[Paper Review] Bogoliubov angle and visualization of particle-hole mixture in superconductors

Kazuhiro Fujita, Ilya Grigorenko+7 more|arXiv (Cornell University)|Sep 4, 2007

Physics of Superconductivity and Magnetism4 citations

TL;DR

This paper introduces the Bogoliubov angle (BA) as a novel local observable in superconductors, defined as the angle quantifying the particle-hole admixture in Bogoliubov quasiparticles. By measuring the ratio of tunneling currents at positive and negative biases in scanning tunneling microscopy (STM), the BA can be mapped spatially and energetically, enabling direct visualization of particle-hole duality and superconducting correlations, even in the pseudogap state.

ABSTRACT

Superconducting excitations --Bogoliubov quasiparticles -- are the quantum mechanical mixture of negatively charged electron (-e) and positively charged hole (+e). Depending on the applied voltage bias in STM one can sample the particle and hole content of such a superconducting excitation. Recent Scanning Tunneling Microscope (STM) experiments offer a unique insight into the inner workings of the superconducting state of superconductors. We propose a new observable quantity for STM studies that is the manifestation of the particle-hole dualism of the quasiparticles. We call it a {\em Bogoliubov angle}. This angle measures the relative weight of particle and hole amplitude in the superconducting (Bogoliubov) quasiparticle. We argue that this quantity can be measured locally by comparing the ratio of tunneling currents at positive and negative biases. This Bogoliubov angle allows one to measure directly the energy and position dependent particle-hole admixture and therefore visualize robustness of superconducting state locally. It may also allow one to measure the particle-hole admixture of excitations in normal state above critical temperature and thus may be used to measure superconducting correlations in pseudogap state.

Motivation & Objective

• To introduce a new local observable—Bogoliubov angle (BA)—that quantifies the particle-hole admixture in superconducting quasiparticles.
• To enable direct experimental visualization of the particle-hole duality in Bogoliubov quasiparticles using scanning tunneling microscopy (STM).
• To develop a method for measuring the energy- and position-dependent particle-hole content in superconducting states, including above the critical temperature and in the pseudogap regime.
• To establish a link between the BA and the effective spin model via the Anderson mapping, providing a theoretical foundation for the observable.
• To provide a practical, iterative numerical framework for computing the BA in inhomogeneous superconducting systems.

Proposed method

• Define the Bogoliubov angle as $\Theta_{n}({\bf r}_{i}) = \arctan\left(\left(\frac{|u_{n}({\bf r}_{i})|^{2}}{|v_{n}({\bf r}_{i})|^{2}}\right)^{1/2}\right)$, where $u_n$ and $v_n$ are the particle and hole amplitudes of the quasiparticle wavefunction.
• Propose that the ratio of tunneling currents at positive and negative biases in STM directly measures the particle-hole weight ratio, enabling local extraction of the BA.
• Use the eigenstates of the Bogoliubov-de Gennes equations to compute $u_n({\bf r}_i)$ and $v_n({\bf r}_i)$, which are then used to calculate the BA at each spatial site and energy level.
• Implement an iterative numerical scheme to solve the self-consistent Bogoliubov-de Gennes equations, including a mixing parameter $\alpha$ to stabilize convergence.
• Apply a convergence criterion based on relative deviation $S^n = \max |\Delta^{n} - \Delta^{n-1}| / \max |\Delta^{n}|$, adjusting $\alpha$ dynamically to ensure convergence within $\epsilon = 10^{-3}$.
• Map the BA across real space and energy to visualize the spatial variation of particle-hole admixture, particularly in inhomogeneous superconducting states.

Experimental results

Research questions

• RQ1 Can the particle-hole admixture in Bogoliubov quasiparticles be measured locally in superconductors using STM?
• RQ2 How does the Bogoliubov angle vary spatially and energetically in superconducting states, especially near impurities or in inhomogeneous regions?
• RQ3 Can the Bogoliubov angle be used to probe superconducting correlations in the pseudogap state, where long-range order is absent?
• RQ4 What is the relationship between the Bogoliubov angle and the effective spin model via the Anderson mapping?
• RQ5 How can the BA be extracted experimentally from STM tunneling spectra using bias-dependent current ratios?

Key findings

• The Bogoliubov angle provides a direct, local measure of the relative particle and hole components in superconducting quasiparticles, with $\Theta = 0$ indicating a pure hole-like state and $\Theta = \pi/2$ a pure electron-like state.
• The maximum particle-hole admixture occurs at $\Theta = \pi/4 = 45^\circ$, corresponding to equal weights of particle and hole components.
• The ratio of tunneling currents at positive and negative biases allows direct experimental determination of the particle-hole weight ratio, enabling local mapping of the BA.
• The BA can be measured in the normal state above $T_c$ and in the pseudogap state, offering a probe of preformed pairs and residual superconducting correlations.
• Numerical solutions of the Bogoliubov-de Gennes equations converge within 20–40 iterations using a dynamic mixing scheme with $\alpha$ adjusted based on convergence criteria.
• The BA is directly connected to the Anderson effective spin model, where $\sin\Theta_{{\bf k}} = 2u_{{\bf k}}v_{{\bf k}}$ and $\cos\Theta_{{\bf k}} = u^2_{{\bf k}} - v^2_{{\bf k}}$, confirming its theoretical consistency.