[Paper Review] Spectral properties of a mixed singlet-triplet Ising superconductor

Sourabh Patil, Gaomin Tang+1 more|arXiv (Cornell University)|Jul 7, 2023

2D Materials and ApplicationsMaterials Science35 references3 citations

TL;DR

This study investigates the spectral properties of a mixed singlet-triplet Ising superconductor under in-plane magnetic fields, focusing on the mirage gap—arising from finite-energy pairing correlations. It shows that triplet pairing reduces the maximum mirage-gap width and enables finite mirage gaps at high fields when temperature is below the triplet critical temperature, contrasting with the singlet-only case where mirage gaps vanish at critical fields.

ABSTRACT

Conventional two-dimensional superconductivity is destroyed when the critical in-plane magnetic field exceeds the so-called Pauli limit. Some monolayer transition-metal dichalcogenides lack inversion symmetry and the strong spin-orbit coupling leads to a valley-dependent Zeeman-like spin splitting. The resulting spin-valley locking lifts the valley degeneracy and results in a strong enhancement of the in-plane critical magnetic field. In these systems, it was predicted that the density of states in an in-plane field exhibits distinct mirage gaps at finite energies of about the spin-orbit coupling strength, which arise from a coupling of the electron and hole bands at energy larger than the superconducting gap. In this study, we investigate the impact of a triplet pairing channel on the spectral properties, primarily the mirage gap and the superconducting gap, in the clean limit. Notably, in the presence of the triplet pairing channel, the mirage-gap width is reduced for the low magnetic fields. Furthermore, when the temperature is lower than the triplet critical temperature, the mirage gaps survive even in the strong-field limit due to the finite singlet and triplet order parameters. Our work provides insights into controlling and understanding the properties of spin-triplet Cooper pairs.

Motivation & Objective

• To understand the interplay between singlet and triplet pairing channels in Ising superconductors under in-plane magnetic fields.
• To investigate how triplet pairing affects the spectral properties, particularly the mirage gap, which is a signature of finite-energy pairing correlations.
• To determine under what conditions the mirage gap persists in the strong-field regime, especially when the temperature is below the triplet critical temperature.
• To resolve discrepancies in experimental observations of enhanced superconducting gaps in few-layer NbSe2 by including triplet pairing effects.

Proposed method

• The study employs a Bogoliubov–de Gennes Hamiltonian in the Nambu basis to describe the system near the K and K′ valleys, incorporating both singlet and triplet order parameters.
• The model includes spin-orbit coupling (Ising type) and an in-plane magnetic field, with the effective Zeeman term arising from the magnetic field and spin-orbit coupling.
• Self-consistent gap equations are solved numerically to determine the temperature and magnetic field dependence of the singlet (Δs) and triplet (Δt) order parameters.
• The mirage-gap width δ is extracted from the density of states, with analytical approximations used in the high-field limit to derive the asymptotic behavior of Δs and Δt.
• The analysis focuses on the clean limit, assuming no disorder, and uses dimensionless units relative to the singlet critical temperature Tcs.
• The system is analyzed in two regimes: T > Tct (where both order parameters vanish at critical field) and T < Tct (where both remain finite at high fields), with emphasis on the resulting spectral differences.

Experimental results

Research questions

• RQ1 How does the inclusion of a triplet pairing channel modify the maximum width of the mirage gap in an Ising superconductor under in-plane magnetic fields?
• RQ2 What happens to the mirage gap at high in-plane magnetic fields when the temperature is below the triplet critical temperature Tct?
• RQ3 Why do experimental observations of superconducting gaps in few-layer NbSe2 exceed predictions based on singlet pairing alone?
• RQ4 Under what conditions does the mirage gap survive in the strong-field limit, and how does this differ from the singlet-only case?
• RQ5 How is the interplay between singlet and triplet order parameters governed by the self-consistent gap equations in the presence of an in-plane magnetic field?

Key findings

• The maximum mirage-gap width achievable by tuning the in-plane magnetic field decreases with increasing triplet critical temperature Tct.
• When the temperature is above Tct, the mirage gaps vanish at the critical magnetic field, consistent with the singlet-only case where both Δs and Δt vanish at the same field.
• When the temperature is below Tct, both the singlet and triplet order parameters remain finite even at very high in-plane magnetic fields, leading to a finite mirage gap.
• In the high-field limit, the singlet order parameter scales as Δs ∝ βsoΔt / Bx, showing that Δs remains finite as long as Δt is non-zero.
• The mirage gap width δ remains finite at high fields due to the non-vanishing order parameters, particularly when T < Tct, indicating robust finite-energy pairing correlations.
• The analytical derivation confirms that the mirage gap persists in the strong-field regime due to the coupling between singlet and triplet channels, which stabilizes the gap structure.