Nonlinear electromagnetic response and Higgs-mode excitation in BCS superconductors with impurities

We reveal that due to the presence of disorder oscillations of the order parameter amplitude called the Higgs mode can be effectively excited by the external electromagnetic radiation in usual BCS superconductors. This mechanism works for superconductors with both isotropic s-wave and anisotropic, such as d-wave, pairings. The nonlinear response in the presence of impurities is captured by the quasiclassical formalism. We demonstrate that analytical solutions of the Eilenberger equation with impurity collision integral and external field drive coincide with the exact summation of ladder impurity diagrams. Using the developed formalism we show that resonant third-harmonic signal observed in recent experiments is naturally explained by the excitation of Higgs mode mediated by impurity scattering.

Figure 1

Two possible processes contributing to the order parameter excitation at the double frequency of external field described by the vector potential ${\mathit{A}}_{\mathrm{\Omega }}$. (a) The second-order diagram with two current vertices $•$ corresponding to the light-matter coupling linear by the external field ${\mathit{A}}_{\mathrm{\Omega }}$. The impurity ladder insertion is shown by the shaded gray area. (b) The first-order diagram with momentum-independent density vertex. The order parameter vertex ${\mathcal{D}}_0$ is defined below in Eq. (6).

Figure 2

Diagrams with current vertices $•=e\mathit{v}/c$ contributing to nonlinear response. (a) Direct third-order coupling to the external field ${\mathit{A}}_{\mathrm{\Omega }}$. The dashed lines show impurity scattering correction in Born approximation averaged by Gaussian disorder. Shaded regions show impurity ladders. (b) Coupling to the external field through the excitation of the order parameter oscillation ${\mathrm{\Delta }}_{2\mathrm{\Omega }}$ shown by the dash-dotted line and the corresponding vertex is $⊳={\widehat{\tau }}_1$. The filled triangle corresponds to the order parameter vertex modified by the polarization bubble insertions which yield the Higgs mode excitation.

Figure 3

Order parameter vertex $\mathcal{D}$ (filled triangle) corrected by the polarization operator insertions. $⊳={\widehat{\tau }}_1, ⊲={\widehat{\mathcal{D}}}_0$, dashed region corresponds to the impurity ladder insertion.

Figure 4

Diagram contributing to the THG current response through the density modulation due to the direct coupling to the vector potential ${\mathit{A}}_{\mathrm{\Omega }}$ denoted by wavy lines. Density vertices $◯={\widehat{\tau }}_3(e^2/2m{c}^2)$ are the same as in Fig. 1.

Figure 5

Diagrammatic representation of the first-order corrections due to (a) interaction with external field ${\mathit{A}}_{\mathrm{\Omega }}$ through the current vertex and (b) interaction with the order parameter perturbation ${\mathrm{\Delta }}_{2\mathrm{\Omega }}$ through the order parameter vertex dressed by the impurity ladder shown by the dashed region. (c) Diagrammatic equation for the correction (b).

Figure 6

(a) The second-order correction $G_{AA}$ due to the direct coupling to vector potential ${\mathit{A}}_{\mathrm{\Omega }}$ through current vertices, ${\omega }_k=\omega +(3-k)\mathrm{\Omega }$. (b) Diagrammatic equation to determine the second-order correction for the propagator with current vertices modified by impurity scattering in ladder approximation. (c) The third-order correction $G_{AAA}$. (d) Correction that is generated by the combined order parameter and vector potential perturbations.

Figure 7

(a),(b),(d),(e) Absolute values of the THG current amplitudes $|I_{AAA}|, |I_H|$, and $|I_{AAA}^{\left(d\right)}|$ as functions of $\mathrm{\Omega }$ for different values of the impurity scattering rates ${\left({\tau }_{\mathrm{imp}}{T}_c\right)}^{-1}$ varying between the clean and dirty regimes. (c),(f) Dependencies of $|I_{AAA}|$ and $|I_H|$ on the external frequency and scattering rate. The Dynes parameter $\mathrm{\Gamma }=0.01$ and $T=0.1T_c$ for all plots.

Figure 8

Absolute values of the THG current components $j_{AAA}, j_H$ as functions of $T,\mathrm{\Omega }$, the Dynes parameter $\mathrm{\Gamma }=0.1$. Left panels (a),(c),(e): the contribution of $j_{AAA}$ current. Right panels (b),(d),(f): the contribution of $j_H$ current. In (c) and (d) the curves correspond to different values of temperature. In (e) and (f) the curves correspond to the different frequencies. The currents are normalized by the amplitude $j_{0D}$, see Eqs. (76) and (77).