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Deviations of the Exciton Level Spectrum in Cuprous Oxide from The Deviations of the exciton level spectrum in Cu2O from the hydrogen series F. Sch¨one,S.-O. Kr¨uger,P. Gr¨unwald, H. Stolz, and S. Scheel Institut f¨urPhysik, Universit¨atRostock, Albert-Einstein-Strasse 23, D-18059 Rostock, Germany M. Aßmann, J. Heck¨otter,J. Thewes, D. Fr¨ohlich, and M. Bayer Experimentelle Physik 2, Technische Universit¨atDortmund, D-44221 Dortmund, Germany Recent high-resolution absorption spectroscopy on excited excitons in cuprous oxide [Nature 514, 343 (2014)] has revealed significant deviations of their spectrum from that of the ideal hydrogen- like series. Here we show that the complex band dispersion of the crystal, determining the kinetic energies of electrons and holes, strongly affects the exciton binding energies. Specifically, we show that the nonparabolicity of the band dispersion is the main cause of the deviation from the hydrogen series. Experimental data collected from high-resolution absorption spectroscopy in electric fields validate the assignment of the deviation to the nonparabolicity of the band dispersion. PACS numbers: 71.35.Cc, 78.20.-e, 32.80.Ee, 33.80.Rv I. INTRODUCTION the band dispersion. Fluorescence spectra of atomic systems are an ex- II. VALENCE BAND DISPERSIONS IN tremely rich source of information about the interactions CUPROUS OXIDE of electrons and the nuclei, and their study laid the foun- dations of the development of quantum mechanics. The Cuprous oxide (Cu O) was historically the first ma- basic hydrogen dependence of the electron binding ener- 2 2 terial in which excitons were observed [7{9]. Their dis- gies of 1=n already turns out to be insufficient for al- covery, combined with the availability of exceptionally kali, i.e. hydrogen-like, atoms where the polarisability of pure natural crystals [10], sparked considerable interest the ionic core modifies the Coulomb potential felt by the in light-matter interactions in condensed-matter systems. valence electron. In semiconductor physics, the exciton Cuprous oxide is endowed with a relatively large Ryd- concept translates the bound states of an electron-hole berg energy of around 86 meV. The crystal environment pair onto a hydrogen-like series, where the crystal envi- is taken into account by the permittivity " that, for crys- ronment is included in the effective masses and the dielec- tals with cubic (Oh) symmetry such as Cu2O, is typi- tric function of the material [1{3]. However, the periodic cally isotropic. This means that the exciton spectrum arrangement of the crystal constituents breaks the rota- should simply be a scaled hydrogen spectrum. However, tional symmetry of atoms, and leads to semiconductor- recent high-precision measurements of the P -exciton en- specific effects such as detailed level splittings [4]. These ergies of the yellow series of Cu2O showed a systematic splittings demonstrate deviations from the Rydberg for- deviation of the observed spectrum from the hydrogen mula. analogy expectation [11], in addition to the splitting of Here we calculate the correct binding energies of Cu2O the excitons with different angular momenta for low n [4]. from the band structure and cast them into the form of Several effects such as the influence of a frequency- and single-parameter corrections to the Rydberg series. In wavevector-dependent permittivity "(k;!), the coupling contrast to atoms, the origin of the excitonic correction is to LO-phonons as well as exchange interactions have all not a modification of the Coulomb potential, but the de- been proposed as causes for that deviation [12]. In two- viation from the parabolic band dispersion and hence the dimensional materials such as transition metal dichal- kinetic energy of electrons and holes. Based on the group- gogenides (TMDC), the nonlocal nature of the Coulomb theoretical band Hamiltonian of Suzuki and Hensel (SH) screening causes similar effects [13{15]. However, as we [5] adapted to the exact band structure of cuprous oxide will show here, the deviation of the exciton binding en- derived using density functional theory [6],we compute ergies of the yellow series in bulk cuprous oxide from the the exciton binding energies and extract from them the hydrogenic Rydberg series is a result of the nonparabolic relevant corrections to the Rydberg series. Our theoret- dispersion of the highest valence band (with symmetry + ical results are compared to experimental data collected Γ7 at the Brillouin zone center). by high-resolution absorption spectroscopy. In combina- The description of excitons in a semiconductor requires tion with electric fields, we are able to extract the binding the detailed knowledge of the electronic band structure. arXiv:1511.05458v2 [cond-mat.mes-hall] 22 Feb 2016 energies of excitons of S-, P -, D- and F -type, extending The common approach in solid-state physics is to ap- previous studies to far higher principal quantum num- proximate the extrema of valence and conduction bands bers. The excellent agreement between theory and ex- via parabolic shapes. In the effective mass approxima- periment corroborates our assignment of the deviations tion these parabolas are then interpreted as the kinetic of the exciton level spectrum to the nonparabolicity of energy terms of electron and hole, respectively. A simple 2 two-band model yields the Wannier equation, which is reported recently are neglected [18]. analogous to the Schr¨odinger equation for hydrogen, and whose bound state solutions are the excitons [16] follow- ing the simple Rydberg formula. Additional interactions such as spin-orbit and inter- band interactions lead to further splitting and deforma- tion of the relevant bands under consideration. In Cu2O, + the highest valence band with symmetry Γ5 splits under 1 the spin-orbit interaction Hso = 3 ∆ (I · σ), where I de- notes the angular-momentum matrices for I = 1 and σ + the Pauli spin matrices, into one (doubly degenerate) Γ7 - + and two (doubly degenerate) Γ8 -bands, associated with total angular momenta J = 1=2 and J = 3=2, respec- tively [5]. The value of the spin-orbit splitting in Cu2O is ∆ = 131 meV. From symmetry arguments [17] it fol- lows that band interactions can be taken into account in a 6 × 6-matrix Hamiltonian that includes powers of the momentum k up to second order [5], 2 FIG. 1: Relevant band dispersions along two particular di- ~ 2 Hk = [A1 + B1(I · σ)] k rections in the Brillouin zone. Black lines: band dispersions 2me from spin-DFT [6]. Blue dashed line: parabolic model for the + 2 1 2 1 2 lowest conduction (Γ6 ) band with an effective electron mass + A2 Ix − I + B2 Ixσx − I · σ kx + c:p: ∗ 3 3 me = 0:98 me. Red dashed lines: best fits of the dimension- less parameters fA1···3;B1···3g in Hk [Eq. (1)] to the band structure. The inset shows the uppermost valence bands and + [A3(IxIy + IyIx) + B3(Ixσy + Iyσx)] fkxkyg + c:p: ; the corresponding fits in the vicinity of the Γ-point. (1) where c:p: stands for cycling permutations and fkxkyg = (kxky + kykx)=2. The band interaction Hamiltonian III. EXCITON BINDING ENERGIES Hk contains a total of six dimensionless parameters fA1···3;B1···3g, which can be obtained by matching the In the following we restrict our analysis to the excitons resulting energy dispersions to the exact band structure + + of the yellow series (Γ6 ⊗Γ7 ) while an investigation of the derived from spin-DFT calculations [6]. Note that DFT + + excitons of the green series (Γ6 ⊗Γ8 ) would follow a sim- does not give the correct gap energy, which can be ob- (7) tained experimentally. ilar path. We first split the valence band dispersion ESH The fit was performed for the highest valence band into a parabolic part Th near k = 0 and a nonparabolic + contribution ∆Th(kh). Adding the kinetic energy Te of (Γ5 ) which splits due to the spin-orbit interaction into + + the conduction band as well as the Coulomb interaction an energetically higher Γ7 and the two lower-energy Γ8 bands. This yields three individual energy dispersions Ve−h to the valence band dispersion results in a Wan- (i) nier equation that, due to the nonparabolic valence band E (k)(i = 7; 8hh; 8lh) for each direction in the Bril- SH dispersion, no longer resembles a simple Schr¨odinger-type louin zone. A least-squares fit to the Γ+ band that 7 equation. After transformation to the single-particle (ex- is of interest to us yields the parameters A = −1:76, 1 citon) picture and neglecting the center-of-mass momen- A = 4:519, A = −2:201, B = 0:02, B = −0:022, 2 3 1 2 tum (i.e. setting K = 0), the Hamiltonian takes the and B = −0:202. The lowest conduction band (Γ+) can 3 6 general form easily be approximated by a parabola in the vicinity of the Γ-point and it has an isotropic curvature in all direc- 2 2 + ~ k tions. The bound states formed between the Γ6 and the H = + ∆Th + Ve−h(k)? (2) + + 2µ Γ7 ,Γ8 bands are the excitons of the yellow and green series, respectively. Figure 1 shows the relevant bands with the reduced exciton mass µ defined as usual. Here, and the fits to the Hamiltonian HSH = Hso + Hk. In the symbol ? denotes the convolution operator. the following, we neglect the anisotropy of the band dis- The exciton binding energies En;l are then obtained (i) persions and use a weighted average ESH for simplicity, from the eigenvalue equation HΦ(k) = En;lΦ(k) in mo- with the weights being the respective geometric multi- mentum space which, due to the convolution with the plicities. After this procedure, there is no free parameter Coulomb potential, is in fact an integral equation. With- left. Furthermore, in doing so, the exciton angular mo- out the nonparabolic contribution ∆Th, the solution for mentum l = 0; 1; 2;::: becomes a good quantum number the momentum-space wavefunction Φ(k) is that for hy- for labelling the states.
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