[Paper Review] Anisotropy and superconductivity

Boris Bondarev|arXiv (Cornell University)|Feb 12, 2013

Superconducting Materials and Applications3 citations

TL;DR

This paper proposes that superconductivity arises from anisotropic electron wave-vector distributions near the Fermi surface, driven by electron repulsion in time-reversed states. Using a mean-field approach, it derives a critical current density threshold and shows that anisotropy—specifically, unequal occupation of wave vectors **k** and **−k**—is the microscopic origin of superconducting behavior, with key results linking anisotropy to the formation of Cooper pairs via an effective attractive interaction.

ABSTRACT

The mean field method is applied for analysis of valence electrons in metals. It is shown that at low temperatures electrons have two wave-vector distribution patterns. Isotropic distribution refers to the first pattern. Anisotropic distribution refers to another pattern, particularly to specific wave-vector values occurred nearby the Fermi sphere. It is shown that it is the anisotropy that makes the metal obtain its specific superconductor features.

Motivation & Objective

• To explain the microscopic origin of superconductivity through anisotropic electron wave-vector distributions in momentum space.
• To demonstrate that electron repulsion in **k** and **−k** states leads to anisotropic distribution patterns, which are essential for superconducting behavior.
• To derive a critical current density threshold that destroys the superconducting state, based on electron velocity and density.
• To establish a connection between anisotropy and the emergence of Cooper pairing via an effective attractive interaction in the Hamiltonian.
• To show that the mean-field model with anisotropic distribution explains superconducting current density and critical current limits without relying on ab initio calculations.

Proposed method

• Applies the mean-field method to analyze valence electrons in metals, focusing on wave-vector distribution patterns.
• Distinguishes between isotropic (f = f(a)) and anisotropic (f(a) ≠ f(−a)) electron distributions in momentum space.
• Uses a modified Hamiltonian incorporating electron repulsion (U) and effective attraction (J) between electrons at **k** and **−k** states.
• Derives the superconducting critical current density using the formula $ j_{ ext{кр}} = e n u $, where $ u = rac{1}{ar{v}} rac{ar{ ho}}{n} $, and $ u $ is the critical velocity.
• Applies the Pythagorean theorem to the Fermi sphere to derive $ k_o = rac{ar{v}}{ar{v}} rac{ar{ ho}}{n} $, linking anisotropy to superconducting energy gap.
• Uses the energy width $ ar{ ho} $ of the Fermi surface layer to define the superconducting force displacement $ k_o = rac{ar{v}}{ar{v}} rac{ar{ ho}}{n} $, leading to $ k_o = rac{ar{v}}{ar{v}} rac{ar{ ho}}{n} $.

Experimental results

Research questions

• RQ1 What causes the anisotropic distribution of electron wave vectors in metals at low temperatures?
• RQ2 How does electron repulsion in time-reversed states (**k** and **−k**) lead to superconducting behavior?
• RQ3 What is the critical current density that destroys the superconducting state, and how is it derived from electron velocity and density?
• RQ4 How does the anisotropic wave-vector distribution relate to the formation of Cooper pairs in the superconducting state?
• RQ5 What is the role of the effective attractive interaction J in the Hamiltonian, and how does it emerge from electron repulsion?

Key findings

• Anisotropic wave-vector distribution, particularly at specific values near the Fermi sphere, is the fundamental origin of superconducting properties.
• The critical current density that destroys superconductivity is given by $ j_{ ext{кр}} = e n u $, where $ u = rac{1}{ar{v}} rac{ar{ ho}}{n} $, and $ u $ is the critical velocity.
• The superconducting force displacement is derived as $ k_o = rac{ar{v}}{ar{v}} rac{ar{ ho}}{n} $, linking anisotropy to the energy width $ ar{ ho} $ of the Fermi surface layer.
• The ratio of maximum superconducting current velocity $ v_{ ext{max}} $ to the critical velocity $ u $ is $ rac{v_{ ext{max}}}{u} = rac{3}{8} rac{ar{ ho}}{ar{ ho}} $, showing $ v_{ ext{max}} o u $, indicating instability under external fields.
• The mean electron energy $ ar{ ho} $ depends on kinetic energy $ ho $ and anisotropy correction $ I w_1( ho) $, with $ ar{ ho} = ho + I w_1( ho) $, showing temperature-dependent anisotropy effects.
• The model Hamiltonian $ ho_{ extbf{k} extbf{k}'} = I ho_{ extbf{k}+ extbf{k}'} - J ho_{ extbf{k}- extbf{k}'} $ shows that effective attraction (J) emerges from electron repulsion, enabling Cooper pair formation.