Deviations of the Exciton Level Spectrum in Cuprous Oxide from The

Deviations of the exciton level spectrum in Cu2O from the hydrogen series

F. Schöne,S.-O. Krüger,P. Grünwald, H. Stolz, and S. Scheel Institut fürPhysik, UniversitätRostock, Albert-Einstein-Strasse 23, D-18059 Rostock, Germany

M. Aßmann, J. Heckötter,J. Thewes, D. Fröhlich, and M. Bayer Experimentelle Physik 2, Technische UniversitätDortmund, 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 ε 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 ε(k, ω), 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 Schrödinger equation for hydrogen, and whose bound state solutions are the excitons [16] following the simple Rydberg formula. Additional interactions such as spin-orbit and interband 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 denotes 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, respectively [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 dimensionless parameters {A1···3,B1···3} in Hk [Eq. (1)] to the band structure. The inset shows the uppermost valence bands and + [A3(IxIy + IyIx) + B3(Ixσy + Iyσx)] {kxky} + c.p. , the corresponding fits in the vicinity of the Γ-point. (1) where c.p. stands for cycling permutations and {kxky} = (kxky + kykx)/2. The band interaction Hamiltonian III. EXCITON BINDING ENERGIES Hk contains a total of six dimensionless parameters {A1···3,B1···3}, 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 Schrödinger-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. Tiny fine structure splittings as drogenic systems [19], and the energy eigenvalues En,l 3 follow a hydrogen-like Rydberg series. Inclusion of the Hamiltonian for the cubic crystal symmetry to an ab ini- term ∆Th prevents one from obtaining an analytic solu- tio calculation of the band structure, thus avoiding any tion. experimental fit parameters at this point. Furthermore, In this case, we make use of a method proposed the solution of the coupled differential equations in the in Ref. [20] by which the original eigenvalue problem k · p-approach requires a set of basis functions that are HΦ(k) = En,lΦ(k) is converted into an auxiliary Stur- arbitrary at this point. On the other hand, the structure mian eigenvalue problem of the Fredholm integral equation yields a natural basis set gn,l. Another advantage of our method is that we 2 2 ~ k describe one source of deviations at one time. Combined + ∆Th − En,l Φ(k) = −λ(En,l)Ve−h(k) ? Φ(k) 2µ with our purely theoretical data we are thus able to gauge (3) the influence of the nonparabolicity on the deviation of with Sturmian eigenvalue λ(En,l) that itself depends exciton binding energies from the Rydberg series. parametrically on the sought energy eigenvalue En,l. The energy eigenvalues of the system now emerge from the λ- Phenomenologically, the energy shift of the excitonic spectrum if the condition λ(En,l) = 1 is met. The advan- Rydberg series can also be traced back to deviations of tage of this method is that Eq. (3) can be transformed, the interaction potential between electron and hole from after separation of radial and angular variables, into a the strict Coulomb potential at short distances (central- Fredholm integral equation of the second kind, whose cell corrections). For excitons in Cu2O this was done for real and symmetric integral kernel can be spectrally de- the first time in Ref. [24]. There, the atomic concepts of composed into eigenfunctions gn,l(k). quantum defects was applied, but only to match exper- In the absence of any nonparabolicity (∆Th = 0), the imental data of the yellow S-excitons, with the assump- eigenfunctions gn,l(k) are orthonormal, and the condition that the quantum defect for the P -excitons is negligi- tion λ(En,l) = 1 yields the standard hydrogenic Rydberg bly small. Besides the nonparabolicity of the bands, the formula [20]. The inclusion of ∆Th results in a mixing binding energy of the excitons can be affected by other of eigenfunctions gn,l(k) with different principal quantum central-cell corrections such as the coupling of electrons numbers n. Therefore, an additional diagonalisation step and holes to LO-phonons, the frequency-dependent di- has to be added. The numerical procedure thus involves: electric function ε(k, ω) and exchange interactions [12]. However, as we are interested in excitons with large prin- 1. solving the integral equation (3) for all principal cipal quantum number n, their low binding energy pre- quantum numbers n with n ≤ nmax for a given an- vents the coupling to LO-phonons. Possible coupling gular quantum number l, with En,l as parameters, to phonons above the band gap is suppressed due the limitation of phonons by the low temperature. In addi- 2. diagonalising the matrix of (non-orthogonal) eigen- tion, a large exciton Bohr radius allows one to assume functions g (k), n,l that the dielectric function can be treated as a constant 3. and extracting the energy eigenvalues that fulfil the ε = 7.5 [25], hence we expect the contribution of the other condition λ(En,l) = 1. central-cell corrections to be negligibly small. Overall, one may expect the nonparabolicity of the valence to be Typically, principal quantum numbers up to nmax = 60 the major source of corrections to the ideal Rydberg se- provided good convergence of the numerical procedure to ries. obtain the exciton spectrum up to n = 25. As the deviations from the ideal Rydberg series are small compared to the binding energies and tend to zero IV. DEVIATION FROM THE RYDBERG for large n, it is convenient to display these corrections SERIES relative to ideal variations of binding energies as

The energy eigenvalues En,l that are obtained from such a diagonalisation do no longer follow the Rydberg ∗ En − En series of a hydrogen atom. In semiconductor physics, the αn,l = , (4) E∗ − E∗ common approach at this point is to employ k · p-theory n+1 n to account for the nonspherical symmetry [21–23]. In- corporating the Luttinger parameters of Cu2O, it allows one to determine the binding energies as demonstrated where En is the actually measured binding energy (differ- ∗ in Ref. [18]. However, a few remarks are in order in com- ence to band gap), while the Eν represent ideal Rydberg parison to our approach. Besides the band gap and the energies Ry/ν2. The results are shown in Fig. 2 (solid Rydberg-energy, applying k · p-theory to cuprous oxide lines) as function of principal quantum number n for the requires the Luttinger parameters and in some cases the lowest angular momentum states with l = 0 ... 3. As ex- dielectric function, which are both determined by fitting pected from the modified Hamiltonian (2), the theoret- the model Hamiltonian to the experimental band struc- ical corrections saturate for large n, and decrease with ture. In contrast, our approach fits the most general increasing l. 4

1 α splittings are observed which ultimately lead to the Stark n,S 0.9 α ladder. From extrapolation towards U = 0, the energies n,P α of these excitons at zero ﬁeld can be accurately assessed. 0.8 n,D α n,F From their order in energy we can attribute them to S- 0.7 and P -excitons, as labeled. To enhance the accuracy of the energy evaluation, we 0.6 have also taken the second derivatives of the absorp-

n,l 0.5 tion spectra, shown in Fig. 4. By doing so, the enor- α mous intensity diﬀerences in absolute transmission are 0.4 leveled out, and the diﬀerent absorption lines become 0.3 much better resolvable, so that high-accuracy determi- nation of the S- and D-exciton energies becomes pos- 0.2 sible: For the D-excitons again the center-of-gravity is 0.1 determined for each principal quantum number. In do-

0 ing so, we are able to extend the reported energies to 0 5 10 15 20 25 30 signiﬁcantly higher principal quantum numbers, for the n S-excitons up to n = 12 compared to 7 in previous re- ports, and up to n = 8 for the D-excitons compared to FIG. 2: Deviations αn,l from the Rydberg series of the yellow 5 previously. The assignment of the states via the sec- excitons in Cu2O as function of principal quantum number n ond derivative is depicted in the transmission spectrum for angular momenta l = 0 ... 3. Solid lines: energy eigen- Fig. 3 by the dotted lines. Note that for higher n these values from Eq. (3). Marker: experimental values (S – black excitons become also optically activated in electric ﬁelds. dots, P – red crosses, D – blue dots, F – green crosses) from However, it is no longer possible to assign them in the absorption spectroscopy. multiplicity of emerging lines, so that also their energies can no longer be determined with the required accuracy for assessing the deviation αn,l. V. TRANSMISSION SPECTROSCOPY

) s t

i 4 n

To determine the energies of exciton states in Cu2O u

. 2 experimentally, we performed high resolution absorption b r

a n = 4 n = 5 ( 0 spectroscopy on a high-quality crystal. The energies of F T S P the P -excitons can be accessed directly by one-photon P S 4 absorption in electric-dipole approximation, as demon- ) V ( strated in [11]. In a strict sense angular momentum is d

l 8 not a good quantum number in crystals, but can be used e i F approximately in high-symmetry cubic crystals with O

h c

i 1 2 r or Td symmetry. The symmetry reduction leads to an t c

e D admixture of P -excitons to the F -excitons so that they l 1 6 F become observable in absorption [4]. As a consequence, E D G the energies of these states can be determined with high 2 0 accuracy [18]. In case they show a fine structure split- 2 . 1 6 5 2 . 1 6 6 2 . 1 6 7 2 . 1 6 8 2 . 1 6 9 E n e r g y ( e V ) ting, we have taken the center of gravity energy of the corresponding multiplet, which is in line with the theoretical model that averages over different k-space directions. FIG. 3: Absorption spectra of the n = 4 and n = 5 excitons in While the energies of excitons with odd-parity envelopes external electric fields. The curve on top shows a cut through can be determined accurately in that way, this is not the contour plot at U = 8 V. possible for states with even parity. Therefore, we applied additionally an electric field along the optical axis, which leads to a mixing of the S-exciton with the P -exciton with zero magnetic quan- VI. RESULTS AND DISCUSSION tum number. Similarly, D-excitons become mixed with P - and F -excitons of the same magnetic quantum num- The experimentally extracted values for the deviations ber. As a consequence, the related features appear in the αn,l of the exciton binding energies from the ideal Ryd- absorption spectra, see Fig. 3, which shows the absolute berg series are shown in Fig. 2, together with the theoret- absorption of the n = 4 and n = 5 excitons as func- ical results for the energy eigenvalues that are obtained tion of the electric field. With increasing field strength from solving Eq. (3). In addition, the S-excitons are af- new features emerge and intensify, as expected from the fected by electron-hole exchange interaction with a com- increased state mixing with the P -excitons. Also line mon scaling of n−3. The experimental binding energies 5

E n e r g y ( e V ) energies from their predicted values, in particular for low 2 . 1 6 8 2 . 1 6 9 2 . 1 7 0 2 . 1 7 1 0 n. This can be attributed to the fact that in this energy range the background permittivity ε(k, ω) is far from con- 2 stant [26], and that the binding energy, especially for 4 n = 2, is close to a phonon resonance. Taking this into

) 6 account will shift the data point closer to the theoreti- V (

d 8 cally expected value. This points to a possible extension l e i of our theory in which a frequency-dependent permittiv- F

1 0 c

i ity could be incorporated in a self-consistent manner by r

t 1 2 c including it both in the Coulomb interaction potential e l

E 1 4 and the Rydberg energy. The observed high-n shift on

1 6 the other hand may be due to the fact that the measure- ment setup had to be altered for the resonances beyond 1 8 n = 10. Thus, a small systematic shift might have oc- 2 0 curred, which features prominently in the deviation parameter αn,S. FIG. 4: Contour plot of the second derivative of an absorp- Although we have focussed solely on the yellow exci- tion spectrum extended over a larger energy range covering ton series of Cu2O, it is clear that our procedure is valid the excitons from n = 5 towards the band gap. The multi- for all semiconductors that possess both spin-orbit in- ple splitting of the states while transforming into the Stark teractions as well as cubic symmetry within their band ladder becomes obvious. structure. In particular, we expect the green exciton series of Cu2O to provide purely negative values for αn,l because the interband coupling results in an increased of the S-excitons, whose deviation αn,S is depicted in + Fig. 2, have thus been shifted downwards by an amount curvature of the Γ8 valence band. of 8.04 meV/n3 (following previous measurements of the Acknowledgments exchange [4]) with respect to the values extracted from the absorption spectra. We thank Ch. Uihlein for his suggestion to analyse the The agreement, in particular of the P -exciton reso- data by use of the 2nd derivative. We gratefully acknowl- nances for large n, is astonishingly good. This implies edge support by the Collaborative Research Centre SFB that nonparabolicity of the bands (in case of Cu2O only 652/3 ’Strong correlations in the radiation ﬁeld’ and the the valence bands) due to interband coupling is the dom- International Collaborative Research Centre TRR 160 inant contribution to the deviation of the exciton reso- ’Coherent manipulation of interacting spin excitations in nances from the hydrogenic Rydberg series. Nonetheless, tailored semiconductors’, both funded by the Deutsche one still observes a systematic deviation of the S-exciton Forschungsgemeinschaft.

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