Semiclassical theory for the orbital magnetic moment of superconducting quasiparticles

Abstract

We study the orbital magnetic moment of Bogoliubov quasiparticles in superconductors with the semiclassical approach. We 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 we find that nontrivial structure in the superconducting pairing gap alone is unable to produce quasiparticle orbital magnetic moment, which is in sharp contrast to the behavior of quasiparticle Berry curvatures. We apply the formula to study a tight-binding model with chiral $d$-wave superconducting gap, and show the influence of orbital magnetic moment on the energy spectrum and local density of states. We also calculate the orbital Nernst effect driven by the interplay between the orbital magnetic moment and the Berry curvature of Bogoliubov quasiparticles.

Figures (3)

• Figure 1: (a) Momentum-space distribution of the Berry curvature $\Omega^{z}(\mathbf{k})$ (in units of $a^{2}$) for the lowest positive-energy band $n=2$. The gray dashed lines denote the boundary of the first Brillouin zone. The high-symmetry points are marked with black dots. The inset is the zoom-in view of the red-dashed region. The parameters are taken as $\mu/t=-0.1$ and $\Delta/t$=0.09. (b) Electronic band structure along the high-symmetry lines. The dashed line indicates the chemical potential with $\mu/t=-0.1$. (c) Momentum-space distribution of the orbital magnetic moment $m^{z}(\mathbf{k})$ (in units of $a^{2}et/\hbar$) for the lowest positive-energy band $n=2$, with the parameters taken the same as in panel (a). (d) Quasiparticle band structure along the high-symmetry lines with parameters taken the same as in panel (a). (e) and (f) The momentum-space distribution of the orbital magnetic moment and the quasiparticle band structure along the high-symmetry lines with a different parameter of $\Delta/t$=0.225. The chemical potential is taken the same as in panel (a).
• Figure 2: (a) The ratio between energy correction $\Delta E_{n\mathbf{k}}$ due to a uniform magnetic field and pairing amplitude $\Delta$ for the positive-energy quasiparticle bands. (b) Field-induced density of states correction $\delta n(E)$ (orange line, left axis) and local density of states $n_{0}(E)$ in the absence of a magnetic field (blue line, right axis). The peaks and dips in $\delta n(E)$ are marked by pink dashed lines and blue dashed lines, respectively. (c) Quasiparticle band structure. The color scale represents the weight of the electron component $|u|^2$ with respect to the local density of states. (d) Zoom-in view of the red-dashed region in panel (c), showing the detailed band structure near the $K$ point and $F$ point. The horizontal dashed lines mark the characteristic energies corresponding to the peaks and dips in the density of states correction $\delta n(E)$ shown in panel (b), using the same color coding. The parameters used are $B_{z}=-(\hbar/a^2e)/500$, $\Delta/t=0.09$ and $\mu/t=-0.1$.
• Figure 3: (a) Temperature dependence of the orbital Nernst coefficient $\eta^{z}$ (in units of $a^{2}etk_B/\hbar^2$) at fixed $\mu/t=-0.1$. (b) Chemical potential dependence of the orbital Nernst coefficient at fixed $k_{B}T/\Delta=0.5$. (c) Quasiparticle band structure along the high-symmetry lines with $\mu/t=-0.015$. This chemical potential corresponds to the position marked by the pink dashed line in panel (b). The inset shows a detailed view near the $K$ point. (d) Quasiparticle band structure with $\mu/t=-0.5$. This chemical potential corresponds to the position marked by the blue dashed line in panel (b). The other parameter used is $\Delta/t=0.09$.