Second-harmonic generation of blue series excitons and magnetoexcitons in Cu 2 O

Nonlinear optical studies of the yellow and green exciton series in ${\mathrm{Cu}}_{2}\mathrm{O}$ have been reported for more than 40 years. Because of the band structure (the two highest even-parity valence and lowest conduction bands), the $S$ excitons of the two lowest exciton series are dipole-forbidden for one-photon absorption and thus dipole-allowed for two-photon absorption. There is an odd-parity higher conduction band that leads with the two even-parity valence bands to the blue and violet exciton series. We report on second-harmonic generation (SHG) of the blue exciton series. The odd-parity $S$-exciton SHG is due to a dipole-quadrupole excitation and a dipole emission process. Because of their high oscillator strength density, polariton effects have to be taken into account, since resonances might be shifted to higher energies by up to $10\phantom{\rule{4pt}{0ex}}\mathrm{me}\mathrm{V}$ compared to the transverse exciton energies. The polariton dispersion for the blue excitons up to $n=4$ is calculated and compared to the experimental results. In magnetic fields up to $10\phantom{\rule{4pt}{0ex}}\mathrm{T}$ applied in a Voigt configuration ($\mathbf{B}\ensuremath{\perp}\mathbf{k}$), SHG of $S$ excitons by a dipole-dipole excitation is observed, which is due to the admixture of dipole-dipole excited $P$ excitons by the effective electric field from the magneto-Stark effect (MSE). From the analysis of the diamagnetic shift and the MSE interaction of the three-level system of $1S, 2S$, and $2P$ excitons, we derive experimental results for the ratio $\ensuremath{\langle}{r}_{n,l}^{2}\ensuremath{\rangle}/{\ensuremath{\mu}}_{X}$ between the average of radius squared for the three states and the reduced exciton mass. For higher principal quantum number states, we observe magnetoexcitons up to $n=8$. We analyze their magnetic field dependence and derive the electron effective-mass values for the crystalline orientations [111], $[1\overline{1}0]$, and [001].