Tuning the electronic band structure of metal surfaces for enhancing high-order harmonic generation

High harmonic generation (HHG) from condensed matter phase holds promise to promote future cutting-edge research in the emerging field of attosecond nanoscopy. The key for the progress of the field relies on the capability of the existing schemes to enhance the harmonic yield and to push the photon energy cutoff to the extreme ultraviolet (XUV, 10-100 eV) regime and beyond towards the spectral "water window" region (282-533 eV). Here, we demonstrate a coherent control scheme of HHG, which we show to give rise to quantum modulations in the XUV region. The control scheme is based on exploring surface states in transition-metal surfaces, and specifically by tuning the electronic structure of the metal surface itself together with the use of optimal chirped pulses. Moreover, we show that the use of such pulses having moderate intensities permits to push the harmonic cutoff further to the spectral water window region, and that the extension is found to be robust against the change of the intrinsic properties of the material. The scenario is numerically implemented using a minimal model by solving the time-dependent Schrodinger equation for the metal surface Cu(111) initially prepared in the surface state. Our findings elucidate the importance of metal surfaces for generating coherent isolated attosecond XUV and soft-x-ray pulses and for designing compact solid-state HHG devices.