Prediction of intrinsic topological superconductivity in Mn-doped GeTe monolayer from first-principles

Topological superconductors (TSCs), characterized by Majorana fermions in their boundary states, hold immense promise for topological quantum computation. However, their experimental realization is challenging due to the intricate interplay of multiple microscopic parameters. Current theoretical approaches often rely on effective models with empirical parameters, lacking material-specificity and accuracy in predicting the superconducting transition temperature (Tc) and the topological properties of the superconducting quasiparticles, especially in the presence of spin-orbit coupling (SOC) and magnetism. This study addresses this gap by developing a first-principles approach to quantify realistic TSC conditions. The approach self-consistently solves the BdG equation using a Wannier function (WF) construction of the band structure, accounting for Rashba SOC, Zeeman splitting, and electron-phonon coupling (EPC). The method's power is demonstrated by applying it to Mn-doped GeTe (Ge1-xMnxTe) monolayer, a known dilute magnetic semiconductor exhibiting superconductivity under hole doping. GeTe, a non-centrosymmetric material, is chosen due to its inherent properties aligning with the prerequisites for Class D TSCs, including inversion symmetry breaking and the possibility of Zeeman gap opening upon Mn doping. The goal is to predict if this material exhibits intrinsic TSC behavior and establish the experimental conditions for its observation, creating a blueprint for future TSC predictions using first-principles calculations.

The discovery of topological superconductors (TSCs) has sparked extensive research due to their potential applications in topological quantum computing. Theoretical understanding of TSCs has advanced, but experimental confirmation remains a significant challenge. First-principles predictions of electronic and topological materials have been successful, but predicting TSCs is more difficult because of the uncertainty in the parameters required for constructing the Bogoliubov-de Gennes (BdG) Hamiltonian. Previous first-principles studies often focused on preconditions for TSC, such as Rashba splitting or topological properties in the normal state, rather than the topology of superconducting quasiparticles. Effective models with empirical parameters, often partially fitting first-principles results, have been used. Number-conserving approaches beyond the mean-field approximation have also been developed but not yet applied to specific materials. Conventional first-principles methods for estimating Tc using McMillan's formula or solving the Migdal-Eliashberg formula are unsuitable for systems with SOC and magnetism. The current work aims to address these limitations by developing a more versatile and accurate first-principles approach.

The researchers developed a first-principles approach to predict the conditions for topological superconductivity by self-consistently solving the Bogoliubov-de Gennes (BdG) equation, incorporating Rashba spin-orbit coupling, Zeeman splitting, and electron-phonon coupling. The methodology employed the following key steps: 1. **First-principles calculations:** Electronic properties of GeTe monolayer and bulk were calculated using the Vienna ab initio simulation package (VASP) within density functional theory (DFT). The generalized gradient approximation (GGA) in the Perdew-Burke-Ernzerhof (PBE) form was used for the exchange-correlation functional. Structural relaxation and self-consistent calculations were performed on a 30x30x1 k-point mesh for the monolayer and an 18x18x18 mesh for the bulk. A plane-wave energy cutoff of 400 eV was used. 2. **Phonon calculations and electron-phonon coupling (EPC):** Phonon spectra and EPC strength were calculated using the Quantum ESPRESSO package, based on density-functional perturbation theory. The Optimized Norm-Conserving Vanderbilt pseudopotentials were used, with a kinetic energy cutoff of 100 Ry for wave functions. A 18x18x1 q-point mesh with a 18x18x1 k-point sampling was used for the dynamic matrix and phonon frequency calculations. A finer 36x36x1 k-point grid was used for EPC calculations to ensure convergence of the density of states. The phonon density of states (DOS) F(ω), the Eliashberg spectral function α²F(ω), and the cumulative frequency-dependent EPC strength λ(ω) were calculated using a 60x60x1 q-point sampling through Fourier interpolation. The EPC strength was computed using equations 5, 6, 7, and 8 from the paper, considering electronic states and phonon modes. 3. **Wannier function construction:** Wannier functions (WFs) were obtained using the WANNIER90 code by fitting the first-principles band structures. These WFs formed the basis for constructing the BdG Hamiltonian. 4. **BdG Hamiltonian construction:** A material-specific BdG Hamiltonian H_BdG(k) was constructed using the Wannier functions. The equation used for BdG Hamiltonian is mentioned in equation 1 of the paper. 5. **Superconducting gap equation:** A self-consistent gap equation (equation 3 in the paper) was formulated to determine the superconducting gap Δ at different temperatures. The Debye temperature θ_D and Coulomb repulsion μ were considered. The gap equation was solved self-consistently. 6. **Zeeman splitting:** The effect of Mn doping was simulated by introducing an out-of-plane Zeeman energy B_z in the Hamiltonian (equation 4 in the paper). The relationship between B_z and Mn concentration was established using the virtual crystal approximation (VCA) and the experimental findings on bulk Ge1-xMnxTe. 7. **Topological characterization:** The topological properties of the superconducting state were characterized by calculating the Chern number using the Berry curvature. Chiral Majorana edge modes were identified by solving the BdG Hamiltonian for a Ge1-xMnxTe nanoribbon. 8. **Phase diagram construction:** A phase diagram was constructed in the parameter space of temperature and Mn concentration to illustrate the superconducting and topological superconducting phases.

The study's key findings are: 1. **GeTe monolayer properties:** The GeTe monolayer exhibits a large Rashba spin splitting, comparable to heavy metals, due to the absence of inversion symmetry. Hole doping moves the Fermi level to the Dirac point, increasing the density of states at the Fermi level. 2. **Zeeman gap opening:** Mn doping introduces a Zeeman gap in the Rashba and Ising bands, with the gap size linearly dependent on the Mn concentration. This was determined using virtual crystal approximation and confirmed by reproducing experimental results for bulk Ge1-xMnxTe. 3. **Superconductivity in GeTe monolayer:** The 0.1-hole doped GeTe monolayer exhibits phonon-mediated superconductivity below ~120 mK, a finding corroborated by the calculated electron-phonon coupling (λ). The superconducting gap Δ and Tc were determined by solving the self-consistent gap equation. 4. **Topological superconductivity in Mn-doped GeTe:** The Mn-doped GeTe monolayer is predicted to be a Class D topological superconductor. The criteria for Class D TSC (δ/2 > Δ, where δ is the Zeeman gap and Δ is the superconducting gap) was met at a Mn concentration exceeding 0.014%. The topological nature was confirmed by calculating the Chern number (-1) and observing chiral Majorana edge modes. 5. **Phase diagram:** A phase diagram showing the superconducting (SC) and topological superconducting (TSC) phases as a function of temperature and Mn concentration was constructed. The TSC phase is predicted to exist within a specific range of Mn concentrations (0.014% < x < 0.017%) and temperatures below the critical temperature Tc. The transition temperature for the TSC phase is estimated to be around 40 mK. The ferromagnetic Curie temperature (TFM) was also considered in constructing the phase diagram and was found to be consistent with the onset of the TSC phase. 6. **Candidate Materials:** Beyond the Mn-doped GeTe monolayer, two more candidate materials for Class D TSCs were suggested: MnBi2Te4/Bi2Te3/NbSe2 heterostructures and Si-terminated HoRh2Si2 surfaces with proximity-induced superconductivity.

The results of this study demonstrate the successful development of a first-principles approach for predicting topological superconductors (TSCs). The prediction of intrinsic topological superconductivity in Mn-doped GeTe monolayer addresses a significant challenge in the field by providing a material-specific and quantitative assessment of TSC conditions. The method's accuracy is supported by its ability to reproduce superconductivity in known materials. The prediction of a relatively low transition temperature for the TSC phase (~40 mK) suggests that experiments should focus on low-temperature measurements to observe the predicted topological phase. The phase diagram generated offers a roadmap for experimental verification. The proposed candidate materials beyond GeTe suggest that the developed methodology is broadly applicable and can guide the search for new TSCs. This work provides valuable insights into the design and realization of TSCs, bridging the gap between theoretical predictions and experimental realization.

This study presents a novel first-principles approach to predict topological superconductors, successfully predicting the emergence of topological superconductivity in Mn-doped GeTe monolayer. The prediction highlights the importance of considering electron-phonon coupling, Rashba spin-orbit coupling and Zeeman splitting in the search for novel TSC materials. The development of this approach opens up exciting avenues for future research, enabling the exploration of various material systems and guiding the design of experiments to verify the predicted topological phases. The identification of additional candidate materials for Class D TSCs further underscores the method's potential in expanding the field of topological superconductivity. Future work could involve refining the theoretical model by incorporating additional factors, such as magnetic anisotropy and defects, to enhance the accuracy of predictions.

The study acknowledges limitations inherent to the computational methods used. The accuracy of the predictions depends on the accuracy of the DFT calculations and the approximations used in modeling the electron-phonon coupling and Coulomb interactions. The use of the virtual crystal approximation for modeling Mn doping may not perfectly capture the effects of Mn atoms in the GeTe lattice. While the study presents a lower-limit estimate for the electron-phonon coupling constant, the precise value might be influenced by many-body interactions and correlation effects not fully captured in this model. These limitations highlight the need for experimental validation of the theoretical predictions.