Coherent control of photocurrents in graphene and carbon nanotubes

Coherent one-photon $\left(2\mathrm{}\omega \right)$ and two-photon $\left(\omega \right)$ electronic excitations are studied for graphene sheets and for carbon nanotubes using a long-wavelength theory for the low-energy electronic states. For graphene sheets we find that a coherent superposition of these excitations produces a polar asymmetry in the momentum space distribution of the excited carriers with an angular dependence that depends on the relative polarization and phases of the incident fields. For semiconducting nanotubes we find a similar effect which depends on the square of the semiconducting gap, and we calculate its frequency dependence. We find that the third-order nonlinearity, which controls the direction of the photocurrent is robust for semiconducting tubes and vanishes in the continuum theory for conducting tubes. We calculate corrections to these results arising from higher-order crystal-field effects on the band structure and briefly discuss some applications of the theory.