PHYSICAL REVIEW B 102, 115203 (2020) Two-photon absorption and second harmonic generation of 1S para- and orthoexcitons in Cu 2 O coupled by a magnetic field A. Farenbruch , 1 D. Fröhlich, 1 D. R. Yakovlev , 1, 2 and M. Bayer 1, 2 1 Experimentelle Physik 2, Technische Universität Dortmund, D-44221 Dortmund, Germany 2 Ioffe Institute, Russian Academy of Sciences, 194021 St. Petersburg, Russia (Received 30 May 2020; revised 4 August 2020; accepted 20 August 2020; published 10 September 2020) We report on two-photon absorption (TPA) and second harmonic generation (SHG) spectroscopy of para- and orthoexcitons in Cu 2 O subject to a strong magnetic field up to 10 T. The magnetic field splits the orthoexciton into its three quasispin components M = 0, ±1 and activates the symmetry and spin forbidden paraexciton by an admixture from the M = 0 component of the orthoexciton. For the excitation of the paraexciton we suggest an alternative mechanism of TPA without an external perturbation. It involves instead of the electric dipole-electric dipole the electric quadrupole-magnetic dipole excitation process. By application of group theory we derive for both mechanisms of TPA and SHG polarization selection rules for each of the four resonances (one para- and three orthoexcitons) and present experimental results for different crystalline orientations which agree perfectly with the derivations from group theory. High spectral resolution of the used SHG technique allows us to refine the exchange splitting between the para- and orthoexciton of ε = 12.120 meV and the g values of the upmost valence band g v =−0.72 ± 0.03 and the lowest conduction band g c = 2.38 ± 0.08. DOI: 10.1103/PhysRevB.102.115203 I. INTRODUCTION The 1S paraexciton of the yellow exciton series of Cu 2 O has gained a lot of interest mainly as a candidate for the ob- servation of Bose-Einstein condensation (BEC). For a recent review on these attempts we refer to Ref. [1]. The paraexciton is the lowest energy exciton in Cu 2 O. From the upmost va- lence band ( + 7 symmetry) and the lowest conduction band ( + 6 symmetry) one derives the threefold orthoexciton ( + 5 symmetry) and the single paraexciton ( + 2 symmetry). The paraexciton is a pure spin triplet state 12.1 meV [2] below the orthoexciton. The small exciton radius of the 1S exciton of 0.7nm [3] leads to the rather large exchange splitting of 12.1 meV. In linear optical spectroscopy the paraexciton can only be excited if an external perturbation like stress [4] or magnetic field [2,5] is applied, which leads to an admixture of the orthoexciton to the pure triplet paraexciton and thus to a nonvanishing oscillator strength. In a high quality natural crystal a very narrow paraexciton resonance of 80 neV was measured [2]. In Ref. [6] details of two-photon magnetoabsorption of para- and orthoexcitons are derived. The low sensitivity of the two-photon absorption (TPA) experiments, however, did not allow us to detect paraexcitons. Resonant two-photon absorption of the 1S paraexciton in a potential trap is reported in Refs. [7–9]. Here the paraexciton gets allowed by strain- induced admixture of the 1S green orthoexciton [4]. TPA for odd parity excitons without external perturbation can be achieved by replacing one of the electric dipole op- erators in the TPA process by a magnetic dipole operator as was shown in alkali halides in Ref. [10] and for GaAs in Ref. [11]. In Cu 2 O, however, we are dealing with even parity excitons. For the direct TPA excitation of the paraexciton ( + 2 symmetry) the other odd parity electric dipole operator ( − 4 symmetry) has to be replaced by the even parity elec- tric quadrupole operator ( + 5 symmetry). There are thus two mechanisms to excite the paraexciton in Cu 2 O: (i) by electric dipole-electric dipole (DD) excitation of the orthoexciton ad- mixture either by a magnetic field or strain or (ii) by direct two-photon electric quadrupole-magnetic dipole (QMD) ex- citation. The latter excitation is expected to be rather weak, since two higher order processes are needed to excite the even parity paraexciton. For both mechanisms (i) and (ii) we will derive the de- tailed polarization dependencies by merely group theoretical techniques. In Ref. [12] we did not take into account the degeneracy of the excited states (e.g., threefold excitons of  + 5 symmetry) for the derivation of polarization selection rules, which was sufficient for the higher excited exciton states (n  3) to distinguish between the different processes considered (magneto-Stark and Zeeman effect). For the 1S excitons, however, one has to take into account the threefold degeneracy of the orthoexciton and the single paraexciton, which are coupled by a magnetic field and split into three separate orthoexciton components (M =±1, M = 0) and the 12.1 meV lower paraexciton. As shown in detail in Ref. [12] we will only use the tables of Koster et al. [13] to derive for any crystalline orientation the polarization dependence for the different excitation channels of TPA and second harmonic generation (SHG) of paraexcitons. The paper is organized as follows: In Sec. II details of the experiments are explained. In Sec. III we derive the mag- netic field dependence of the 1S para- and orthoexcitons and present experimental results. From the spectra we extract the 2469-9950/2020/102(11)/115203(12) 115203-1 ©2020 American Physical Society