{\textbf{k}}}\sum_{nm}\sum_{\textbf{q}}\sum_{\textbf{k}}\frac{|g_{mn,\nu}(\textbf{k},\textbf{q})|^{2}}{\omega_{\textbf{q}\nu}}\frac{1}{2\pi\sigma^{2}}exp\left[-\frac{(\epsilon_{n\textbf{k}}-\epsilon_{F})^{2}+(\epsilon_{m\textbf{k}+\textbf{q}}-\epsilon_{F})^{2}}{\sigma^{2}}\right].

(7)

Here, $N_{\textbf{q}}$ and $N_{\textbf{k}}$ respectively are the total number of q and k grid points, and $\sigma$ is the smearing used to broaden states at the Fermi-level $\epsilon_{F}$ . With infinitely large k- grids, and $\sigma\rightarrow 0$ , the double summation changes back to double delta integration. Moreover, $\lambda$ can also obtained from frequency $\omega$ resolved Eliashberg spectral function as [4]

\lambda=2\int\frac{d\omega\alpha^{2}F(\omega)}{\omega}=\frac{N(\epsilon_{F})<g^{2}>}{M<\omega^{2}>},

(8)

where, $<\omega^{2}>$ average of the square of the phonon frequency $\left(\frac{\int d\omega\omega\alpha^{2}F(\omega)}{\int\frac{d\omega}{\omega}\alpha^{2}F(\omega)}\right)$ , $<g^{2}>$ is average over the Fermi surface of the square of electronic phonon coupling matrix element [4].

Convergence tests for MgB₂ and AlB₂

The ground-state total energy and phonon frequencies at $\Gamma$ -point are well converged for MgB₂ with k-grids of 8 $\times$ 8 $\times$ 6, compared to the dense grid of 24 $\times$ 24 $\times$ 24. K-mesh grid in materials project database for MgB₂ is 8 $\times$ 8 $\times$ 7. Since the superconducting properties of MgB₂ with k-grid of 8 $\times$ 8 $\times$ 8 doesn’t follow the trend of converged result of denser k-mesh with $\sigma\rightarrow 0$ , we chose 8 $\times$ 8 $\times$ 6 so that we can use sufficient q-grids of 4 $\times$ 4 $\times$ 3 instead of using 9 $\times$ 9 $\times$ 9 k-grid and 3 $\times$ 3 $\times$ 3 q-grid. We utilized 8 $\times$ 8 $\times$ 6 k-grid and its multiple to compute the decay parameter. In this work, we are taking $\lambda=$ 0.75 obtained from solving anisotropic Migdal-Eliashberg equation using $\mu^{*}=$ 0.16 as a reference [Comput. Phys. Commun. 209, 116 (2016)]. One can obtain this converged $\lambda$ with denser k- and q- grids and $\sigma\rightarrow 0$ , as shown in Figs. S2 $-$ S4. At last, we check whether the result is transferable to systems with larger unit cell where essentially smaller k-grids provide converged ground-state properties and q-grid, taken half of k-grid, sometime shrinks only to include the $\Gamma$ -point. To do that, we performed EPC calculations with 2 $\times$ 2 $\times$ 2 supercell of MgB₂ which has 24 atoms per cell, and present the results in Fig. S5 and S6. Results confirm that q-grid as half as that of the k- grid can provide converged results even for larger systems. Instead of increasing coarse k- and q- grids, one can also utilize extremely large k- grid for interpolating EPC matrix element to achieve convergence, as shown in Figs. S4 and S7 respectively for MgB₂ and AlB₂. However, it also increases computational complexity for larger systems.

Table ST1: Convergence of the ground-state total energy with respect to K-point mesh

K-mesh	Total Energy (eV/atom)
6 $\times$ 6 $\times$ 4	-615.178
8 $\times$ 8 $\times$ 6	-615.179
8 $\times$ 8 $\times$ 8	-615.180
9 $\times$ 9 $\times$ 9	-615.180
12 $\times$ 12 $\times$ 12	-615.180
16 $\times$ 16 $\times$ 16	-615.180
24 $\times$ 24 $\times$ 24	-615.180

Table ST2: Convergence of phonon frequency (THz) at

\Gamma

-point with respect to K-point mesh

K-mesh	$\omega_{1}$	$\omega_{2}$	$\omega_{3}$	$\omega_{4}$	$\omega_{5}$	$\omega_{6}$	$\omega_{7}$	$\omega_{8}$	$\omega_{9}$
6 $\times$ 6 $\times$ 4	-0.654	-0.654	0.343	10.18	10.18	12.07	20.96	24.38	24.38
8 $\times$ 8 $\times$ 6	-0.49	-0.49	0.48	10.11	10.11	12.11	16.45	16.45	20.75
8 $\times$ 8 $\times$ 8	-0.63	-0.63	0.28	10.1	10.1	12.11	17.43	17.43	20.8
9 $\times$ 9 $\times$ 9	-0.62	-0.62	-0.33	9.99	9.99	12.08	14.36	14.36	20.65
12 $\times$ 12 $\times$ 12	-0.64	-0.64	-0.23	10.06	10.06	12.12	17.13	17.13	20.69
16 $\times$ 16 $\times$ 16	-0.64	-0.64	-0.25	10.05	10.05	12.1	17.2	17.2	20.69
24 $\times$ 24 $\times$ 24	-0.64	-0.64	-0.22	10.05	10.05	12.1	16.99	16.99	20.71

Convergence ansatz

Estimation of SC properties requires the computation of double-delta integration, which also defines the nesting function, around the Fermi level E_F over the entire Brillouin zone [Eq. 3]. However, it has a slow convergence with respect to k- and q- grids. In principle, it requires infinitely dense grids, which makes the computation exorbitantly expensive. Therefore, Gaussian broadening technique [Eq. 5] is employed to compute $\lambda$ with finite k-mesh. To compute T_c, we employed the DFPT calculation with the isotropic Eliashberg approximation. We used a coarse k-grid obtained from the MP database, which provides reasonably accurate ground-state properties. The q-grid was set to half the size of the k-grid, and a broadening parameter ( $\sigma$ ) of 0.01 Ry was utilized with the coarse MP grid, which accurately captures a converged electron-phonon coupling constant ( $\lambda$ ) of MgB₂ computed using the anisotropic Migdal-Eliashberg equations [5]. However, in practice, $|g_{mn,\nu}(\textbf{k},\textbf{q})|^{2}$ is computed for a reasonable coarse k- and q- grids and interpolated the matrix elements to fine k- for any q-point, from 32 $\times$ 32 $\times$ 32 to as high as 60 $\times$ 60 $\times$ 60, to achieve numerical convergence, as implemented in Quantum Espresso (QE) code[3]. Furthermore, an efficient interpolation scheme utilizing both dense k- as well as q-grids has been implemented in EPW package that uses wannier orbitals [5]. However, employing extremely fine grids for interpolation can increase the computational complexity, which is not suitable for highthroughput calculations. Therefore, we have restricted ourselves for choosing fine k-grid only twice of corresponding coarse grid for highthroughput calculations.

Our investigation revealed that a reasonable number of calculations did not converge, leading to inaccurate predictions, while utilizing a coarse k-grid from MP database and fine grid for interpolation only twice that of the k-grid [Table ST3]. For example, AlB₂ was predicted to be a superconductor (T_c $\sim$ 11 K) with the coarse grid [k-grid: 8 $\times$ 8 $\times$ 8, q-grid: 4 $\times$ 4 $\times$ 4, fine-k-mesh: 16 $\times$ 16 $\times$ 16] from MP database, whereas a denser grid twice the size of the coarse grid corrected this inaccuracy and predicted AlB₂ to be a nonsuperconductor, consistent with experimental observations [Fig. S7]. To identify such cases of convergence failure, we examined the variation of T_c with respect to $\sigma$ and developed a simple ansatz based on the converged results of MgB₂, as depicted in Fig. S9. The ansatz involves extracting T_c with a fixed broadening value (as in the case of MgB₂) and estimating the decay parameter (A) in the exponential variation of T_c with $\sigma$ as T ${}_{c}\sim\exp(-A\sigma^{1/3}+B)$ . Unconverged results exhibit larger values of A, which decrease with a denser k-mesh.

The detail theory of EPC calculations within DFPT formalism in presented in the section “Theory of superconductivity: isotropic approximation”. The variation of density of states at the Fermi level ( $N(\epsilon_{F})$ ) with respect to smearing $\sigma$ , depends on Gaussian,

G(\Delta,\sigma)=\frac{1}{\sigma^{2}}exp\left[-\frac{(\epsilon_{n\textbf{k}}-\epsilon_{F})^{2}+(\epsilon_{m\textbf{k}+\textbf{q}}-\epsilon_{F})^{2}}{\sigma^{2}}\right]\sim\frac{1}{\sigma^{2}}exp\left[-\frac{\Delta^{2}}{\sigma^{2}}\right]

(9)

with $\Delta^{2}\sim(\epsilon_{n\textbf{k}}-\epsilon_{F})^{2}+(\epsilon_{m\textbf{k}+\textbf{q}}-\epsilon_{F})^{2}$ also depends on $\sigma$ . Despite of dependency of $\lambda$ on EPC matrix ( $<g^{2}>$ ) and phonon frequency ( $<\omega^{2}>$ ) terms, the variation in $\lambda$ is largely guided by the variation in $N(\epsilon_{F})$ or the Gaussian term G. Fig. S8(a)(a) shows the variation of G with respect to smearing $\sigma$ for different values of $\Delta$ , assuming $\Delta$ independent of $\sigma$ . The effect of $\sigma$ on G dictates the variation of $\lambda$ and hence T_c, with $\Delta$ depends on the materials as well as on size of the grid to compute double delta integration. A larger $\Delta$ represents more states contributing towards the double delta summation within space around the Fermi-level spanned by $\sigma$ . In other word, $\Delta$ will increase with $\sigma$ if the density of states has local maximum close to the Fermi-level, and the variation of EPC properties will be similar to the Gaussian plots corresponding to $\Delta$ $\geq$ 0.05. For Niobium, the variation of EPC properties with respect to $\sigma$ follows similar to blue-dashed curve corresponding to $\Delta$ = 0.01 [3], whereas for MgB₂, the variation follows the Gaussian corresponding to $\Delta$ $<$ 0.01. For $\sigma\rightarrow 0$ and $\Delta\rightarrow 0$ , if $\frac{\Delta}{\sigma}\leq 1$ the Gaussian function recovers the delta function, while for $\frac{\Delta}{\sigma}\geq 1$ the Gaussian drops to zero similar to $\Delta\geq$ 0.01 cases. Our primary focus is more on exponentially decaying cases.

The effect of smearing $\sigma$ on $\omega_{log}$ is insignificant compared to $N(\epsilon_{F})$ or $\lambda$ [2]. According to the Bardeen $–$ Cooper $–$ Schrieffer (BCS) theory [6],

T_{c}\sim\Theta_{D}exp(-1/\lambda)\sim\Theta_{D}exp(-\frac{1}{N(\epsilon_{F})U})

(10)

where $\Theta_{D}$ is the Debye cutoff energy and U is electron-phonon coupling potential. One can established the relation between $\lambda$ or $T_{c}$ on $\sigma$ as $\lambda\sim 1/\sigma^{\alpha}$ and $T_{c}\sim\Theta_{D}exp(-1/\lambda)=exp(-A\sigma^{\alpha}+B)$ with $\Theta_{D}\sim exp(B)\times$ constant. Here parameter $\alpha$ is a constant with a positive value if $T_{c}$ decreases with increasing $\sigma$ and becomes negative otherwise. Here A and B are constants to be determined with A denoting the coefficient of exponential increase (for negative A) or decrease (for positive A) of $T_{c}$ with respect to $\sigma$ . To determine the value of these constants, we perform linear-fit of $logT_{c}$ vs $\sigma^{\alpha}$ for different k- grids for MgB₂, which is frequently studied both theoretically as well as experimentally and often challenging to obtain the converged SC properties [7, 8, 9]. Furthermore, one can capture the behavior of T_c vs $\sigma$ for a wide range of T_c using MgB₂, as shown in Fig. S8(a)(b). We utilize a smaller k-mesh of 6 $\times$ 6 $\times$ 4 and a slightly larger k^′-grid of 8 $\times$ 8 $\times$ 6 for comparison [Fig. S8(a)(b) and (c)].

Fig. S8(a) (b) represents the variation of $T_{c}$ with respect to $\sigma$ , while Fig. S8(a) (c) presents $logT_{c}$ vs $\sigma^{\alpha}$ plot. It exhibits nearly linear behavior up to $\sigma=0.05\,\text{Ry}$ , i.e. $\sigma^{1/3}$ = 0.37, after which it deviates from linearity. The linear-fit has optimal coefficient of determination score (R²-score) for $\alpha<0.5$ on $logT_{c}$ vs $\sigma^{\alpha}$ data for k(charge density and EPC)-q(Phonon and EPC)-2k(interpolating EPC matrix) grids (Fig. S8(a) (c)) [Table within the Fig. S8(a)(c)]. Therefore, we chose $\alpha=1/3$ in this work with R²-score of 0.9987. Besides different choice of $\alpha$ leads to different values of A and B, it doesn’t have a significant role. For coarse grid, the T_c has a larger dependency on $\sigma$ and shows larger exponential decay, compared to more converged calculations on denser grids. The exponential decay parameter “A” decreases from 19 to 10 with grids changing from coarse to dense one. This analysis suggests that A ${}_{MgB_{2}}$ = 12 $-$ 13 can be used as a cutoff for this work, the calculations can be considered unconverged for A $>$ A ${}_{MgB_{2}}$ , while the calculations can be considered converged for A $<$ A ${}_{MgB_{2}}$ for accurate predictions. Fig. S8(a)(d) shows the convergence of the T_c with respect to q-point mesh, keeping k-grid fixed at 12 $\times$ 12 $\times$ 12. A q-point grid as half as that of the ground-state k-point grid is sufficient to provide the converged results for MgB₂. Fig. S8(a) (e) represents the spectral functions for various grids with k=8 $\times$ 8 $\times$ 6. Denser grid (2k) slightly blue shifts the spectral function peaks at lower frequency range (15.5-16.5 THz), while peaks at higher frequency ranges (20-22.5 THz) remain unaffected. This results slight change in $\lambda$ from 0.78 to 0.72 and T_c changing from 20 K to 15-16 K [Figs. S1-S5]. A $\lambda$ of 0.78 agrees with previous theoretical value of $\sim$ 0.75 from anisotropic Migdal-Eliashberg calculation using $\mu^{*}=$ 0.16 [5], $\lambda=$ 0.71 from fully anisotropic SCDFT [10], and $\lambda=$ 0.73 from previous isotropic Eliashberg approximation [11] at slightly larger $\sigma=$ 0.015 Ry. This indicates a k-grid obtained from MP database with $\sigma=0.01$ Ry already provide converged results in the case of MgB₂. However, it is not always the case for other materials. Based on these results, parameters $\sigma$ = 0.01 Ry with $\mu^{*}_{c}=0.16$ seems to be reasonable choice for smearing with 8 $\times$ 8 $\times$ 6 to 16 $\times$ 16 $\times$ 12 k-grids for MgB₂, and for other systems for the sake of comparison. These parameters could also depend on pseudopotentials. In order to achieve converged results, it is necessary to use denser k- and q-grids. Subsequently, a double-delta integration should be performed, selecting the results that correspond to the limit of $\sigma\rightarrow 0$ . Other details convergence tests of MgB₂ with respect to Brillouin-zone sampling are presented in Figs. S2-S7.

In Fig. S8(b), we show two cases of YB₁₂ and C₂I₂Y₂ in which the coarse k- grids from Materials Project and fine k- grid only twice of that result to qualitatively inaccurate results, unusually high T_c for the former while predicting negligible T_c for the latter. With coarse grids, the exponential decay parameter of YB₁₂ is around 38, much larger than the critical value estimated for MgB₂ (A ${}_{MgB_{2}}$ $\sim$ 12). Increasing k-mesh grids two times in each direction, T_c drops from 40.6 K to 1.67 K at $\sigma=$ 0.01 Ry and drops to almost zero for $\sigma>$ 0.01 Ry indicating non-superconductor, which is in good agreement with the experiment [12]. We found a couple of cases such as C₂I₂Y₂ when $T_{c}$ first decreases and increases later with $\sigma$ (we set A = 0 in Fig. S10). Such behavior hasn’t been observed in G( $\Delta$ , $\sigma$ ) vs $\sigma$ plots for a wide range of $\Delta$ (Fig. 8(a) of the main text). The coarse grids result to a very low T_c of 0.97 K for C₂I₂Y₂. However, the denser mesh corrects the behavior with an exponential decay parameter of 12.5 much closer to the reference value of MgB₂. Also, the critical temperature T_c improves to 5.29 K agreeing much more strongly to the experiment [13]. Note that, this analysis only works for compounds with non-zero $T_{c}$ for a wide range of $\sigma$ . For $T_{c}<\sim$ 1 K at $\sigma=0.01$ Ry (computational details for choosing $\sigma=0.01$ Ry), $T_{c}<$ 0.1 K for $\sigma>0.01$ Ry, and rapidly drops to zero for larger $\sigma$ , we identify the compound to be non-superconducting with the value of A as zero (A = 0 $<$ A ${}_{MgB_{2}}$ , already converged). To summarize the ansatz, we present a simple schematics of the process shown as in Fig. S9.

Table ST3: Comparing results between coarse and fine grids for compounds with A

>

{}_{MgB_{2}}

;

T^{eff}_{c}

is the critical temperature calculated from efficient EPC calculations, using k-point mesh grid from MP database;

T^{Corr}_{c}

is the critical temperature calculated using denser k-point mesh and fine k- mesh grids (2 times of MP database K-point mesh), while q-grid is kept fixed;

Compound	$T^{eff}_{c}$ (K)	$T^{Corr}_{c}$ (K)	$T^{Expt}_{c}$ (K)
YB₁₂(Fm-3m)	40.6	1.67	0 [12]²²2No transition observed above 2.5 K
CaNiBN (P4/nmm)	0.96	2.43	2.2 [15]
BaC6 (P6₃/mmc)	6.50	1.70	0 [16]
La3PbC (Pm-3m)	3.20	0.50	0 [17]³³3No transition observed above 1 K
La3SnC (Pm-3m)	5.09	0.81	0 [17] ⁴⁴4No transition observed above 1 K
CsC8 (P6/mmm)	5.97	0.4	0.13 [20]
AlB2 (P6/mmm)	11.3	0.3	0 [21]
C2I2Y2 (C2/m)	0.97	5.29	9.97 [13]
C2Cl2Y2 (C2/m)	0.508	2.73	2.3 [22]

Table ST4: Comparing results between coarse and fine grids for compounds with A

<

{}_{MgB_{2}}

;

T^{eff}_{c}

is the critical temperature calculated from efficient EPC calculations, using k-point mesh grid from MP database;

T^{Corr}_{c}

is the critical temperature calculated using denser k-point mesh and fine k- mesh grids (2 times of MP database K-point mesh), while q-grid is kept fixed; Here results do not change much qualitatively from coarse to denser mesh. Vertical line in the table after LaB6 separates data from accurate to inaccurate qualitative predictions, compared to experimental results.

Compound	$T^{eff}_{c}$ (K)	$T^{Corr}_{c}$ (K)	$T^{Expt}_{c}$ (K)
WB (Cmcm)	5.69	4.9	2.8 [23]
ReB2 (P6₃/mmc)	0	0	0 [24]
LaB6 (Pm-3m)	0.42	0	0.005 [25]
Ta2C (P-3m1)	0.0	0.0	0 [26]
YIr3B2 (P6/mmm)	5.5	5.54	N/A ⁵⁵5Not reported
RuB2 (Pmmn)	0.59	0.32	1.5 [28]
VC (Fm-3m)	31.67	19.9	3.2 [29]
ZnNi3C (Pm-3m)	15.37	15.4	0 [30]⁶⁶6No transition observed above 2.0 K
YRh3B2 (P6/mmm)	5.8	4.2	0 [32] ⁷⁷7No transition observed above 1.5 K
YRu3B2 (P6/mmm)	3.7	2.12	0 [34]⁸⁸8No transition observed above 1.2 K

Next, we present results obtained from EPC calculations for 113 known boron and carbon superconducting (N = 53) and non-superconducting (N = 60) compounds from SuperCon database [36]. Fig. S10 represents the distribution of $T_{c}$ with respect to exponential decay parameter A for efficient calculations for $\sigma=0.01$ Ry with $\mu^{*}_{c}=0.16$ . Based on the convergence check ansatz, we found that around 15 % (NoConv) of the results are not fully converged, while 70 % of them are qualitatively inaccurate (NoConv-False), compared to available experimental results. Similarly, we have 12 % of cases that have A $<$ A ${}_{MgB_{2}}$ with qualitatively inaccurate predictions (Conv-False), which could be attributed from either inaccuracy of approximations to compute $T_{c}$ or inaccuracies within available experimental results. This work not only addresses the 15 % (NoConv) cases but also assists in validating true (Conv-True) results.

We present a few qualitatively inaccurate predictions with A $>$ A ${}_{MgB_{2}}$ and improved results with a denser grid in Table ST3, while Table ST4 shows unaffected results (qualitatively) with denser grids for the cases with A $<$ A ${}_{MgB_{2}}$ . For example, AlB₂ is predicted to be superconductors with a coarse grid, while it changes to a non-superconductor with a denser grid, consistent with the experimental prediction. Similarly, $T_{c}$ of superconductors such as CaNiBN, C2I2Y2, and C2Cl2Y2 change from below 1 K (non-superconductor) to a larger value, and show a better agreement with the experiment. On the contrary, the qualitative results remain unchanged for converged calculations represented by A $<$ A ${}_{MgB_{2}}$ , regardless of their agreement with the experimental references.

Application of convergence ansatz on known compounds of SuperCon

In this section, we compare the computed $T_{c}$ with available experimental results, as shown in Fig. S11 for efficient EPC calculations with and without improved data for A $>$ A ${}_{MgB_{2}}$ (left panel) and A $<$ A ${}_{MgB_{2}}$ (right panel) cases respectively. We also highlight some significantly deviated results with red rectangles for efficient calculations. With improved data, computed critical temperatures $T_{c}$ have much better agreement compared to the references, as indicated by the red rectangle enclosing more data points (right panel of Fig. S11), compared to the plot on the left. The results obtained from the ab-initio calculations demonstrate a notable level of accuracy with the mean absolute error (MAE) of 2.21 K for the critical temperature, compared to experiments, highlighting the reliability and robustness of this ansatz test. This is in comparison to the MAE of 3.18 K observed in efficient calculations. Please note that we have exclusively incorporated DFPT T_c results for systems that are dynamically stable. Despite the general agreement between DFPT and experimental findings, there exist notable discrepancies. Instances like ZnNi3C (Pm-3m), with a calculated T_c of 15 K, deviate enormously from the experimental values of 0 K [37]. A comparable investigation of ZnNi3C by Hoffmann et al.[38], utilizing different pseudopotentials from the PSEUDODOJO project[39], revealed phonon instabilities in these compounds. Notably, these unstable phonon modes were identified to possess significant mode-resolved EPC strength[40], a finding deserving attention, as we address these concerns in the “Imaginary Phonon Modes and Superconductivity” section. Similarly, Nb2InC (P6₃/mmc) with a calculated T_c of 0.34 K, VC (Fm-3m) at 20 K, and WB (I4₁/amd) at 18.5 K, present a problematic cases. Experimental T_c measurements stand at 3.2 K [29], 4.3 K [41], and 7.5 K [42] for VC, WB, and Nb2InC, respectively. These discrepancies can be ascribed to the constraints of both theoretical approaches and experimental procedures. For instance, older experimental investigations initially reported critical temperatures that were later rectified by more recent experiments or vice versa. For instance, in Ref [43], superconductivity below 4.7 K was claimed in certain YB₁₂ samples, yet subsequent experiments, such as those in [44], failed to corroborate these findings. Furthermore, employing more sophisticated theoretical methodologies like solving the many-body anisotropic Migdal-Eliashberg equations or utilizing SCDFT to estimate T_c can significantly enhance the accuracy of calculations compared to isotropic methods. As an illustration, when examining MgB₂, T_c calculated using isotropic Eliashberg theory falls within the 15-20 K range. However, in Ref. [45], T_c values of 34-42 K were reported over a range of $\mu^{*}$ values, demonstrating the improvement achievable with these advanced techniques.

In Table ST5, we show some of the inaccurate results predicting nonsuperconductors with 2D q- grids (4 $\times$ 4 $\times$ 1). Emergence of strong el-ph coupling and T_c with denser 3D q- grid (3 $\times$ 3 $\times$ 3) is unlikely for such nonsuperconducting predictions with T_c $<$ 0.01 K for $\sigma$ $>$ 0.01 Ry [Fig. S9].

Table ST5: Comparison of superconducting properties predicting nonsuperconductors that do not agree with the experiments. Denser k- and q- grids do not improve the results.

Compound	Grids	$\lambda$	$\omega_{log}$ (K)	$T_{c}$ (K)	$T^{Expt}_{c}$ (K)
Nb2InC	12 $\times$ 12 $\times$ 12, 3 $\times$ 3 $\times$ 3	0.39	285	0.1	7.5 [46]
	8 $\times$ 8 $\times$ 2, 4 $\times$ 4 $\times$ 1	0.43	297	0.34
Ti2InC	12 $\times$ 12 $\times$ 12, 3 $\times$ 3 $\times$ 3	0.18	355	0.0	3.1 [47]
	16 $\times$ 16 $\times$ 4, 4 $\times$ 4 $\times$ 1	0.13	352	0.0

Table ST6: Mean Absolute Errors (MAEs) for the 10% test set in each training iteration with different sizes running for 3000 epochs.

Training size	CGCNN				ALIGNN
	$\lambda$	$\omega_{log}$ (k)	T_c (K)	T ${}^{\prime}_{c}$ (K)	$\lambda$	$\omega_{log}$ (K)	T_c (K)	T ${}^{\prime}_{c}$ (K)
250 (Run1)	0.15	72	1.8	1.3	0.15	56	1.7	2.2
323 (Run2)	0.27	85	6.4	5.5	0.16	83	3.0	3.3
351 (Run3)	0.31	70	3.9	6.5	0.24	60	3.0	4.4

DFPT results for dynamically stable compounds

Table ST7: EPC properties of dynamically stable compounds

ID	Compound	SG	FE (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
mp-2252	B2Sc1	191	-0.84	0.34	606	0.032
mp-10020	C1Sc1	225	-0.14	0.61	442	5.086
mp-29941	C1Sc2	164	-0.42	0.24	351	0.0
mp-1232372	B2C2Sc2	194	-0.6	0.21	565	0.0
mp-27693	B8C8Sc4	55	-0.46	0.45	560	1.092
mp-10343	B1C2Sc2	139	-0.57	0.41	467	0.383
mp-10139	B1Sc3Sn1	221	-0.61	0.1	331	0.0
mp-12062	Ag1B2	191	0.52	0.88	568	21.563
mp-7817	B12Y1	225	-0.24	0.48	605	1.69
mp-1084	B12Zr1	225	-0.21	0.61	306	3.409
mp-345	B1Hf1	225	-0.41	0.51	281	1.225
mp-1890	B4Mo4	141	-0.5	0.4	364	0.228
mp-999198	B2Mo2	63	-0.49	0.63	338	4.398
mp-451	B1Zr1	225	-0.37	0.47	383	0.946
mp-763	B2Mg1	191	-0.13	0.78	734	20.027
mp-450	B2Nb1	191	-0.69	0.72	365	7.68
mp-1773	B4Re2	194	-0.43	0.22	409	0.0
mp-1077	B4Ru2	59	-0.29	0.42	364	0.356
mp-1472	B2Zr1	191	-0.99	0.14	581	0.0
mp-2680	B6La1	221	-0.56	0.26	343	0.0
mp-2203	B6Y1	221	-0.4	1.97	81	9.999
mp-1079500	B2Ca2N2Ni2	129	-0.97	0.53	464	2.474
mp-1106180	B6Re14	186	-0.22	0.72	180	3.83
mp-15671	B2Re6	63	-0.2	0.72	209	4.4
mp-2850	B4Os2	59	-0.21	0.61	263	3.07
mp-21502	B4Rh8	62	-0.21	0.65	183	2.692
mp-7857	B4Ti4	62	-0.83	0.27	443	0.0
mp-20881	B2La2N2Ni2	129	-1.05	0.64	301	4.241
mp-9219	B2C1La1Pt2	139	-0.64	0.76	262	6.732
mp-3465	B2La1Rh3	191	-0.67	0.84	174	5.859
mp-6114	B2La3N3Ni2	139	-1.15	0.61	362	4.056
mp-20234	B4Li8Pt12	212	-0.57	1.31	96	7.615
mp-4984	B4Mo10Si2	140	-0.4	0.81	257	7.838
mp-1189073	B8Os4Sc4	62	-0.68	0.63	330	4.305
mp-1105309	B4Ge2Ta10	140	-0.59	0.29	234	0.0
mp-1078866	B4C4Y2	127	-0.44	0.52	455	2.323
mp-15955	B4C4Y2	131	-0.3	0.5	392	1.531
mp-1024989	B2C1Pd2Y1	139	-0.5	1.14	249	15.971
mp-5984	B8Rh8Y2	137	-0.59	0.77	223	5.866
mp-1190832	B16Ru4Y4	55	-0.62	0.28	501	0.0
mp-980205	B16Os4Y4	55	-0.57	0.28	404	0.0
mp-2091	B4V6	127	-0.72	0.23	473	0.0
mp-260	B2Cr2	63	-0.52	0.45	412	0.68
mp-9973	B2V2	63	-0.85	0.23	508	0.0
mp-1183252	B1Ir1	187	-0.2	0.47	214	0.539
mp-567164	B2Rh2	194	-0.39	0.15	313	0.0
mp-1063752	B2Rh2	194	-0.18	0.91	173	7.127
mp-4472	B2C2Mo4	63	-0.25	0.78	199	5.414
mp-10112	B2Ir3La1	191	-0.65	0.62	151	1.828
mp-2536	B2Ni4	140	-0.29	0.22	316	0.0
mp-1080664	B4Cr4	141	-0.53	0.27	449	0.0
mp-1994	B2Hf1	191	-1.02	0.16	518	0.0
mp-2331	B4Mo2	166	-0.43	0.4	437	0.239
mp-20689	B4Nb6	127	-0.63	0.26	365	0.0
mp-13415	B4Ta6	127	-0.67	0.24	306	0.0

Table ST8: EPC properties of dynamically stable compounds

ID	Compound	SG	FE (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
mp-1491	B2V1	191	-0.74	0.29	494	0.001
mp-8431	B4Ca2Rh4	70	-0.61	0.26	325	0.0
mp-1008487	B2W2	63	-0.36	0.68	274	4.902
mp-10142	B4Ta3	71	-0.77	0.36	357	0.046
mp-865	Ca1B6	221	-0.41	0.16	538	0.0
mp-14445	B4Ca2Ir4	70	-0.68	0.26	338	0.0
mp-944	Al1B2	191	-0.04	0.4	564	0.295
mp-10852	B4C4La2	127	-0.44	0.38	420	0.113
mp-7783	B4C4La2	131	-0.31	0.5	459	1.862
mp-2967	B2Co2La1	139	-0.48	0.31	304	0.002
mp-5992	B2C1Ir2La1	139	-0.57	0.36	250	0.035
mp-568083	B2C1La1Ni2	139	-0.44	0.33	299	0.009
mp-6794	B2C1La1Rh2	139	-0.6	0.35	284	0.022
mp-571428	B6Mg3Ni9	181	-0.38	0.2	353	0.0
mp-4938	B2Co3Sc1	191	-0.55	0.37	275	0.07
mp-6939	B4Ir4Sr2	70	-0.63	0.28	351	0.0
mp-7348	B4Rh4Sr2	70	-0.56	0.23	335	0.0
mp-3515	B2Co2Y1	139	-0.56	0.24	383	0.0
mp-1024941	B2Rh3Y1	191	-0.71	0.76	162	4.181
mp-4382	B2Ru3Y1	191	-0.47	0.62	172	2.121
mp-9956	Al2C2Cr4	194	-0.17	0.24	403	0.0
mp-3271	Al1C1Ti3	221	-0.59	0.28	271	0.0
mp-1025497	Al2C2V4	194	-0.52	0.23	425	0.0
mp-4448	Y3Al1C1	221	-0.42	0.1	254	0.0
mp-1190604	Al8C4Nb12	213	-0.46	0.35	256	0.022
mp-1190760	Al8C4Ta12	213	-0.45	0.4	191	0.111
mp-1214417	Ba2C12	194	-0.09	0.53	300	1.709
mp-9961	C2Cd2Ti4	194	-0.54	0.28	233	0.0
mp-21075	C1Hf1	225	-0.94	0.13	457	0.0
mp-1096993	C2Hf2	194	-0.74	0.46	416	0.865
mp-1002124	Hf1C1	216	-0.3	0.05	424	0.023
mp-10611	C1La3Pb1	221	-0.43	0.49	151	0.516
mp-1206443	C1La3Sn1	221	-0.49	0.53	146	0.806
mp-5443	C2Nb4Sn2	194	-0.42	0.45	271	0.527
mp-20661	C2Pb2Ti4	194	-0.57	0.31	321	0.004
mp-631	C1Ti1	225	-0.81	0.16	601	0.0
mp-1282	C1V1	225	-0.41	1.14	312	19.989
mp-20648	C4V8	60	-0.47	0.3	348	0.001
mp-1008632	C1V2	164	-0.45	0.24	319	0.0
mp-2795	C1Zr1	225	-0.8	0.15	484	0.0
mp-1014307	C2Zr2	194	-0.64	0.41	480	0.321
mp-570112	C4Cr6	63	-0.07	0.55	343	2.407
mp-28861	C8Cs1	191	-0.05	0.4	766	0.438
mp-568643	C16Rb2	70	-0.04	0.5	755	2.902
mp-16290	C1Ni3Zn1	221	-0.06	1.94	126	15.406
mp-1066566	C2Ni1Y1	38	-0.33	0.47	395	1.074
mp-1079635	C2Ga2Mo4	194	-0.12	0.77	249	6.627
mp-2305	C1Mo1	187	-0.08	0.18	512	0.0
mp-1552	C4Mo8	60	-0.11	0.7	272	5.298
mp-1221498	C1Mo2	164	-0.05	0.6	287	3.119
mp-910	C1Nb1	225	-0.46	1.11	309	18.924
mp-1094093	C2Nb2	194	-0.35	1.14	300	19.284
mp-999388	C2Nb2	194	-0.33	0.84	455	15.269
mp-999377	C2Nb2	194	-0.09	0.71	320	6.7
mp-1086	C1Ta1	225	-0.58	0.68	236	4.239

Table ST9: EPC properties of dynamically stable compounds

ID	Compound	SG	FE (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
mp-1009832	C1Ta1	216	-0.06	0.69	213	4.034
mp-7088	C1Ta2	164	-0.6	0.22	250	0.0
mp-2034	C4W8	60	-0.02	0.77	212	5.512
mp-33065	C2W4	58	-0.01	0.79	213	6.119
mp-2367	C2La1	139	-0.16	0.5	258	0.978
mp-313	C2Y1	139	-0.19	0.83	282	9.383
mp-1208630	C12Sr2	194	-0.04	0.67	310	5.291
mp-1018048	C2La1Ni1	38	-0.25	0.71	346	6.943
mp-1206284	Br2C2La2	12	-1.12	0.7	174	3.416
mp-20315	C2In2Ti4	194	-0.67	0.13	352	0.0
mp-643367	Br2C2Y2	12	-1.08	0.77	251	6.614
mp-37919	Br2C2Y2	59	-0.08	1.21	109	7.789
mp-23062	C2I2Y2	12	-0.86	0.77	196	5.177
mp-1206889	C2Cl2Y2	12	-1.44	0.6	258	2.725
mp-6576	B2C1Ni2Y1	139	-0.51	0.59	315	3.159
mp-2192	B2Pt2	194	0.07	1.24	188	13.786
mp-1009218	C1Mo1	216	0.5	1.71	103	11.153
mp-632442	Al4C3	1	0.31	0.6	256	2.704
mp-1076	KB6	221	-0.03	0.51	823	3.715
mp-1078278	CrB4	58	-0.31	0.25	622	0.0
mp-1079437	FeB4	58	-0.17	0.9	468	18.723
mp-1080111	B3Mo	166	-0.31	0.29	574	0.001
mp-1106184	MnB4	14	-0.29	0.14	559	0.0
mp-1213975	CaB4	127	-0.39	0.25	551	0.0
mp-1228730	B24Mo7	187	-0.14	1.33	359	28.901
mp-2315	NaB15	74	-0.05	0.1	763	0.0
mp-262	Na3B20	65	-0.06	0.4	782	0.513
mp-27710	CrB4	71	-0.3	0.23	755	0.0
mp-576	B13C2	166	-0.07	0.97	835	39.304
mp-637	YB4	127	-0.59	0.43	388	0.457
mp-7283	LaB4	127	-0.56	0.28	465	0.0
mp-1218188	SrLaB12	123	-0.51	0.36	306	0.041
mp-1232339	LiC12	63	-0.0	0.43	1055	1.24
mp-1227841	BaLaB12	123	-0.49	0.36	304	0.052
mp-1001581	LiC6	191	-0.0	0.39	1043	0.397
mp-1021323	LiC12	191	-0.01	0.33	949	0.035
mp-1001835	LiB	194	-0.17	0.21	328	0.0
mp-1002188	TcB	187	-0.28	0.4	445	0.267
mp-1009695	CaB2	191	-0.14	0.68	444	7.604
mp-1224328	HfNbB4	191	-0.86	0.33	462	0.013
mp-1019317	TcB2	194	-0.44	0.27	497	0.0
mp-1025170	Ti3B4	71	-0.93	0.25	517	0.0
mp-1217974	TaB4W3	25	-0.48	0.6	277	2.921
mp-1080021	Nb2B3	63	-0.75	0.45	409	0.777
mp-1217965	TaB2Mo	38	-0.66	0.57	348	2.788
mp-1102394	Nb5B6	65	-0.77	0.4	398	0.209
mp-1108	TaB2	191	-0.65	1.22	253	18.116
mp-1217924	TaNbC2	166	-0.52	1.41	326	28.387
mp-1217818	TaVB4	191	-0.69	0.6	387	4.047
mp-1206441	V5B6	65	-0.83	0.26	488	0.0
mp-1207750	Y5(SiB4)2	127	-0.68	0.33	401	0.014
mp-1217023	TiB2W	38	-0.78	0.3	419	0.001
mp-1079333	B2CN	51	-0.53	0.46	1045	2.381
mp-13854	B3Ru2	194	-0.33	0.14	445	0.0
mp-14019	NiB	63	-0.24	0.24	376	0.0

Table ST10: EPC properties of dynamically stable compounds

ID	Compound	SG	FE (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
mp-1216709	TiNbB2	38	-0.81	0.31	472	0.005
mp-20857	CoB	62	-0.4	0.11	357	0.0
mp-2580	NbB	63	-0.77	0.31	415	0.005
mp-1215250	ZrB4Mo	191	-0.62	1.3	193	15.047
mp-569803	B2W	194	-0.33	0.48	323	0.999
mp-999120	TcB	194	-0.22	0.42	378	0.335
mp-1215178	ZrTiB4	191	-1.0	0.09	617	0.0
mp-9208	V2B3	63	-0.8	0.29	488	0.001
mp-999118	TcB	194	-0.12	1.32	332	26.488
mp-12054	Cr2B3	63	-0.43	0.56	358	2.753
mp-10114	ScB4Ir3	176	-0.59	0.18	258	0.0
mp-9880	YB2C	135	-0.52	0.32	506	0.012
mp-995282	LiAlB4	55	-0.05	0.5	693	2.659
mp-1008527	B2CN	115	-0.44	0.8	644	18.759
mp-1207086	MgAlB4	191	-0.15	0.34	769	0.059
mp-1215209	ZrTaB4	191	-0.84	0.39	436	0.177
mp-1215223	ZrNbB2	38	-0.71	0.35	404	0.04
mp-1216966	TiCrB4	191	-0.68	0.59	339	3.385
mp-1217028	TiB2Mo	38	-0.82	0.25	461	0.0
mp-1220697	Nb2AlB6	191	-0.47	0.54	431	2.781
mp-1224283	HfTaB4	191	-0.85	0.41	423	0.354
mp-1224291	HfTaB2	38	-0.77	0.4	354	0.199
mp-1224347	HfNbB2	38	-0.76	0.37	387	0.073
mp-1226228	CrB2Mo	38	-0.48	0.62	331	3.919
mp-13341	YB4Rh	55	-0.61	0.62	483	6.008
mp-568985	La2B3Br	187	-0.84	0.35	212	0.018
mp-569002	Y3ReB7	63	-0.33	0.81	330	9.953
mp-1008526	B2CN	156	-0.42	1.11	566	34.9
mp-569121	La2B3Cl	174	-1.02	0.41	209	0.158
mp-1191641	YVB4	55	-0.7	0.3	473	0.001
mp-1216398	VCrB2	38	-0.69	0.39	460	0.19
mp-1216475	V3ReB4	25	-0.72	0.39	391	0.169
mp-1216667	TiVB4	191	-0.9	0.27	516	0.0
mp-1217026	TiB4Mo	191	-0.68	0.69	348	6.374
mp-1217095	Ti3B4Mo	6	-0.83	0.28	454	0.0
mp-1217898	TaTiB4	191	-0.88	0.29	441	0.001
mp-1217997	Ta3AlB8	191	-0.5	0.54	316	1.858
mp-1220349	NbB2W	38	-0.58	0.57	341	2.717
mp-1220351	NbVB4	191	-0.71	0.51	421	1.948
mp-1220383	NbB4Mo3	38	-0.56	0.58	354	3.101
mp-1008525	B2CN	160	-0.43	1.66	577	60.733
mp-10057	VCoB3	63	-0.62	0.26	438	0.0
mp-1226242	CrB2W	38	-0.43	0.74	290	6.877
mp-1228634	B4MoIr	187	-0.25	0.5	280	1.054
mp-1196778	Y2B6Ru	55	-0.66	0.22	511	0.0
mp-22709	TaNiB2	62	-0.63	0.3	324	0.001
mp-569116	Sc2B6Rh	55	-0.73	0.52	414	2.079
mp-1224274	HfTiC2	166	-0.83	0.42	464	0.462
mp-1215222	ZrNbC2	166	-0.66	0.73	397	8.92
mp-1224334	HfNbC2	166	-0.73	0.65	385	5.753
mp-1018050	CrC	187	-0.0	0.23	610	0.0
mp-1215480	Zr3NbC4	166	-0.73	0.77	346	9.324
mp-1216691	TiVC2	166	-0.62	0.53	446	2.47
mp-1215174	ZrTiC2	123	-0.71	0.25	461	0.0
mp-1215218	ZrMoC2	166	-0.35	1.79	120	13.656

Table ST11: EPC properties of dynamically stable compounds

ID	Compound	SG	FE (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
mp-38818	HfNbB4	71	-0.86	0.4	420	0.254
mp-1086667	ScNiC2	129	-0.28	0.52	368	1.864
mp-1217975	Ta3TiB4	25	-0.84	0.26	394	0.0
mp-9530	Y4C7	14	-0.28	0.13	323	0.0
mp-9459	Y4C5	55	-0.33	0.84	225	7.678
mp-1188534	ScCrC2	59	-0.34	0.62	389	4.881
mp-1224170	HfZrC2	123	-0.87	0.25	483	0.0
mp-1334	Y2C	166	-0.29	0.29	201	0.0
mp-1232379	YBC	194	-0.29	0.28	427	0.0
mp-29896	Y2B3C2	65	-0.5	0.6	375	3.915
mp-612670	YNiBC	129	-0.52	0.2	421	0.0
mp-567692	B4C3Ni4Y3	139	-0.51	0.31	393	0.003
mp-12737	B2C1Rh2Y1	139	-0.6	0.38	285	0.098
mp-1087495	Tc3B	63	-0.27	0.8	254	7.66
mp-978989	Tc7B3	186	-0.32	0.68	227	4.037
mp-1068296	Fe(BW)2	71	-0.43	0.45	279	0.465
mp-11750	Ti6Si2B	189	-0.66	0.22	333	0.0
mp-1238800	CaBC	194	-0.07	0.42	501	0.494
mp-27261	Ba7(BIr)12	166	-0.48	0.47	233	0.626
mp-29980	Nb4B3C2	63	-0.59	1.01	219	11.258
mp-29979	Nb3B3C	63	-0.66	1.22	229	16.438
mp-541849	Al3(BRu2)2	123	-0.59	0.26	308	0.0
mp-1188408	Zr5Sn3B	193	-0.68	0.45	202	0.343
mp-1206909	CaBPd3	221	-0.56	1.1	103	6.187
mp-9985	NbNiB	63	-0.56	0.34	256	0.018
mp-1080829	Ti6Ge2B	189	-0.62	0.21	305	0.0
mp-1025192	Ta4C3	221	-0.45	1.46	138	12.644
mp-1215211	ZrNbB4	191	-0.85	0.31	479	0.005
mp-1220641	Nb3B4W3	38	-0.43	0.46	299	0.66
mp-10721	Ti2C	227	-0.64	0.14	386	0.0
mp-27919	Ti8C5	166	-0.72	0.22	455	0.0
mp-1218000	Ta4C3	160	-0.55	0.52	183	0.919
mp-1220752	Nb10(SiB)3	42	-0.65	0.24	310	0.0
mp-1189539	Hf2Al3C5	194	-0.13	0.47	1454	3.645
mp-9958	Ti2GeC	194	-0.81	0.48	276	0.847
mp-1025524	Zr2TlC	194	-0.63	0.21	188	0.0
mp-1216707	TiNbC2	166	-0.67	0.73	422	9.623
mp-1079992	Zr2PbC	194	-0.66	0.4	230	0.146
mp-1207413	Zr5Sn3C	193	-0.68	0.53	180	1.013
mp-1217106	Ti2C	166	-0.63	0.35	357	0.037
mp-1217822	TaVC2	166	-0.46	0.8	343	10.296
mp-12990	Ti2AlC	194	-0.7	0.25	371	0.0
mp-3871	Ti2SnC	194	-0.73	0.24	353	0.0
mp-1025427	Ta2GaC	194	-0.52	0.45	231	0.385
mp-1078712	Hf2TlC	194	-0.63	0.21	171	0.0
mp-1079076	Hf2PbC	194	-0.63	0.41	194	0.15
mp-1079908	Ti2SiC	194	-0.79	0.4	334	0.205
mp-1220365	NbVC2	166	-0.39	0.83	365	11.893
mp-21023	Ti3SnC2	194	-0.79	0.18	421	0.0
mp-22144	Ta2InC	194	-0.38	0.37	233	0.062
mp-4893	Hf2SnC	194	-0.78	0.31	205	0.003
mp-5659	Ti3SiC2	194	-0.82	0.37	345	0.081
mp-1092281	Ti2TlC	194	-0.57	0.14	295	0.0
mp-3747	Ti3AlC2	194	-0.76	0.26	342	0.0
mp-3886	Zr2AlC	194	-0.64	0.56	163	1.181

Table ST12: EPC properties of dynamically stable compounds

ID	Compound	SG	FE (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
mp-13137	Hf2CS	194	-1.51	0.19	303	0.0
mp-1220725	Nb2CN	166	-0.8	0.88	310	11.685
mp-1216616	V2CN	166	-0.8	1.0	322	16.224
mp-1214755	BPd6	15	-0.2	0.16	169	0.0
mp-7424	BPd2	58	-0.27	0.42	159	0.16
mp-1078540	Ni6Ge2B	189	-0.31	0.34	190	0.011
mp-1078623	Zr2BIr6	225	-0.81	0.36	184	0.03
mp-12073	Ba(BIr)2	139	-0.5	0.37	269	0.051
mp-7349	Ba(BRh)2	139	-0.48	0.29	295	0.001
mp-7705	NbFeB	187	-0.12	1.76	162	18.063
mp-1215258	ZrBeB	187	-0.58	0.45	436	0.751
mp-1208348	Ta5Ga3B	193	-0.42	0.31	190	0.002
mp-605839	Li2B2Rh3	55	-0.5	0.27	302	0.0
mp-8308	Ca3Ni7B2	166	-0.34	0.13	305	0.0
mp-1206490	Nb2B2Mo	127	-0.64	0.28	367	0.0
mp-31052	LaBPt2	180	-0.92	0.28	142	0.0
mp-1223681	La2(Ni2B)3	44	-0.33	0.46	227	0.517
mp-1097	B2Ta2	63	-0.81	0.41	366	0.297
mp-28930	C16K2	70	-0.03	0.57	1102	8.897
mp-7832	B4W4	141	-0.37	0.45	301	0.523
mp-28613	B3Li3Pt9	189	-0.51	0.4	152	0.101
mp-569759	B4Rh8Zn5	65	-0.42	0.26	223	0.0
mp-571419	Al4C5Zr2	166	-0.35	0.36	384	0.06
mp-1207385	Al8C8Zr2	164	-0.25	0.26	481	0.0
mp-1189895	B2Ge6Ta10	193	-0.45	0.61	180	2.075
mp-10140	B1Sc3Tl1	221	-0.36	0.53	219	1.247
mp-1216165	B1Si6Y10	162	-0.65	0.4	169	0.092
mp-20175	C2In2Sc4	194	-0.55	0.21	171	0.0
mp-20983	C2In2V4	194	-0.34	0.35	349	0.03
mp-1224263	B4Hf1Ti1	191	-1.02	0.1	592	0.0
mp-1224184	B4Hf1Zr1	47	-1.0	0.15	507	0.0
mp-8307	B2Ca2Ni8	191	-0.3	0.33	131	0.004
mp-4079	Al1C1Sc3	221	-0.59	0.04	299	0.047
mp-1103814	C3K5N6	229	-0.52	0.08	199	0.0
mp-1224285	C2Hf1Ta1	166	-0.81	0.64	346	4.709
mp-1215219	C2Ta1Zr1	166	-0.74	0.65	370	5.558
mp-570499	B2La5N6	12	-1.51	0.25	350	0.0
mp-569935	B2La3N4	71	-1.5	0.26	378	0.0
mp-1223086	C6La2Y1	12	-0.16	0.56	271	2.049
mp-1221519	C1Mo2N1	25	-0.35	0.37	472	0.105
mp-1222150	Al1B10Mg4	191	-0.14	0.59	768	7.488
mp-1189984	C8Mo4Y4	62	-0.24	0.61	409	4.562
mp-3380	C8La4Rh4	76	-0.33	0.27	259	0.0
mp-4262	BeAlB	216	-0.05	0.16	578	0.0
mp-5971	YBPt2	180	-1.0	0.18	191	0.0
mp-9596	La(BIr)4	86	-0.58	0.47	201	0.503
mp-1105186	Cu3B5Pt9	189	-0.21	0.4	164	0.099
mp-1106165	Nb5Si3B	193	-0.69	0.46	282	0.638
mp-1106398	V5Ge3B	193	-0.46	0.38	303	0.11
mp-1188194	Ta3B2Ru5	127	-0.53	0.33	146	0.006
mp-1188856	V5Si2B	140	-0.39	0.37	302	0.067
mp-1216445	V9Cr3B8	10	-0.7	0.25	459	0.0
mp-1216643	V10Si6B	162	-0.63	0.33	323	0.014
mp-1220688	Nb3Re3B4	38	-0.35	0.53	265	1.47
mp-1226327	Cr3B4Mo3	38	-0.29	0.78	289	8.091

Table ST13: EPC properties of dynamically stable compounds

ID	Compound	SG	FE (eV/atom)	$\lambda$	$\omega_{log}$ (K)	T_c (K)
mp-29723	LaB2Ru3	191	-0.41	0.67	148	2.477
mp-22759	CoBW	62	-0.43	0.37	342	0.071
mp-28786	Zn(BIr)2	139	-0.32	0.08	319	0.0
mp-9999	Ni(BMo)2	71	-0.48	0.3	352	0.001
mp-1076987	TaNiB	63	-0.62	0.3	253	0.001
mp-3348	LiBIr	70	-0.51	0.16	325	0.0
mp-2760	Nb6C5	12	-0.55	0.51	393	1.632
mp-32679	Nb10C7	12	-0.48	0.45	301	0.537
mp-1226378	Cr2C	164	-0.04	0.38	376	0.118
mp-2318	Nb2C	164	-0.45	0.31	285	0.002
mp-974437	Re2C	194	-0.03	0.32	348	0.007
mp-10037	AlCo3C	221	-0.22	0.1	212	0.0
mp-9987	Nb2PC	194	-0.75	0.44	345	0.493
mp-21003	Y2ReC2	62	-0.42	0.29	282	0.0
mp-28767	Sc5Re2C7	65	-0.48	0.21	384	0.0
mp-567462	Sc3RhC4	12	-0.5	0.68	393	6.91
mp-7130	ScRu3C	221	-0.28	0.4	227	0.12
mp-996161	Nb3AlC2	194	-0.52	0.36	384	0.048
mp-996162	Nb2AlC	194	-0.51	0.46	337	0.768
mp-1078811	Nb2GeC	194	-0.51	0.45	307	0.531
mp-1080835	V2GaC	194	-0.52	0.27	375	0.0
mp-1189574	YWC2	62	-0.27	0.54	389	2.392
mp-4992	ScCrC2	194	-0.34	0.62	415	4.966
mp-8044	V2PC	194	-0.69	0.4	409	0.227
mp-10046	V2AsC	194	-0.53	0.47	354	0.878
mp-1212439	Hf5Al3C	193	-0.47	0.2	2988	0.0
mp-1217764	Ta2CN	123	-0.82	2.7	127	19.495
mp-37179	Ta2CN	141	-0.84	1.94	162	19.757
mp-4384	Nb2CS2	166	-1.12	1.03	222	11.827
mp-559976	Ta2CS2	164	-1.19	0.55	251	1.71
mp-995201	Ti5Si3C	193	-0.82	0.55	253	1.784
mp-1025441	Ta2AlC	194	-0.52	0.44	272	0.374
mp-1079546	Nb2GaC	194	-0.52	0.44	263	0.424
mp-1220371	NbAlVC	164	-0.47	0.44	362	0.491
mp-3732	Ti2CS	194	-1.42	0.2	473	0.0
mp-4563	Ti3TlC	221	-0.44	0.29	229	0.0
mp-1216139	Y4C4I3Br	8	-0.91	0.84	174	5.825
mp-1220491	Nb6V2(CS2)3	12	-1.09	0.54	278	1.768
mp-1220693	Nb2CuCS2	156	-0.9	0.85	207	7.255
mp-1215225	ZrTaCN	160	-1.15	0.78	301	8.323
mp-1025205	Y2Re2Si2C	12	-0.62	0.73	172	3.887
mp-1215184	ZrTiCN	160	-1.33	0.43	437	0.5
mp-1224279	HfTiCN	160	-1.41	0.4	420	0.267
mp-1009894	Zr1C1	216	-0.19	0.01	444	0.405
mp-1068661	ZrBRh3	221	-0.75	0.01	197	0.176
mp-1145	B2Ti1	191	-1.06	0.11	638	0.0
mp-1009817	C1Ta1	187	-0.16	1.54	235	22.8
mp-1542	YB2	191	-0.56	0.42	473	0.453
mp-1216692	TiNbB4	191	-0.89	0.29	481	0.001
mp-4613	Zr2SnC	194	-0.79	0.32	259	0.004
mp-1232384	ZrBC	194	-0.32	1.12	320	19.998

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Machine-learning Guided Search for Phonon-mediated Superconductivity in Boron and Carbon Compounds

Abstract

I Introduction

II Results

II.1 Machine learning guided workflow

II.2 Overview of DFPT calculations

II.3 Training and testing ML models

II.4 Comparison of ML models

II.5 Dynamically stable systems

II.6 Imaginary phonon modes and superconductivity

III Discussion

IV METHODS

IV.1 Convergence with respect to Brillouin zone sampling (k-point mesh)

IV.2 Computational details

V Data availability

VI Acknowledgements

VII Author contributions

VIII Competing interests

IX References

References

Supplementary Materials for “Machine-learning Guided Search for Phonon-mediated Superconductivity in Boron and Carbon Compounds”

Statistics for compounds with unknown Tc

Theory of superconductivity: Isotropic Approximation

Convergence tests for MgB2 and AlB2

Convergence ansatz

Application of convergence ansatz on known compounds of SuperCon

DFPT results for dynamically stable compounds

References

References

Statistics for compounds with unknown T_c

Convergence tests for MgB₂ and AlB₂