Anisotropic effective exciton mass in Cu ₂ O

Abstract

The degeneracy of the yellow 1S orthoexcitons in Cu2O is lifted by wave vector dependent electron-hole exchange. This interaction scales quadratically with k and strongly depends on the direction of the wave vector. Therefore it is interpreted as an anisotropic correction of the effective exciton mass. This can explain the discrepancies in the measurements of the exciton mass in Cu2O reported so far.Spins are coupled via exchange interaction. For exciton physics the exchange between the coupled electron-hole pairs is of increased interest. Despite being studied intensively the understanding of electronhole exchange even in bulk materials is far from being complete. The electron-hole exchange interaction depends on the full band structure and hence on the wave vector k of the exciton center-of-mass motion. However, normally it is approximated as a spin-spin interaction, which is independent of the wave vector. This represents a severe simplification of the exchange interaction.Recently the wave-vector-dependence of the electron-hole exchange has been derived and experimentally verified for the yellow 1s orthoexciton in Cu 2 O.[1] In this article we will summarize the essential results and discuss the impact of wave-vector-dependent exchange for the interpretation of the effective exciton mass.In Cu 2 O excitonic transitions with holes in the Γ + 7 band and electrons in the Γ + 6 band give the socalled yellow exciton series. One finds four 1S excitons that are split by k-independent exchange into the threefold Γ + 5 states termed orthoexcitons and a single Γ + 2 level referred to as paraexciton. In the following we will solely concentrate on the orthoexcitons. In lowest order, the orthoexcitons couple to light via quadrupole interaction. Exchange interaction of order k 0 leaves the orthoexcitons degenerate. When calculating exchange of higher order in k one finds that the k-linear terms vanish. The calculation of the terms of order k 2 is described in great detail in Ref. [1,2]. Summarizing one finds the long range exchange term

Optical properties of Rydberg excitons and polaritons

“…[23], [24]). As was shown by Dasbach et al [25] (see also [26]), the total exciton mass in the [0, 0, 1] direction equals 3 m 0 , whereas the value in the [1, 1, 0] direction is equal 0.66 (for example, [7]) or 0.5587 ( [12]), see Table I. The effective mass anisotropy plays an important role in the description of the optical properties of excitons.…”

Impact of the valence band structure ofCu2Oon excitonic spectra

“…In this paper will also show that a finite momentum K of the center of mass leads to terms in the Hamiltonian, which were initially assigned to the exchange interaction [23,24]. These terms are of the correct order of magnitude to describe the K-dependent experimentally observed line splitting of the 1S exciton [23,24].…”

“…Solving the eigenvalue problem (25), we obtain the coefficients c of the basis functions according to Eq. (24). We can now compare the coefficients of those basis functions with a fixed value of F t ≤ 4 to the coefficients of the functions |A, Γ i given in Ref.…”

“…For this reason the momentum of the center of mass K is a constant of motion [43]. About ten years ago the K-dependent line splitting of the 1S exciton state was observed and explained in terms of a K-dependent short-range exchange interaction of the form [23,24,42] …”

“…These terms are of the correct order of magnitude to describe the K-dependent experimentally observed line splitting of the 1S exciton [23,24].…”