Superconductors’ Pairing Gap Cannot Produce Magnetic Moments

Scientists have long sought to understand the fundamental properties of quasiparticles within superconducting materials, and a new study from Jian-hua Zeng, Zhongbo Yan, and Zhi Wang at the School of Physics, Sun Yat-sen University, in collaboration with Qian Niu from the Department of Physics, University of Science and Technology, Anhui, offers significant insight into the orbital magnetic moment of these elusive particles. Their research details a semiclassical theory describing this magnetic moment, derived by analysing energy corrections to quasiparticle wavepackets in a magnetic field, and verified using a full mechanical calculation. This work is particularly noteworthy because it demonstrates that the structure of the superconducting pairing gap alone cannot generate a quasiparticle orbital magnetic moment, a finding that sharply contrasts with the behaviour of quasiparticle Berry curvatures, and opens avenues for exploring novel phenomena like the orbital Nernst effect driven by the interplay of magnetic moment and Berry curvature.

This surprising finding challenges conventional understanding of particle behaviour within these materials and opens avenues for manipulating their properties. Understanding this subtle difference could unlock advanced applications in areas like energy transport and quantum computing.

Scientists derive the orbital magnetic moment of a quasiparticle wavepacket by considering the energy correction of the wavepacket to the linear order of the magnetic field. The semiclassical result is further verified by a linear response calculation with a full quantum mechanical method. From the analytical expression, they find that nontrivial structure in the superconducting pairing gap alone is unable to produce quasiparticle orbital magnetic moment. This is in sharp contrast to the behaviour of quasiparticle Berry curvatures.

Quasiparticle magnetic moments emerge from energy corrections and charge non-conservation

Calculations reveal an orbital magnetic moment for Bogoliubov quasiparticles, quantified as −e 2mRe ⟨Dkφn,k| × pk|φn,k⟩ |k=kc — where Dk is the gauge invariant k-space derivative and pk represents the k-dependent momentum operator. Here, this expression defines the magnetic moment of a quasiparticle wavepacket, dependent on the momentum-space centre kc, and detailed a chiral p-wave superconducting pairing alone cannot generate a nonzero orbital magnetic moment. A result contrasting sharply with the behaviour of quasiparticle Berry curvatures.

Instead, the orbital magnetic moment arises from the interaction between the wavepacket’s energy correction and the applied magnetic field. The project establishes a distinction between orbital magnetic moment and orbital angular momentum in superconducting quasiparticles, stemming from the charge non-conservation inherent in the mean-field BdG Hamiltonian.

By employing a semiclassical approach alongside a linear response calculation, The project verifies the derived formula for the orbital magnetic moment. For a tight-binding model exhibiting chiral d-wave pairing. Numerical studies show how the orbital magnetic moment modulates the energy spectrum and the local density of states — consideration of a honeycomb lattice with chiral d-wave pairing allows for mapping the momentum-space distribution of the orbital magnetic moment. In turn, this then influences spectroscopic and transport responses within the superconductor.

At the core of the semiclassical theory lies a local approximation of the BdG Hamiltonian, mirroring techniques used for Bloch electrons — once the quasiparticle wavepacket is constructed using BdG Hamiltonian eigenstates. Meanwhile, the energy is evaluated to first order in the perturbation gradients, yielding a correction term ΔEn,kc, and the orbital magnetic moment is defined as the coefficient of this energy correction with respect to a homogeneous magnetic field. By applying a circular gauge for the vector potential, A(rc) = (B × rc)/2, the gradient expansion of the local Hamiltonian is obtained, with the first-order energy correction expressed as ΔEn,kc= −B · mn(kc). Where mn(kc) represents the orbital magnetic moment.

Orbital magnetic moments and their contribution to electronic and magnetotransport properties

Scientists investigate the impact of the orbital magnetic moment on the energy spectrum and local density of states, and calculate the orbital Nernst effect driven by the interaction between the orbital magnetic moment and the Berry curvature of Bogoliubov quasiparticles. At the same time, the orbital magnetic moment plays an important role in the magnetic and optical properties of Bloch electrons.

Modern understanding of the orbital magnetic moment for Bloch electrons has been formulated by methods such as the Wannier function approach, the linear response approach, and the semiclassical strategy. In the semiclassical picture, the orbital magnetic moment of Bloch electrons originates from the self-rotation of the wavepackets and is a important part of the orbital magnetization of the electron system, which might be even larger than spin-induced magnetization.

Here, the orbital magnetic moment is also directly linked to various magnetic phenomena such as magnetoresistivity, Zeeman splitting, gyrotropic magnetic effect, and magnetic susceptibility. Also, it plays a key role in transport phenomena, such as the nonlinear thermoelectric Hall effect. In turn, the valley Hall effect, and the orbital Hall effect. In superconducting systems, Bogoliubov quasiparticles behave similarly to Bloch electrons with a band structure determined by the Bogoliubov-de Gennes (BdG) Hamiltonian.

Since the BdG Hamiltonian is constructed by both the electron Hamiltonian and the superconducting gap function, the structure of the quasiparticle band is shaped by the interaction of the electron Bloch functions and the superconducting gap symmetry. Meanwhile, this interaction induces rich topological phases which support exotic boundary modes such as the chiral Majorana mode, and exhibits nontrivial geometric properties such as Berry curvatures, which can induce an anomalous Hall response for quasiparticles even in the topological trivial state.

Theoretical interest has already been drawn to this topic, with the orbital angular momentum of superconducting quasiparticles derived using a Wannier function approach, and the resulting orbital magnetization decomposed into local and itinerant contributions. At the same time, the orbital magnetic moment has also been discussed from the perspective of conserved charge current in superconductors.

A linear response approach has investigated the orbital magnetization and calculated various effects such as the orbital Edelstein effect — unlike the Bloch electrons, different methods for studying the orbital magnetic moment of superconducting quasiparticles often lead to different results. The main difficulty lies in the charge non-conservation of the mean-field BdG Hamiltonian, and this brings a discrepancy between the charge distribution and the probability distribution of a quasiparticle wavepacket. Scientists verify the semiclassical result with a linear response approach. Calculating the linear response to a gauge field and taking the long-wavelength limit to obtain the orbital magnetic moment that couples with the static magnetic field.

Quasiparticle orbital magnetism and its potential for thermoelectric devices

Once established as a purely theoretical curiosity, The effort of quasiparticles within superconductors is now yielding insights with potential for device applications. Recent work detailing the orbital magnetic moment of these quasiparticles moves beyond simply cataloguing their properties and begins to explore how these intrinsic characteristics might be manipulated.

For years, understanding the behaviour of these particles, born from the complex interaction of electrons in superconducting materials. Proved difficult because their fleeting existence and subtle interactions demanded new theoretical approaches. Calculating this orbital moment isn’t merely an academic exercise. Beyond fundamental materials science, the ability to control quasiparticle behaviour opens doors to novel electronic devices.

Specifically, the calculated ‘orbital Nernst effect’, where a temperature gradient induces an electrical current, suggests possibilities for new types of thermoelectrics, potentially converting waste heat into usable energy. By realising this potential requires overcoming significant hurdles, as the theoretical models rely on idealised systems and the impact of material imperfections or complex interfaces remains largely unexplored.

This the structure of the superconducting gap itself isn’t enough to generate the orbital magnetic moment, unlike earlier work focusing on quasiparticle Berry curvature. Instead, it highlights the importance of external factors and the interaction between different quasiparticle properties. Experimental verification remains a challenge, demanding techniques sensitive enough to probe these subtle magnetic signatures.

Future investigations could explore how these orbital moments interact with other quasiparticle phenomena, potentially leading to even more complex and useful effects. The field is poised to move beyond prediction towards realisation, with initial efforts focused on identifying materials exhibiting these properties. The next phase will involve engineering systems to enhance and control them, bridging the gap between theoretical calculations and actual device fabrication. Demanding close collaboration between physicists and materials scientists.