[Paper Review] Superconducting critical temperature of hole doped blue phosphorene

Davoud Nasr Esfahani, Reza Asgari|arXiv (Cornell University)|Oct 16, 2017

2D Materials and Applications1 references3 citations

TL;DR

This study theoretically investigates hole-doped blue phosphorene using density functional theory and the Migdal-Eliashberg formalism with vertex corrections, predicting a high superconducting critical temperature (Tc) ranging from 40 K to 100 K due to strong electron-phonon coupling. The inclusion of vertex corrections alters Tc by ±20 K depending on doping level, confirming robust superconductivity even beyond standard Eliashberg theory.

ABSTRACT

We theoretically explore the superconducting critical temperature of hole doped blue phosphorene. Implementing the density functional theory calculations, we show that for the hole doped blue phosphorene, the isotropic superconducting state is induced owing to the quite strong electron-phonon coupling. The theory is based on the Migdal-Eliashberg formalism and the critical temperature is obtained through set-of-equations, self-consistency. In addition, we include a vertex correction diagram to the Migdal-Eliashberg formalism. The inclusion of the vertex correction beyond the Migdal-Eliashberg formalism changes the $T_c$ about $\pm20$K, depending on the level of the doping. Our accurate numerical results show that the superconducting critical temperature is still quite high, even in the cases that the vertex correction is implemented.

Motivation & Objective

• To investigate the superconducting critical temperature (Tc) in hole-doped blue phosphorene using first-principles calculations.
• To determine whether strong electron-phonon coupling (EPC) can induce high-Tc superconductivity in this 2D semiconductor.
• To assess the impact of vertex corrections beyond the Migdal-Eliashberg formalism on Tc predictions.
• To evaluate the role of doping level and electronic density of states (DOS) in tuning Tc.
• To provide a quantitative prediction of Tc for experimental verification in 2D superconducting materials.

Proposed method

• Employed density functional theory (DFT) to compute electronic and phononic structures of hole-doped blue phosphorene.
• Applied the multi-band Migdal-Eliashberg formalism to solve for Tc self-consistently using electron-phonon coupling matrix elements.
• Incorporated vertex corrections via second-order self-energy diagrams to improve accuracy beyond standard Eliashberg theory.
• Used Gaussian broadening and fine k/q-mesh convergence tests to ensure numerical stability of λ, α²F, and Tc.
• Calculated electron-phonon coupling strength (λ) and spectral function α²F as key inputs for Tc determination.
• Compared results using constant DOS (ConstDOS) and variable DOS (VarDos) approximations to assess Tc sensitivity.

Experimental results

Research questions

• RQ1 Can hole doping induce a high-Tc superconducting state in blue phosphorene due to strong electron-phonon coupling?
• RQ2 How does the inclusion of vertex corrections in the Migdal-Eliashberg formalism affect the predicted Tc?
• RQ3 What is the dependence of Tc on the Fermi energy shift (δEF) and hole doping density?
• RQ4 How do different DOS approximations (ConstDOS vs. VarDos) influence the Tc prediction?
• RQ5 Is the predicted Tc robust and experimentally verifiable across varying doping levels and numerical convergence parameters?

Key findings

• The superconducting critical temperature (Tc) in hole-doped blue phosphorene ranges from 40 K to 100 K for hole densities between 5×10¹³ cm⁻² and 3.8×10¹⁴ cm⁻².
• The inclusion of vertex corrections in the Migdal-Eliashberg formalism shifts Tc by ±20 K depending on the doping level (δEF), indicating significant many-body effects.
• Tc is highest at lower hole doping densities, with the maximum Tc occurring near δEF = -0.055 eV.
• The VarDos approach yields more accurate Tc predictions than ConstDOS, especially near the valence band maximum.
• Numerical convergence is achieved with 800×800 k-mesh and 200×200 q-mesh, with α²F and λ stable under varying Gaussian broadening σ.
• Strong electron-phonon coupling (λ) is confirmed, with peak coupling near the valence band maximum, supporting high-Tc superconductivity.