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We have investigated both inter-band and intra-band second order nonlinear optical conductivity based on the velocity correlation formalism and the spectral expansion technique. We propose a scenario in which the second order intra-band process is nonzero while the inter-band process is zero. This occurs for a band structure with momentum asymmetry in the Brillouin zone. Very low-energy photons are blocked by the Pauli exclusion principle from participating in the inter-band process; however, they are permitted to participate in the intra-band process, with the band smeared by some impurity scattering. We establish a connection between the inter-band nonlinear optical conductivity in the velocity gauge and the shift vector in the length gauge for a two-band model. Using a quasiclassical kinetic approach, we demonstrate the importance of intra-band transitions in high harmonic generations for the single tilted Dirac cone model and hexagonal warping model. We confirm that the Kramers-Kronig relations break down for the limit case of ($\omega$, $-\omega$) in the nonlinear optical conductivity. Finally, we calculate the superconducting transition temperature of NbN and the dielectric function of AlN, and the resistance of the NbN/AlN junction. The natural non-linearity of the Josephson junction brings a Josephson plasma with frequency in the Terahertz region., Comment: 11 pages, 10 figures

We investigate the nonlocal optical conductivity of Fermi surface nesting materials which support charge density waves or spin density waves. The nonlocal optical conductivity contains information of correlations in electron fluids which could not be accessed by standard optical probes. Half metal emerges from doping a charge density wave and similarly spin-valley half metal emerges from doping a spin density wave. Based on the parabolic band approximation, we find the Drude peak is shifted to higher frequency and splits into two peaks in the nonlocal optical conductivity. We attribute this to the two Fermi velocities in the half-metal or spin-valley half metal states., Comment: 16 pages, 12 figures, to appear on Science China Physics, Mechanics & Astronomy