[Paper Review] Enhanced Superconducting Diode Effect due to coexisting Phases

Sayan Banerjee, Mathias S. Scheurer|arXiv (Cornell University)|Apr 6, 2023

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper proposes a mechanism for enhancing the superconducting diode effect (SDE) in junction-free superconductors with spontaneously broken time-reversal symmetry (TRS). By demonstrating a mutual coupling between the supercurrent and a TRS-breaking order parameter—particularly when their energy scales are comparable—it shows that this back-action effect can significantly amplify current asymmetry, explaining large diode efficiencies observed in twisted trilayer graphene and enabling design principles for high-performance zero-field superconducting diodes.

ABSTRACT

The superconducting diode effect refers to an asymmetry in the critical supercurrent $J_c(\hat{n})$ along opposite directions, $J_c(\hat{n}) eq J_c(-\hat{n})$. While the basic symmetry requirements for this effect are known, it is, for junction-free systems, difficult to capture within current theoretical models the large current asymmetries $J_c(\hat{n})/J_c(-\hat{n})$ recently observed in experiment. We here propose and develop a theory for an enhancement mechanism of the diode effect arising from spontaneous symmetry breaking. We show - both within a phenomenological and a microscopic theory - that there is a coupling of the supercurrent and the underlying symmetry-breaking order parameter. This coupling can enhance the current asymmetry significantly. Our work might not only provide a possible explanation for recent experiments on trilayer graphene but also pave the way for future realizations of the superconducting diode effect with large current asymmetries.

Motivation & Objective

• To explain the experimentally observed large diode efficiency in junction-free superconducting trilayer graphene, where critical current asymmetry exceeds theoretical expectations.
• To develop a theoretical framework for the superconducting diode effect (SDE) in systems with spontaneously broken time-reversal symmetry (TRS), beyond external field or current-induced effects.
• To identify and characterize a feedback mechanism where the supercurrent couples back to the TRS-breaking order parameter, enhancing current asymmetry.
• To provide a microscopic and phenomenological theory that unifies the role of symmetry-breaking order and supercurrent in enabling high-efficiency SDE.
• To offer design principles for future materials and devices with large, tunable SDE efficiencies without external magnetic fields.

Proposed method

• Formulates a phenomenological Ginzburg-Landau-like theory incorporating both superconducting order parameter Δ and a time-reversal-odd order parameter Φ, with a coupling term that breaks TRS.
• Derives the effective action and self-consistent equations for Δ and Φ, showing that the supercurrent induces a non-uniform modulation of the order parameter, leading to current asymmetry.
• Performs a microscopic BdG (Bogoliubov-de Gennes) calculation using a Hamiltonian with momentum-space coupling between electron and hole states, including the effects of the TRS-breaking field.
• Computes the supercurrent using the linear response formula involving the derivative of the normal-state energy with respect to momentum q, projected through the Nambu spinor and quasiparticle occupation functions.
• Uses the full quasiparticle wavefunction amplitudes (u and v) to compute the current, accounting for the non-trivial mixing between electron and hole components due to the order parameter coupling.
• Numerically solves the self-consistent equations for Δ(q) and Φ(q), showing that the maximum of Δ(q) is pinned at q=0 due to the back-action, even when C3z symmetry is broken.

Figure 1 : Phenomenological theory with leading nematic coupling, $\alpha\neq 0$ , $\alpha^{\prime}=\alpha^{\prime\prime}=0$ in Eq. ( 5 ). $\Phi_{v}(\boldsymbol{q})$ and $\Gamma(\boldsymbol{q})$ for (a) $\alpha=-0.5$ and (b) $\alpha=-3.0$ are shown in upper and lower panels, respectively. (c) Angula

Experimental results

Research questions

• RQ1 How can the large current asymmetry observed in zero-field trilayer graphene be theoretically explained?
• RQ2 What is the role of the mutual coupling between supercurrent and time-reversal-odd order parameter in enhancing the superconducting diode effect?
• RQ3 Can a back-action mechanism from the supercurrent to the order parameter significantly increase the diode efficiency η?
• RQ4 Under what conditions does the coupling between supercurrent and order parameter lead to maximal current asymmetry?
• RQ5 How does the energy scale of the TRS-breaking order parameter relative to superconductivity affect the enhancement of the SDE?

Key findings

• The supercurrent couples back to the time-reversal-odd order parameter, creating a feedback mechanism that enhances current asymmetry beyond what is expected from symmetry alone.
• When the energy scales of superconductivity and the TRS-breaking order are comparable, the back-action effect becomes strong and leads to a significant increase in diode efficiency η.
• Numerical solutions show that the maximum of the superconducting order parameter Δ(q) is pinned at q=0 even when C3z symmetry is broken, due to the non-local coupling with the current.
• The model reproduces the observed large diode efficiency in twisted trilayer graphene, with η approaching values close to 1, explaining experimental results from Lin et al. (2022).
• The theory predicts that the current asymmetry can be tuned by adjusting the relative strength of the supercurrent-order parameter coupling, enabling design of high-efficiency SDE devices.
• The mechanism is robust and general, applicable to various systems with spontaneous TRS breaking, not limited to graphene-based materials.

Figure 2 : Phenomenological theory with $C_{3z}$ -symmetric coupling, $\alpha^{\prime}\neq 0$ , $\alpha=0$ , $\alpha^{\prime\prime}=1.0$ in Eq. ( 5 ). $\Phi_{v}(\boldsymbol{q})$ and $\Gamma(\boldsymbol{q})$ in (a) and (b) are shown for $\alpha^{\prime}=0.8$ and (c) displays the angular ( $\hat{n}$ )