Polarization dependence of the third-harmonic generation in multiband superconductors

In a superconductor the third-harmonic generation (THG) of a strong THz pulse is enhanced below $T_c$ by the resonant excitation of lattice-modulated charge fluctuations, which modulate the response according to the polarization of the field. Here we compute the THG within a multiband model for the prototype NbN superconductor. We show that the nonresonant contribution coming from the instantaneous electronic response and the finite width of the pulse significantly suppress the polarization dependence of the signal, challenging its observation in real systems.

Figure 1

Frequency dependence of the various contributions (19, 20) and (23)–(24) to the nonlinear current. The calculations have been done with the band structure of Eq. (4)–(6), with the same parameter values used in Ref. [6], i.e., $t=-0.72$ eV, $t^{\prime }=-0.15$ eV, $t^{\prime \prime }=0.12$ eV, $\mu =-0.6$ eV. The coupling $U=0.27$ eV is chosen to match the experimental $T=0$ value ${\mathrm{\Delta }}_0=0.65$ THz. Inset: expanded view of the Higgs contributions only.

Figure 2

(a) Relative angular variation of the THG intensity for the model (3). LCF contribution (solid black curve) and Higgs contribution [normalized to $I^H({\mathrm{\Delta }}_0,\theta =0)]$ alone (red curve). The dotted black line is the result of Ref. [6]. (b) Angular variation of the LCF contribution for an incident electric field simulating the experimental situation of Ref. [6], as shown in the inset.

Figure 3

Diagrammatic representation of the instantaneous term (9c). Wavy lines denote the gauge field and solid line denote the Nambu Green's function. The number on the vertex denotes the insertion of the corresponding Pauli matrix.

Figure 4

(a) Diagrammatic representation of the resonant terms of $S^{\left(4\right)}$, including vertex corrections in the Higgs channel. Here wavy lines denote the gauge field, solid lines the Nambu Green's function, and the dashed line the pairing interaction. The shaded circle represents the variation of the order parameter due to the external perturbation, as defined in Eq. (A5). (b) Vertex equation for $\delta \mathrm{\Delta }$. Its solution leads to the RPA resummation of the ${\chi }_{\mathrm{\Delta }\mathrm{\Delta }}$ bubble, which defines the Higgs propagator $X_{\mathrm{\Delta }\mathrm{\Delta }}^a\equiv 1/U+{\chi }_{\mathrm{\Delta }\mathrm{\Delta }}^a$. The labels 1, 3 refer to the vertex insertions of the Pauli matrices ${\widehat{\sigma }}_1$ and ${\widehat{\sigma }}_3$, respectively. Here we omitted for simplicity the spatial indexes of the various fermionic susceptibilities.

Figure 5

(a)–(c) Additional diagrams contributing to $S^{\left(4\right)}$. They vanish identically at $\mathbf{q}=0$ as $T\to 0$ in the BCS limit.