[Paper Review] Phases and phase transitions in spin-triplet ferromagnetic superconductors

D. V. Shopova, Dimo I. Uzunov|arXiv (Cornell University)|Apr 12, 2004

Physics of Superconductivity and Magnetism5 references3 citations

TL;DR

This paper investigates the thermodynamic stability and phase transitions in spin-triplet ferromagnetic superconductors using a quasi-phenomenological Ginzburg-Landau theory. It demonstrates that superconductivity in materials like UGe₂, ZrZn₂, and URhGe is triggered by spontaneous ferromagnetic order (M-triggering), leading to a stable coexistence phase where superconductivity and ferromagnetism coexist, with phase transitions from normal to coexistence being first-order and from ferromagnetic to coexistence potentially first- or second-order depending on material parameters.

ABSTRACT

Recent results for the coexistence of ferromagnetism and unconventional superconductivity with spin-triplet Cooper pairing are reviewed on the basis of the quasi-phenomenological Ginzburg-Landau theory. New results are reported. The results are discussed in view of applications to metallic compounds as UGe2, URhGe, ZrZn2.

Motivation & Objective

• To understand the thermodynamic behavior of spin-triplet superconductors where ferromagnetism and unconventional superconductivity coexist.
• To clarify the role of spontaneous magnetization (M) in triggering superconductivity in the absence of an external magnetic field.
• To analyze the stability and order of phase transitions in the coexistence phase of ferromagnetic superconductors.
• To assess the influence of Cooper pair and crystal anisotropies on phase diagram structure and thermodynamic properties.
• To provide a theoretical framework applicable to real materials like UGe₂, ZrZn₂, and URhGe with high ferromagnetic transition temperatures.

Proposed method

• A quasi-phenomenological Ginzburg-Landau free energy functional is used, incorporating coupling terms between the ferromagnetic order parameter M and superconducting order parameters ψ₁, ψ₂.
• The model includes symmetry-conserving terms such as Mψ₁ψ₂ (γ-term) and M²ψ₁ψ₂ (γ₁-term), crucial for stabilizing the coexistence phase.
• Phase existence and stability conditions are derived from the free energy functional, leading to algebraic equations (Eqs. 54–58) for the order parameters as functions of M and temperature.
• The analysis considers both uniform and non-uniform phases, with a focus on the stability of the mixed FM–FS phase in the temperature range T_s < T < T_f.
• The effects of anisotropies (Cooper pair and crystal) are incorporated to refine the phase diagram and assess their impact on ground state degeneracy and thermodynamics.
• Theoretical results are compared with experimental observations in UGe₂, ZrZn₂, and URhGe, particularly regarding the hierarchy T_f ≫ T_s and the absence of a standard Meissner transition.

Experimental results

Research questions

• RQ1 How does spontaneous ferromagnetic order (M) trigger superconductivity in spin-triplet superconductors without an external magnetic field?
• RQ2 What determines the stability of the coexistence phase between ferromagnetism and superconductivity in these materials?
• RQ3 What is the nature of the phase transition from the normal state to the coexistence phase?
• RQ4 How do Cooper pair and crystal anisotropies affect the phase diagram and thermodynamic properties?
• RQ5 Why is the superconducting transition temperature T_s significantly lower than the ferromagnetic transition temperature T_f in materials like UGe₂?

Key findings

• The coexistence phase of ferromagnetism and spin-triplet superconductivity is thermodynamically stable, while other superconducting phases are either unstable or metastable under typical conditions.
• The phase transition from the normal state to the coexistence phase is of the first order, as confirmed by the analysis of the free energy and order parameter profiles.
• The transition from the ferromagnetic phase to the coexistence phase can be either first- or second-order, depending on the material-specific parameters such as γ₁ and γ.
• The inclusion of the γ₁-term in the free energy is essential for stabilizing the ferromagnetic order down to absolute zero, as observed in real materials like UGe₂.
• Cooper pair and crystal anisotropies refine the phase diagram and reduce ground state degeneracy but do not drastically alter the overall stability domains or thermodynamic behavior.
• The model explains the experimental observation that superconductivity in UGe₂, ZrZn₂, and URhGe coexists with ferromagnetism across a wide temperature range below T_s ≈ 1 K, with T_f ≫ T_s, and is driven by the M-triggering mechanism.