Dipole-allowed generation of the yellow-series excitons inCu2Odue to an applied electric field

A. R. H. F. Ettema, J. Versluis
2003 Physical Review B (Condensed Matter)  
The electron bands of Cu 2 O near the band gap have been calculated for the undistorted cubic crystal with O h symmetry and a tetragonally distorted case with C 4v symmetry. The tetragonal structure has a distortion of the linear Cu crystal field that represents the structure of a polarized crystal in an electric field. The symmetry of the bands changes in such a way that the dipole forbidden transition of the yellow series excitons in the undistorted cubic structure becomes dipole allowed in
more » ... e tetragonal structure. The energy of the excitons becomes lower in the polarized crystal. These changes in the band gap properties make it possible to create an exciton trap in thin Cu 2 O films with a scanning tunneling microscopy tip while the excitons can be resonantly created with a laser via a dipole allowed transition. The excitons in Cu 2 O have gained much attention during the last decade because of the prosperity of exciton Bose-Einstein condensation ͑BEC͒ in Cu 2 O. The small mass m ex ϭ2.7m 0 , the small bohr radius of 7 Å, the large binding energy of 150 meV, and the long lifetime of 10 s make the exciton system in Cu 2 O a promising candidate to achieve BEC. 1, 2 Besides the excitons in Cu 2 O, the exciton system in Al-GaAs multiple quantum well structures has progressed fast during the last year with respect to BEC. 3 With the observation of high-density exciton lakes in in-plane potential traps, whereby the photoluminescence experiments show that the quasi-two-dimensional excitons form a statistically degenerated Bose gas. 4 A macroscopically ordered state is observed in the luminescence ring that has a fragmentation pattern of circular structures that forms a periodic array over lengths up to 1 mm. 5, 6 Despite the prosperous characteristics of the exciton system in Cu 2 O and AlGaAs quantum wells, BEC has not been realized due to the difficulty to create sufficient excitons in order to reach the critical density needed for BEC. 7,8 The long radiative lifetime in Cu 2 O is related inversely proportional to the oscillator strength of the corresponding transition. One of the main problems in achieving the critical density in Cu 2 O is given by the symmetry of the conductionband minimum ͑CBM͒ and the valence-band maximum ͑VBM͒. The symmetry of the electron bands does not allow a dipole transition from the VBM to the CBM. 9 Therefore different kinds of techniques have been used to circumvent this problem. In various studies the exciton properties were examined as a function of applied stress. 10-12 By applying stress to the semiconductor host lattice the band gap is reduced and excitons diffuse into the region of lowest energy. Although the para-and orthoexcitons show a strong mixing and the exciton energy levels show some dispersion upon stress, the bands remain in a dipole forbidden symmetry. The yellow series excitons in Cu 2 O can be formed resonantly via a quadrupole transition, a two-photon process, or a one-photon absorption accompanied with the absorption or emission of an optical phonon. [13] [14] [15] It is obvious that the transition probability is strongly reduced by the simultaneous interaction of two particles with a valence-band electron in order to create an exciton. Once approaching high exciton densities in Cu 2 O, the kinetics of this exciton system become complicated due to the Auger decay process at high densities. 16 -18 This decay process occurs with a constant rate but is strongly density dependent. In order to overcome the decay rate of the excitons, an efficient and fast exciton generation process seems to be even more imperative to reach sufficient exciton densities for BEC. In this paper we present results from band structure calculations and a group theory symmetry analysis which show that the symmetry problem to create the excitons via a dipole transition can be lifted with the Stark effect caused by an applied electric field. The electric field polarizes the crystal and distorts the crystal field of the Cu atoms resulting in a removal of the inversion symmetry. The distortion in this study is applied along the ͑100͒ axis of the crystal but the distortion can be applied in any other direction including ͑110͒ and the ͑111͒ direction. By making use of group theory compatibility tables the appropriate representations can be found for similar distortions in other directions. The energy, symmetry, and degeneracy of the bands are, due to the distortion, changed in such a way that the transition between the VBM and the CBM becomes dipole allowed. Moreover, the band gap decreases slightly by the applied field which makes it possible to create the yellow exciton series resonantly via a dipole transition and collect the excitons in a trap if thin Cu 2 O films are subjected to the inhomogeneous electric field of a scanning tunneling microscopy ͑STM͒ tip. The calculations in this study were performed using the full potential linearized augmented plane-wave ͑FLAPW͒ method of the WIEN2K code. 19 This code solves the Kohn-Sham equations inside the atomic spheres and augments the numerical radial wave functions with plane waves outside the spheres. The basis set of orbitals is split in core states and valence states at an energy of 6 Ry. For the atoms in Cu 2 O the O 1s, Cu 1s, 2s, and 2p form the core states that are treated inside the sphere only with a spherical potential. The radii of the muffin-tin spheres were taken 1.7 Å for both atoms. The electron-electron interactions are calculated within the generalized gradient approximation ͑GGA͒. Without external electric field the compound Cu 2 O has a cubic structure with the Cu atoms forming an fcc lattice, the PHYSICAL REVIEW B 68, 235101 ͑2003͒
doi:10.1103/physrevb.68.235101 fatcat:fj35znlz7bbxxkuaoefsy6qt3e