The even exciton series in Cu 2 O

Transcription

1 The even excitn series in Cu O Frank Schweiner, Jörg Main, and Günter Wunner Institut für Theretische Physik, Universität Stuttgart, Stuttgart, Germany Christph Uihlein Experimentelle Physik, Technische Universität Drtmund, 44 Drtmund, Germany (Dated: February 7, 07 Recent investigatins f excitnic absrptin spectra in cuprus xide (Cu O have shwn that it is indispensable t accunt fr the cmplex valence band structure in the thery f excitns. In Cu O parity is a gd quantum number and thus the excitn spectrum falls int tw parts: The diple-active excitn states f negative parity and dd angular mmentum, which can be bserved in ne-phtn absrptin (Γ 4 symmetry and the excitn states f psitive parity and even angular mmentum, which can be bserved in tw-phtn absrptin (Γ + 5 symmetry. The unexpected bservatin f D excitns in tw-phtn absrptin has given first evidence that the dispersin prperties f the Γ + 5 rbital valence band is giving rise t a cupling f the yellw and green excitn series. Hwever, a first theretical treatment by Ch. Uihlein et al. [Phys. Rev. B 3, 73 (98] was based n a simplified spherical mdel. The bservatin f F excitns in ne-phtn absrptin is a further prf f a cupling between yellw and green excitn states. Detailed investigatins n the fine structure splitting f the F excitn by F. Schweiner et al. [Phys. Rev. B 93, 9503 (06] have prved the imprtance f a mre realistic theretical treatment including terms with cubic symmetry. In this paper we shw that the even and dd parity excitn system can be cnsistently described within the same theretical apprach. Hwever, the Hamiltnian f the even parity system needs, in cmparisn t the dd excitn case, mdificatins t accunt fr the very small radius f the yellw and green S excitn. In the presented treatment we take special care f the central-cell crrectins, which cmprise a reduced screening f the Culmb ptential at distances cmparable t the plarn radius, the exchange interactin being respnsible fr the excitn splitting int rth and para states, and the inclusin f terms in the furth pwer f p in the kinetic energy being cnsistent with O h symmetry. Since the yellw S excitn state is cupled t all ther states f psitive parity, we shw hw the central-cell crrectins affect the whle even excitn series. The clse resnance f the S green excitn with states f the yellw excitn series has a strng impact n the energies and scillatr strengths f all implied states. The cnsistency between thery and experiment with respect t energies and scillatr strengths fr the even and dd excitn system in Cu O is a cnvincing prf fr the validity f the applied thery. PACS numbers: y, 7.0.Nr, 7.70.Gm, k I. INTRODUCTION Excitns are the quanta f fundamental ptical excitatins in bth insulatrs and semicnductrs in the visible and ultravilet spectrum f light. The Culmb interactin between electrn and hle leads t a hydrgen-like series f excitnic states []. Cuprus xide (Cu O is a prime example where ne can even identify fur different excitnic series (yellw, green, blue, and vilet being related t the tw tpmst valence bands and the tw lwest cnductin bands []. Recently, the yellw series culd be fllwed up t a spectacular high principal quantum number f n = 5 []. This utstanding experiment has launched the new field f research f giant Rydberg excitns and led t a variety f new theretical and experimental investigatins n the tpic f excitns in Cu O [3? 6]. Cu O has ctahedral symmetry O h s that the symmetry f the bands can be assigned by the irreducible representatins Γ ± i f O h. The yellw and green excitn series share the same threefld degenerate Γ + 5 rbital valence band state. This state splits due t spin-rbit interactin int an upper twfld degenerate Γ + 7 valence band (yellw series and a lwer furfld degenerate Γ + 8 valence band (green series. The band structure f bth bands is essentially determined by the anistrpic dispersin prperties f the rbital state. The threefld degeneracy f the rbital state is lifted as sn as a nn-zer k vectr gets invlved, with new eigenvectrs depending n the rientatin f k. A cnsequence f the splitting f the rbital state is a partial quenching f the spin-rbit interactin. This k dependent quenching is nt nly respnsible fr a remarkable nn-parablicity f the tw tp valence bands but leads likewise t a k dependent mixing f the Γ + 7 and Γ+ 8 Blch states and can thus cause a mixing f the yellw and green excitn series. A mixing f bth series is favred by the large Rydberg energy f apprximately 00 mev, a crrespnding large excitn extensin in k space and the small spin-rbit splitting f nly 30 mev. A Hamiltnian that is able t cpe with a cupled system f yellw and green excitns must take explicit care f the dispersin prperties f the rbital valence band state and has t include the spin-rbit interactin. Such a kind f Hamiltnian was first intrduced by Uih-

2 lein et al. [7] fr explaining the unexpected fine structure splitting bserved in the tw-phtn absrptin spectrum f Cu O. They used a simplified spherical dispersin mdel fr the Γ + 5 rbital valence band with an identical splitting int lngitudinal and transverse states independent f the rientatin f k. This simplificatin had the appealing advantage that the ttal angular mmentum remains a gd quantum number s that the excitn prblem culd be reduced t calculate the eigenvalues f a system f cupled radial wave functins. A prblem in their paper is the incrrect ntatin f the S green and S yellw excitns states. Bth ntatins need t be exchanged t be cnsistent with their calculatins. Althugh the spherical mdel can explain many details f the experimental findings, ne has t be aware f its limitatins. A mre realistic Hamiltnian being cmpliant with the real band structure by including terms f cubic symmetry has already prved its validity by explaining the puzzling fine structure f the dd parity states in Cu O [0]. The intentin f this paper is t shw that the same kind f Hamiltnian can likewise describe the fine splitting f the even parity excitns. Hwever, when cmparing the even parity and dd parity excitn systems, it is bvius that the even excitn system is a much mre challenging prblem. One reasn fr this is the clse resnance f the green S excitn with the even parity states f the yellw series with principal quantum number n. This requires a very careful calculatin f the binding energy f the green S excitn. Furthermre, the binding energy f the yellw S excitn is much larger than expected frm a simple hydrgen like series, inter alia, due t a less effective screening f the Culmb ptential at distances cmparable t the plarn radius. Mrever, a breakdwn f the electrnic screening is expected at even much shrter distances, but a prper treatment is exceeding the limits f the cntinuum apprximatin. Hence, we intrduce in this paper a δ-functin like central cell crrectin term that shuld accunt fr all kinds f shrt range perturbatins affecting the immediate neighbrhd f the central cell. The magnitude f this term is treated as a free parameter that can be adjusted t the experimental findings. It is imprtant t nte that a change f this parameter leads t a significant shift f the green S excitn with respect t the higher rder states f the yellw series and has therefre a high impact n the energies and the cmpsitins f the resulting cupled excitn states. Taking this in mind it is fundamental that ne can likewise achieve a match t the relative scillatr strengths f the invlved states. Dealing with the even parity system f Cu O is als cnfrnting us with the prblem f a prper treatment f the S excitn with respect t its very small radius since a small excitn radius means a large extensin f the excitn in k space. The challenge is therefre t meet the band structure f the valence band in a much larger vicinity f the Γ pint. Fr cping with this situatin, we include in the kinetic energy f the hle all terms in the furth pwer f p being cmpliant with the ctahedral symmetry f Cu O. The parameters f these terms are carefully adjusted t get a best fit t the band structure in the part f the k space being relevant fr the S excitn. Despite f all these mdificatins, it is imprtant t nte that the Hamiltnian is essentially the same as the ne being applied t the dd excitn system [0]. The fundamental mdificatins presented in this paper are irrelevant fr the dd parity system because f their δ functin like nature r their specific frm affecting nly excitn states with a small radius. Hence, we present a cnsistent theretical mdel fr the cmplete excitn spectrum f Cu O. Cmparing ur results t experimental data, we can prve very gd agreement as regards nt nly the energies but als the scillatr strengths since ur methd f slving the Schrödinger equatin allws us als t calculate relative scillatr strengths fr ne-phtn and twphtn absrptin. This agreement between thery and experiment is imprtant nt nly fr the investigatin f excitn spectra in electric r cmbined electric and magnetic fields. A crrect theretical descriptin f excitns is indispensable if Rydberg excitns will be used in the future in quantum infrmatin technlgy, r used t attain a deeper understanding f quasi-particle interactins in semicnductrs [, 6]. Furthermre, this agreement is a prerequisite fr a future search fr exceptinal pints in the excitn spectrum [8]. The paper is rganized as fllws: Having presented the Hamiltnian f excitns in Cu O when cnsidering the cmplete valence band structure Sec. II, we discuss all crrectins t this Hamiltnian due t the small radius f the S excitn in Sec. III. In Sec. IV we shw hw t slve the Schrödinger equatin using a cmplete basis and hw t calculate relative scillatr strengths fr ne-phtn and tw-phtn absrptin. In Sec. V we discuss the cmplete yellw and green excitn spectrum f Cu O paying attentin t the excitn states with a small principal quantum number and especially t the green S excitn state. Finally, we give a shrt summary and utlk in Sec. VI. II. HAMILTONIAN In this sectin we present the Hamiltnian f excitns in Cu O, which accunts fr the cmplete valence band structure f this semicnductr. This Hamiltnian describes the excitn states f dd parity with a principal quantum number n 3 very well [0, 4]. Hwever, fr the excitn states f even parity and fr the P excitn crrectins t this Hamiltnian are needed, which will be described in Sec. III. The lwest Γ + 6 cnductin band in Cu O is almst parablic in the vicinity f the Γ pint and the kinetic energy can be described by the simple expressin H e (p e = p e m e, (

3 3 with the effective electrn mass m e. Since Cu O has cubic symmetry, we use the irreducible representatins Γ ± i f the cubic grup O h t assign the symmetry f the bands. Due t interband interactins and nnparablicities f the three uppermst valence bands in Cu O, the kinetic energy f the hle is given by the mre cmplex expressin [9, 0, 4], H h (p h = H s + ( / m 0 { (γ + 4γ p h + (η + η p h (I S h 6γ ( p h I + c.p. η ( p h I S h + c.p. γ 3 ({p h, p h } {I, I } + c.p. η 3 ({p h, p h } (I S h + I S h + c.p.} ( with {a, b} = (ab + ba, the free electrn mass m 0, and c.p. denting cyclic permutatin. The quasi-spin I = describes the threefld degenerate valence band and is a cnvenient abstractin t dente the three rbital Blch functins xy, yz, and zx [8]. The parameters γ i, which are called the first three Luttinger parameters, and the parameters η i describe the behavir and the anistrpic effective mass f the hle in the vicinity f the Γ pint. The spin-rbit cupling, which enters Eq. (, is given by H s = ( 3 + I S h. (3 In a first apprximatin, the interactin between the electrn and the hle is described by a screened Culmb ptential V (r e r h = e 4πε 0 ε s r e r h (4 with the dielectric cnstant ε s = 7.5. Fr small relative distances r = r = r e r h crrectins t this ptential and t the kinetic energies H e (p e and H h (p h are needed, which will be described in Sec. III and which will be dented here by V CCC. After intrducing relative and center f mass crdinates [9, 0] and setting the psitin and mmentum f the center f mass t zer, the cmplete Hamiltnian f the relative mtin finally reads [7, ] H = E g + V (r + H e (p + H h (p + V CCC (5 with the energy E g f the band gap. III. CENTRAL-CELL CORRECTIONS Due t its small radius, the S excitn in Cu O is an excitn intermediate between a Frenkel and a Wannier excitn []. Hence, apprpriate crrectins are needed t describe this excitn state crrectly. The crrectins, which allw fr the best pssible descriptin f the excitn prblem within the cntinuum apprximatin f the slid, are called central-cell crrectins and have first been treated by Uihlein et al [7, ] and Kavulakis et al [3] fr Cu O. While Uihlein et al [7] accunted fr these crrectins nly in a simplified way by using a semi-empirical cntact ptential V = V 0 δ(r, the treatment f Kavulakis et al [3] did nn accunt fr the band structure and the effect f the central-cell crrectins was discussed nly n the S state and nly using perturbatin thery. By cnsidering the cmplete valence band structure f Cu O in cmbinatin with a nnperturbative treatment f the central-cell crrectins, we present a mre accurate treatment f the whle yellw excitn series in Cu O. Crrectins beynd the frame f the cntinuum apprximatin will nt be treated here. Hwever, these crrectins may describe remaining small deviatins between experimental and theretical results. The central-cell crrectins as discussed in Ref. [3] cmprise three effects, which are (i the appearance f terms f higher-rder in the mmentum p in the kinetic energies f electrn and hle, (ii the mmentum- and frequency-dependence f the dielectric functin ε, and (iii the appearance f an exchange interactin, which depends n the mmentum f the center f mass. A. Band structure f Cu O Since the radius f the yellw S excitn is small, the extensin f its wave functin in mmentum space is accrdingly large. Hence, we have t cnsider terms f the furth pwer f p in the kinetic energy f the electrn and the hle. The inclusin f p 4 terms in Eqs. ( and ( leads t an extended and mdified Hamiltnian in the sense f Altarelli, Baldereschi and Lipari [, 4 7] r Suzuki and Hensel [8]. The extended Hamiltnian must be cmpatible with the symmetry O h f the crystal and transfrm accrding t the irreducible representatin Γ +. All the terms f the furth pwer f p span a fifteen-dimensinal space with the basis functins p i, p 3 i p j, p i p j, p i p j p k (6 with i, j, k {,, 3} and i j k i. Including the quasi spin I and using grup thery, ne can find six linear cmbinatins f p 4 terms, which transfrm accrding t Γ + [9] (see Appendix A. Using the results f Appendix A, we can write the kinetic energy f the electrn and the hle as

4 H e (p e = {( + λ a p e p m e + λ a [ p ep e + c.p. ]} (7 e 4 and H h (p h = H s + { (γ 4 + 4γ ( + ξ a p h p m h + ξ a [ p hp h + c.p. ] 0 6γ ( + ξ 3 a p h [ p h I + c.p. ] γ 3 ( + ξ 4 a p h [ph p h {I, I } + c.p.] + (η + η [ p h I S h ] η [ p hi S h + c.p. ] η 3 [p h p h (I S h + I S h + c.p.] 6ξ 5 a [( p 4 h + 6p hp h3 I + c.p. ] ξ 6 a [( p h + p h 6p h3 ph p h {I, I } + c.p. ]} (8 with the lattice cnstant a and the unknwn parameters λ i and ξ i. Nte that the values f parameters η i are smaller than the Luttinger parameters γ i (see Table I. Hence, we expect the terms f the frm p 4 IS h t be negligibly small. After replacing H e (p e H e ( k and H h (p h H h ( k, we can determine the eigenvalues f these Hamiltnians and fit them as in Ref. [9] fr k < π/a t the band structure f Cu O btained via spin density functinal thery calculatins [30]. T btain a reliable result, we perfrm a least-squares fit with a weighting functin. Even thugh the excitn grund state will shw deviatins frm a pure hydrgenlike S state, we expect that the radial prbability density can be described qualitatively by that functin. Hence, we use the mdulus squared f the Furier transfrm Φ S (k = F (Ψ S (k f the hydrgen-like functin Ψ S (r = πa 3 S e r/a S (9 as the weighting functin fr the fit. It reads [3] Φ S (k dr Ψ S (r e ikr (π 3 = 8a 3 S π ( + k a S 4 (0 dashed lines [9]. The values f the fit parameters are λ =.09 0, ξ =.389 0, λ =.05 0 ξ 4 =.58 0 ξ = , ξ 5 = ξ 3 =.53 0, ξ 6 = ( As can be seen, e.g., frm Fig., the fit including the quartic terms is nly slightly better than the fit with the quadratic terms fr small k. A clear difference between the fits can be seen nly fr large values f k as regards the valence bands: Since sme f the pre-factrs f the quartic terms are psitive, the energy f the valence bands in the fitted mdel increases fr larger values f k. Cnsidering the minr differences between the fits fr small k and the small extensin f the S excitn functin in k space even fr a S = a (see, e.g., Fig., the quartic terms will hardly affect this excitn state and can be neglected. These arguments still hld if, e.g., a S = 0.a is assumed. In the wrk f Ref. [3] the intrductin f p 4 terms seemed necessary t explain the experimentally bserved large mass f the S excitn. Hwever, the experimental bservatins are already well described by quadratic terms in p when cnsidering the cmplete valence band structure [0]. As we already stated in Ref. [3], a simple restrictin t the Γ + 7 band neglecting the Γ+ 8 band and cnsidering the nnparablicity f the Γ + 7 band via p4 terms as has been dne in Ref. [3] des nt treat the prblem crrectly. with the Bhr radius a S f the S excitn state. Althugh we d nt a priri knw the true value f a S, the experimental value f the binding energy f the S state [, 7] as well as the calculatins f Ref. [3] indicate that it is f the rder f the lattice cnstant a, fr which reasn we assume a S = a = 0.47 nm [3 34]. The results f the fit are depicted as red slid lines in Figs.,, and 3. Fr a cmparisn, we als shw the fit neglecting the quartic terms in the mmenta (blue B. Dielectric cnstant In the case f the S excitn in Cu O the relative mtin f the electrn and the hle is sufficiently fast that phnns cannt fllw it and crrectins n the dielectric cnstant need t be cnsidered. In general, the electrn and the hle are cupled t lngitudinal ptical phnns via the Fröhlich interactin [35, 36] and t lngitudinal acustic phnns via

5 5 (a 4.5 (a E c (k [ev] Φ S (k [a 3 ] E c (k [ev] Φ S (k [a 3 ] (b 0 (b E v (k [ev] E v (k [ev] Γ k [π/a] X Γ k [π/a] M FIG.. Fits t the band structure btained via spin density functinal thery calculatins [30] (black linespints fr (a cnductin band and (b valence bands f Cu O fr the [00] directin using the expressins (7 and (8 (red lines. The green slid line shws the functin Φ S (k fr a S = a in units f a 3. One can see that the differences between the fit using quartic terms and the fit f Ref. [9] (blue dashed lines neglecting these terms are small in the range f extensin f Φ S (k. Nte that Φ S(k is nt shwn in the lwer panel fr reasns f clarity. the defrmatin ptential cupling [36, 37]. While in the case f ptical phnns the ins f the slid are displaced in anti-phase and thus create a diple mment in the unit cell f a plar crystal, the ins are displaced in phase in the case f acustic phnns and n diple mment is created. Hence, ne expects that the interactin between electrn r hle and ptical phnns is much larger than the interactin with acustic phnns in plar crystals [38, 39]. If the frequency f the relative mtin f electrn and hle is high enugh s that the ins f the slid cannt fllw it, the Culmb interactin between electrn and hle is screened by the high-frequency r backgrund dielectric cnstant ε b [, 40]. This dielectric cnstant describes the electrnic plarizatin, which can fllw the mtin f electrn and hle very quickly [4]. Fr lwer frequencies f the relative mtin the cntributin f the phnns t the screening becmes imprtant and the dielectric functin ε becmes frequency dependent. In many semicnductrs the frequency f the relative mtin in excitn states with a principal quan- FIG.. Same as Fig. fr the [0] directin. tum number f n is s small that the lw-frequency r static dielectric cnstant ε s can be used [39], which invlves the electrnic plarizatin and the displacement f the ins [4]. Nte that we use the ntatin ε b, ε s instead f ε, ε 0 t avid the risk f cnfusin with the electric permittivity ε 0 [39, 4]. The transitin frm e /4πεε s r t e /4πεε b r, which takes place when the frequency f the electrn r the hle is f the same size as the frequency f the phnn [4], had been investigated in detail by Haken in Refs. [38, 4 45]. He cnsidered at first the interactin between electrn r hle and the phnns and then cnstructed the excitn frm the resulting particles with plarisatin cluds, i.e., the plarns. The change f the Culmb interactin between bth particles was then explained in terms f an exchange f phnns, i.e., f virtual quanta f the plarizatin field [4]. The final result fr the interactin in the transitin regin between e /4πεε s r and e /4πεε b r was the scalled Haken ptential [, 3, 39, 43 45], [ V (r = e + ( ] 4πε 0 r ε s ε e r/ρ h + e r/ρe. ( Here ρ e and ρ h dente the plarn radii ρ e/h = m e/h ω LO (3 with the frequency ω LO f the ptical phnn and ε is

6 6 (a E c (k [ev] (b E v (k [ev] Γ k [π/a] R FIG. 3. Same as Fig. fr the [] directin. Φ S (k [a 3 ] value f E g fr Cu O has been determined in Ref. [] frm the experimental excitn spectrum, the plarn effect is already accunted fr in the band gap energy [39]. Nte that the abve results were derived in the simple band mdel and by assuming nly ne ptical phnn branch cntributing t the Fröhlich interactin. There are theries fr plarns in the degenerate band case [47 49] but t the best f ur knwledge there is n mdel accunting fr mre than ne ptical phnn branch [3, 4, 46]. This cmplicates the crrect treatment f Cu O, where tw LO phnns cntribute t the Fröhlich interactin. Furthermre, the Haken ptential ( cannt describe the nn-culmbic electrn-hle interactin fr very small values f r, which is due t the finite size f electrn and hle []. The cnditins f validity f the ptential ( have, e.g., been discussed by Haken in Ref. [4]. When treating the Haken ptential numerically fr different plar crystals, the experimental and theretical binding energies f the excitn states smetimes d nt agree, fr which reasn crrectins, smetimes phenmenlgically, t the Haken ptential have been intrduced [50 53] leading t clearly better results. One f these refined frmulas is the ptential prpsed by Pllmann and Büttner [46, 53] given by ε = ε b ε s. (4 e V (r = 4πε 0 r [ + ( mh ε s ε m e r/ρ h m ] e e r/ρe, (5 m Nte that in the result f Haken [38, 46] the plarn masses m i instead f the bare electrn and hle masses have t be used in the plarn radii and the kinetic energies. Furthermre, the lattice relaxatin due t the interactin f excitns and phnns decreases the band gap energy fr electrns and hles. Hwever, since the in which the bare electrn and hle masses have t be used and where m is given by m = m h m e. Hence, we take the statements given abve as a reasn t prpse the fllwing phenmenlgical ptentials fr Cu O, which are mtivated by the frmula f Haken and by the frmula f Pllmann and Büttner: e V H (r = 4πε 0 r [ + (e r/ρ ε s ε h + e r/ρe + ( ] ε e r/ρ h + e r/ρe (6a and [ V PB (r = e + ( 4πε 0 r ε s ε + ( ε m 0 m 0 m e γ e r/ρ h m 0 m 0 m e γ e r/ρ h m e γ e r/ρe m 0 m e γ m e γ m 0 m e γ e r/ρe ]. (6b Here we use and ε i = ε bi ε si (7 γ ρ ei =, ρ hi =, (8 m e ω LOi m 0 ω LOi

7 7 where the energies f the phnns and the values f the dielectric cnstants are given by [3] and ω LO = 8.7 mev, ω LO = 87 mev (9 ε s = 7.5, ε b = ε s = 7., ε b = (0 As has been dne in Ref. [46] fr pervskite CH 3 NH 3 PbI 3, we use V H r V PB in the Schrödinger equatin withut an additinal fit parameter and find ut which f these ptentials describes the excitn spectrum f Cu O best. Since fr the plarn radii ρ e and ρ h.6a ρ 4.4a hlds, we expect the Haken r the Pllmann-Büttner ptential t have a significant influence n the excitn states with n. As the Fröhlich cupling cnstant is small in Cu O, i.e., it is α F < 0. fr the tw ptical phnns and bth the electrn and the hle [30], the bare electrn and hle masses differ frm the plarn masses by at mst 3%. Hence, we can calculate with the bare masses when using V H. Besides the frequency dependence f the dielectric functin als its mmentum dependence becmes imprtant if the excitn radius is n the rder f the lattice cnstant. This mmentum dependence f the dielectric functin arises frm the electrnic plarizatin [3, 54]. When treating the excitns f Cu O in mmentum space, the wave functins f the n states are lcalized abut k = 0 s that fr these states the k dependence f ε is nt imprtant. Hwever, fr the S state a S a hlds and thus this state is screened by ε at higher mmenta k [3]. Cnsidering the Culmb interactin fr the S excitn in k space, V (k, ω = (π 3 e ε 0 ε(k, ωk, ( Kavulakis et al [3] derived a crrectin term by assuming ε(k, ω ε b d(ka + d(ka ε b ε b ( valid fr E g / ω ω LO with a small unknwn cnstant d. Inserting Eq. ( in Eq. ( and Furier transfrming the secnd expressin, ne btains the fllwing crrectin term t the Culmb interactin: e V d = da ε 0 ε V uc δ(r = V 0 V uc δ(r. (3 b Fllwing the calculatin f Ref. [54] n the dielectric functin and using the lwest Γ 8 cnductin band and the highest Γ + 7 valence band, Kavulakis et al [3] estimated the value f d t d 0.8 [3]. Nte that in general a Krnecker delta wuld appear in Eq. (3 []. Hwever, as we treat the excitn prblem in the cntinuum apprximatin, this Krnecker delta is replaced by the delta functin times the vlume V uc = a 3 f ne unit cell. Thus, the parameter V 0 has the unit f an energy. We have already stated abve that the Haken ptential cannt describe the electrn-hle interactin crrectly fr very small r. Therefre, we nw assume that the ptential (3 is nt nly due t the mmentum dependence f the dielectric functin but that it als accunts fr deviatins frm the Haken ptential at small r. Hence, we will treat Ṽ0 as an unknwn fit parameter in the fllwing. C. Exchange interactin In the Wannier equatin r Hamiltnian f excitns the exchange interactin is generally nt included but regarded as a crrectin t the hydrgen-like slutin []. Recently, we have presented a cmprehensive discussin f the exchange interactin in Cu O [3]. We culd shw, in accrdance with Ref. [3], that crrectins t the exchange interactin due t a finite mmentum K f the center f mass f the excitn are negligibly small. Hence, nly the K independent part f the exchange interactin [3, 7, 48, 55] H exch = J 0 ( 4 S e S h V uc δ(r (4 needs t be cnsidered. Within the simple hydrgen-like mdel the exchange interactin wuld nly affect the ns excitn states as these states have a nnvanishing prbability density at r = 0. Hwever, when cnsidering the cmplete valence band structure, the excitn states with even r with dd values f L are cupled, and thus the exchange interactin will affect the whle even excitn series. It is well knwn frm experiments that the splitting between the yellw S rth and the yellw S para excitn amunts t abut mev [56 58]. Hence, we have t chse the value f J0 such that this splitting is reflected in the theretical spectrum. D. Summary Fllwing the explanatins given in Secs. III B and III C, the term V CCC in the Hamiltnian f Eq. (5 takes ne f the fllwing frms:

8 [ VCCC H = e 4πε 0 r + ε (e r/ρ h + e r/ρe [ V 0 + J 0 ( 4 S e S h ] V uc δ(r, + (e ] r/ρ ε h + e r/ρe 8 (5a [ ( VCCC PB = e m 0 4πε 0 r ε e r/ρ h m 0 m e γ + ( m 0 ε e r/ρ h m 0 m e γ ( 4 + [ V 0 + J 0 ] S e S h V uc δ(r, m e γ e r/ρe m 0 m e γ m e γ m 0 m e γ e r/ρe ] (5b [cf. Eqs. (6a, (6b, (3, and (4]. Nte that while the peratrs with δ (r affect nly the excitn series with even values f L, the Haken r Pllmann and Büttner ptential affect all excitn states [4]. A cmparisn f ur results with the experimental values f Refs. [, 3, 7, 59, 60] will allw us, in Sec. V, t determine the size f the unknwn parameters V 0 and J 0. IV. EIGENVALUES AND OSCILLATOR STRENGTHS In this sectin we describe hw the Schrödinger equatin crrespnding t the Hamiltnian (5 is slved in a cmplete basis. Furthermre, we discuss hw t calculate scillatr strengths fr tw-phtn absrptin. An apprpriate basis t slve the Schrödinger equatin has been presented in detail in Ref. [0]. Hence, we recapitulate nly the mst imprtant pints. As regards the angular mmentum part f the basis, we have t cnsider that the different peratrs in the Hamiltnian cuple the quasi spin I, the hle spin S h, and the angular mmentum L f the excitn. Hence, we intrduce the effective hle spin J = I + S h, the angular mmentum F = L+J, and the ttal angular mmentum F t = F + S e. Fr the radial part f the excitn wave functin we use the Culmb-Sturmian functins [6] U NL (r = N (α NL (ρl e ρ L L+ N (ρ (6 with ρ = r/α, an arbitrary cnvergence r scaling parameter α, the assciated Laguerre plynmials L m n (x, and a nrmalizatin factr N (α NL = [ α 3 N! (N + L + (N + L +! ]. (7 The radial quantum number N is related t the principal quantum number n via n = N + L +. Finally, we use the fllwing ansatz fr the excitn wave functin Ψ = c NLJF FtM Ft Π, (8a NLJF F tm Ft Π = N, L; (I, S h J; F, S e ; F t, M Ft (8b with real cefficients c. The parenthesis and semiclns in Eq. (8b are meant t illustrate the cupling scheme f the spins and the angular mmenta. Since the z axis is a furfld axis, it is sufficient t use nly M Ft quantum numbers which differ by ±4 in Eq. (8. We nw express the Hamiltnian (5 in terms f irreducible tensrs [6, 6, 63]. Inserting the ansatz (8 in the Schrödinger equatin HΨ = EΨ and multiplying frm the left with anther basis state Π, we btain a matrix representatin f the Schrödinger equatin f the frm Dc = EMc. (9 The vectr c cntains the cefficients f the ansatz (8. All matrix elements, which enter the symmetric matrices D and M and which have nt been treated in Ref. [0], are given in Appendix D. The generalized eigenvalue prblem (9 is finally slved using an apprpriate LA- PACK rutine [64]. The material parameters used in ur calculatin are listed in Table I. Nte that the presence f the delta functins in Eq. (5 makes the whle prblem mre cmplicated than in Ref. [0] since nt nly the eigenvalues but als the wave functins at r = 0 have t cnverge. Hwever, fr a specific value f α it is nt pssible t btain cnvergence fr all excitn states f interest. Therefre, we slve the Schrödinger equatin initially withut the δ(r dependent terms. We then select the cnverged eigenvectrs and with these we set up a secnd generalized eigenvalue prblem nw including the δ(r dependent terms. This prblem is again slved using an apprpriate LAPACK rutine [64] and prvides the crrect cnverged eigenvalues f the cmplete Hamiltnian (5. Having slved the eigenvalue prblem, we can use the eigenvectrs t determine relative scillatr strengths.

9 9 TABLE I. Material parameters used in the calculatins. Instead f the band gap energy E g =.708 ev f Ref. [] a slightly smaller value is used t btain a better agreement with experimental values in Sec. V. band gap energy E g =.70 ev electrn mass m e = 0.99 m 0 [65] spin-rbit cupling = 0.3 ev [9] valence band parameters γ =.76 [9, 0] γ = [9, 0] γ 3 = [9, 0] η = 0.00 [9, 0] η = [9, 0] η 3 = [9, 0] lattice cnstant a = nm [66] dielectric cnstants ε s = 7.5 [40] ε b = ε s = 7. [40] ε b = 6.46 [40] energy f Γ 4 -LO phnns ωlo = 8.7 mev [3] ω LO = 87 mev [3] The determinatin f relative scillatr strengths in ne-phtn absrptin has been presented in detail in Refs. [0, 4]. While in ne phtn absrptin excitns f symmetry Γ 4 are diple-allwed [0], the selectin rules fr tw-phtn absrptin [67 69] are different and excitns f symmetry Γ + 5 can be ptically excited. When cnsidering ne-phtn absrptin ne generally treats the peratr Ap with the vectr ptential A f the radiatin field in first rder perturbatin thery. The diple peratr then transfrms accrding t the irreducible representatin D f the full rtatin grup. In tw-phtn absrptin ne needs the peratr Ap twice and thus the prduct D D = D 0 D D has t be cnsidered [9]. In Cu O the reductin f these irreducible representatins by the cubic grup O h has t be cnsidered and ne btains Γ 4 Γ 4 = Γ+ Γ+ 4 ( Γ + 3 Γ+ 5. (30 In tw-phtn absrptin the spin S = S e + S h = 0 remains unchanged and the excitn state must have an L = 0 cmpnent. Hence, the crrect expressin fr the relative scillatr strength is given by f rel lim T, M Ft Ψ (r, (3 r 0 with the wave functin Ψ f Eq. (8 and the state F t, M Ft T, which is a shrt ntatin fr (S e, S h S, I; I + S, L; F t, M Ft = (/, / 0, ;, 0; F t, M Ft. (3 Nte that the cupling scheme f the spins and angular mmenta in Eq. (3 given by S e + S h = S (I + S + L = F t (33 is different frm the ne f Eq. (8b due t the requirement that S must be a gd quantum number. It can be shwn that the state, M Ft T transfrms accrding t the irreducible representatin Γ + 5 f O h [9], fr which reasn nly excitn states f this symmetry can be excited in tw-phtn absrptin. By chsing particular directins f the plarizatin f the light, e.g., by chsing ne phtn being plarized in x directin and ne phtn being plarized in y directin, nly ne cmpnent f the Γ + 5 excitn states, the xy cmpnent, can be excited ptically. We cnsider this case in the fllwing and hence use M F t = 0 in Eq. (3. Finally, we wish t nte that the excitn states f symmetry Γ + 5 can weakly be bserved in ne-phtn absrptin in quadruple apprximatin [7]. V. RESULTS AND DISCUSSION In this sectin we determine the values f the parameters J 0 and V 0 and discuss the cmplete excitn spectrum f Cu O. The parameter J 0 describes the strength f the exchange interactin. It is well knwn that the exchange interactin mainly affects the S excitn and that the splitting between the rth and the para excitn state amunts t.8 mev [7, 56 59]. By chsing J 0 = 0.79 ± ev (34 we btain the crrect value f this splitting irrespective f whether using the Haken r the Pllman-Büttner ptential [cf. Eq. (5]. The figures 4 and 5 shw the effect f the crrectin with the cefficient V 0 n the spectrum fr the Haken and the Pllman-Büttner ptential, respectivley. As can be seen frm these figures, the exchange splitting f the S state hardly changes when varying the value V 0. Hence, we can determine V 0 almst independently f J 0. T find the ptimum value f V 0, we cmpare ur results t the energies f the even parity excitn states given in Refs. [9,, 7, 59, 60]. Hwever, we can see frm Figs. 4 and 5 that there is n value f V 0 fr which all theretical results take the values f the experimentally determined energies. This is nt unexpected since the central-cell crrectins are nly an attempt t accunt fr the specific prperties f the S excitn within the cntinuum limit f Wannier excitns and are nt an exact descriptin f this excitn state. Hence, we d nt expect a perfect agreement between thery and experiment. Small deviatins frm the experimental values culd als be explained by small uncertainties in the Luttinger parameters γ i, η i [9, 0] r the band gap energy []. On the ther hand, it is als pssible that the experimental values are affected by uncertainties. This can be seen, e.g., when cmparing the slightly different experimental results f Refs. [9] and [, 60].

10 0 4D y f rel 0 0 4D y f rel 0 0 E [ev] D y 4S y 3D y 3D y 3S y 0 - E [ev] D y 4S y 3D y 3D y 3S y S g.5 S g E [ev] S y 0 - E [ev] S y E [ev] S y p S y V 0 [ev] 0-4 E [ev] S y p S y V 0 [ev] 0-4 FIG. 4. Behavir f the even excitn states as functins f V 0 when using VCCC H [see Eq. (5a]. The clr bar shws the relative scillatr strengths fr tw-phtn absrptin. The blue straight lines dente the psitin f the diple-allwed Γ + 5 S and D excitn states bserved in the experiment. We als shw the psitin f the yellw S para excitn ( Sy p. The gray area indicates the ptimum range f V 0 = ± 0.07 ev, where the rati f the relative scillatr strengths f the yellw S and the green S state amunts t 6. The effect f the central-cell crrectins n the whle even excitn spectrum is evident. Fr further infrmatin see text. Nte that the almst perfect agreement between theretical and experimental results in Refs. [7, ] culd nly be btained by taking als γ, µ and as fit parameters t the experiment. Hwever, these parameters are cnnected t the band structure in Cu O [30] and cannt be chsen arbitrarily [3, 0]. FIG. 5. Same calculatin as in Fig. 4 but with VCCC PB [see Eq. (5b]. One can see nly slight differences fr the n = and n = excitn states when cmparing the results t Fig. 4. The gray area indicates the ptimum range f V 0 = ± 0.07 ev. It can be seen frm Figs. 4 and 5 that the scillatr strength f the excitn state at E.43 ev changes rapidly with increasing V 0. Frm the experimental results f Refs. [7, ] we knw that the tw excitn states at E =.378 ev and E =.544 ev are well separated frm the ther excitn states and that the phnn backgrund is small. Hence, the rati f the relative twphtn scillatr strengths can be calculated quite accurately t 6. We nw chse the value f V 0 such that the rati f the calculated tw-phtn scillatr strengths reaches the

11 0 [ev] (a y g y g 3y µ (b δ (c VH (d V0 [ev] J0 [ev] (e (f [ev] Spg exp 0.30 Sy Spy E [ev] (g Spy (h Sy Py Sg..35 E [ev] E [ev].67 FIG. 6. Excitn spectrum f the even (blue and dd (red excitn states when increasing all material parameters frm zer H (tp t their crrect values (bttm and using VCCC [cf. Eq. (5b]. If all material parameters except fr γ0 are set t zer, ne btains a hydrgen-like spectrum, fr which the yellw (y and green (g excitn states are degenerate [ = 0 in (a]. When increasing the spin-rbit cupling cnstant, this degeneracy is lifted and the green excitn states are shifted twards higher energies (a. Nte that we increase in tw steps t its true value f = 0.3 ev fr reasns f clarity. One can then fllw these states frm (b t (g. Since the effect f the parameters η0, ν and τ n the excitn spectrum is small they are immediately set frm zer t their crrect values between (c and (d. In (g and (h the para and rth excitn states are dented by an upper index p and. The final results at the bttm f (g, which are als listed in Table III, can then be cmpared t the psitin f the excitn states btained frm experiments (h. Nte that due t the marked anticrssing [green arrw in the secnd panel f (g] the assignment f the green S state and the yellw S state changes.

12 TABLE II. Decmpsitin f the irreducible representatins f the rtatin grup r the angular mmentum states by the cubic grup O h. Nte that the quasi-spin I already enters the mmentum F via J. The irreducible representatins dente the symmetry f the envelpe functin (L, the cmbined symmetry f envelpe and hle (F r the cmplete symmetry f the excitn (F t. L F = L + J ( J = Ft = F + S e 0 Γ + Γ+ 7 0 Γ + Γ + 5 radius a exc f the S rth excitn and t the crrect assignment f the n = excitn states. T determine the Bhr radius a exc, we evaluate Ψ r Ψ = c N LJF F tm Ft c NLJF FtM Ft N NLJF F tm Ft j= α (R j NL N + L + j + δ N, N+j (37 with the wave functin Ψ f Eq. (8 and cmpare the result with the frmula [70] r = a [ exc 3n L (L + ] (38 Γ 4 Γ 7 3 Γ 8 0 Γ Γ 5 Γ 4 Γ 3 Γ 5 knwn frm the hydrgen atm, where we set n = and L = 0. Nte that the functin (R j NL in Eq. (37 is taken frm the recursin relatins f the Culmb- Sturmian functins in the Appendix f Ref. [0]. We btain a exc (S nm.86 a (39 Γ + 3 Γ+ 5 3 Γ+ 8 Γ + 5 Γ + 3 Γ+ 4 when using the Haken ptential r a exc (S 0.80 nm.90 a (40 3 Γ Γ 4 Γ 5 5 Γ+ 6 Γ+ 8 5 Γ 6 Γ 8 7 Γ 6 Γ 7 Γ 8 Γ + 3 Γ+ 4 3 Γ + Γ+ 4 Γ+ 5 Γ 3 Γ 4 3 Γ Γ 4 Γ 5 3 Γ Γ 4 Γ 5 4 Γ Γ 3 Γ 4 Γ 5 L F = L + J ( J = 3 Ft = F + S e 0 Γ + 3 Γ+ 8 same value and btain Γ + 5 Γ + 3 Γ+ 4 V 0 = ± 0.07 ev (35 when using the Haken ptential [cf. Eq. (5a] r V 0 = ± 0.07 ev (36 when using the Pllmann-Büttner ptential [cf. Eq. (5b]. Nte that the errr bars fr V 0 are chsen such that the rati f the scillatr strengths lies between 4 and 8. Having determined the mst suitable values f V 0 and J 0, we can nw turn ur attentin t the excitn Bhr when using the Pllmann-Büttner ptential. In bth cases the radius f the S rth excitn is large enugh that the crrectins t the kinetic energy discussed in Sec. III A can certainly be neglected. Let us nw prceed t the crrect assignment f the n = excitn states. Since in the investigatin f Uihlein et al [7, ] the wrng values fr the Luttinger parameters were used (cf. Ref. [0], it is nt clear whether the state at E =.544 ev can still be assigned as the yellw S rth excitn state and the state at E =.378 ev as the green S rth excitn state when using the crrect Luttinger parameters. T demnstrate frm which hydrgen-like states the experimentally bserved excitn states riginate, we find it instructive t start frm the hydrgen-like spectrum with almst all material parameters set t zer and then increase these material parameters successively t their true values. This is shwn in Fig. 6. At first all material parameters except fr γ are set t zer, s that a true hydrgen-like spectrum is btained, where the yellw (y and green (g excitn states are degenerate. This spectrum is shwn in the panel (a f Fig. 6. When increasing the spin-rbit cupling cnstant in Fig. 6(a, the degeneracy between the green and the yellw excitn series is lifted. The increase f the Luttinger parameters µ and δ in the panels (b and (c furthermre lifts the degeneracy between the excitn states f different angular mmentum L. The Haken ptential des nt change degeneracies but slightly lwers the energy f the excitn states in Fig. 6(d. The exchange energy described by the cnstant J 0 lifts the degenercy between rth and para excitn states in Fig. 6(e. As

13 3 TABLE III. Cmparisn f calculated energies E ther t experimental values E exp (References given behind experimental values when using the central-cell crrectins with the Haken ptential (5a. The even excitn states are listed in blue text and the dd excitn states in red text. Nte that we use E g =.70 ev instead f E g =.708 ev [] t btain a better agreement. The assignment f the states in the first clumn is mtivated by Fig. 6 but is generally nt instructive due t the large deviatins frm the hydrgen-like mdel. Hence, we als give the symmetry f the states. In the case f the P and F excitns we d nt give the symmetry f the cmplete excitn state but nly the cmbined symmetry f envelpe and hle. As regards the 5G excitns we nly give the average energy f the states f symmetry Γ + 5. The value in the furth clumn gives the relative scillatr strength in ne-phtn absrptin (np, nf excitns; see Ref. [0] r in tw-phtn absrptin (ns, nd, ng excitns; see Eq. (3. Nte that due t the interactin with the S g state the scillatr strength f the S y state is much smaller than expected when assuming tw independent, i.e., green and yellw, series. The value in the last clumn indicates the percentage f the J = 3/ cmpnent f the state, i.e., the green part. Nte that due t the interactin between the yellw and the green excitn series the green S state is spread ver several yellw excitn states. The green states with n are lcated far abve the states listed here. State E exp [ev] E ther [ev] f rel gp [%] State E exp [ev] E ther [ev] f rel gp [%] S y Γ +.0 [59] D y Γ + 3/4.669 [, 60] S y Γ [7] D y Γ [, 60] D y Γ + / S g Γ + 3/ F y Γ F y Γ [3] S y Γ [7] F y Γ [3] S y Γ F y Γ [3] P y Γ [] F y Γ [3] P y Γ D y Γ + 3/ D y Γ [, 60] S g Γ [7] S y Γ S y Γ S y Γ [, 60] S y Γ [, 60] P y Γ [] P y Γ [] P y Γ P y Γ D y Γ + 3/4.684 [, 60] D y Γ + 3/4.683 [, 60] D y Γ [, 60] D y Γ [, 60] D y Γ + / D y Γ + / F y Γ D y Γ + 3/ F y Γ [3] D y Γ [, 60] F y Γ [3] F y Γ [3] S y Γ F y Γ [3] S y Γ [, 60] D y Γ + 3/ P y Γ [] Ḡy Γ P y Γ D y Γ [, 60]

14 4 TABLE IV. Same cmparisn as in Table III but when using the central-cell crrectins with the Pllmann-Büttner ptential (5b. Especially fr the states with n < 3 differences in the calculated energies can be bserved when using the different crrectins (5a r (5b. Nte that fr each n the relative scillatr strength f ne nd state is larger than the relative scillatr strengths f the ns state in accrdance with the experimental results f Ref. [7]. State E exp [ev] E ther [ev] f rel gp [%] State E exp [ev] E ther [ev] f rel gp [%] S y Γ +.0 [59] D y Γ + 3/4.669 [, 60] S y Γ [7] D y Γ [, 60] D y Γ + / S g Γ + 3/ F y Γ F y Γ [3] S y Γ [7] F y Γ [3] S y Γ F y Γ [3] P y Γ [] F y Γ [3] P y Γ D y Γ + 3/ D y Γ [, 60] S g Γ [7] S y Γ S y Γ S y Γ [, 60] S y Γ [, 60] P y Γ [] P y Γ [] P y Γ P y Γ D y Γ + 3/4.684 [, 60] D y Γ + 3/4.683 [, 60] D y Γ [, 60] D y Γ [, 60] D y Γ + / D y Γ + / F y Γ D y Γ + 3/ F y Γ [3] D y Γ [, 60] F y Γ [3] F y Γ [3] S y Γ F y Γ [3] S y Γ [, 60] D y Γ + 3/ P y Γ [] Ḡy Γ P y Γ D y Γ [, 60]

15 5 the peratr δ(r affects nly the states f even parity (blue lines, the energy f the dd excitn states (red lines remains unchanged in Fig. 6(f. Nte that we increase in tw steps t its true value f = 0.3 ev fr reasns f clarity. Hence, at the bttm f Fig. 6(g all material values have been increased t their true values. Fr a cmparisn, we shw in panel (h the psitin f the experimentally bserved states. Fllwing the excitn states frm panel (a t (g, it is pssible t assign them with the ntatin nl p/ y/g, where the upper index dentes a para r an rth excitn state and the lwer index a yellw r a green state. The results presented in Fig. 6 suggest t assign the excitn state at E =.378 ev t the green S rth excitn state. Hwever, ne can bserve an anticrssing between the green S state and the yellw S state, which is indicated by a green arrw in Fig. 6(g. Hence, the assignment has t be changed. As a prf, we can calculate the percentage f the J = 3/ cmpnent f these states, i.e., their green part, by evaluating with the prjectin peratr P = 3/ M J = 3/ gp = Ψ P Ψ (4 3 3, M J, M J (4 and the excitn wave functin Ψ (see als Appendix C. The green part gp f the state at E.544 ev is distinctly higher (gp 40% than the green part f the excitn state at E.378 ev (gp %. Hwever, since als gp 40% is significantly smaller than ne, we see that the assignment f this excitn state as the grund state f the green series is questinable and shws the significant deviatins frm the hydrgen-like mdel. The green S excitn state is distributed ver the yellw states. Nte that in Ref. [7] als the state f higher energy had a larger green part than the state f lwer energy. Hwever, in Fig. f Ref. [7] the assignment is reversed since the limit f µ 0 was used t designate the states. It seems bvius that a similar anticrssing between the green S state and the yellw S state was disregarded. A cnsiderable effect f the interactin between the green and yellw series is the change in the scillatr strength f the states. The scillatr strength f the S y state is much smaller than expected when assuming tw independent, i.e., green and yellw, series [7, ] (cf. als Tables III and IV. Fr reasns f cmpleteness, we give the size f the green S and the yellw S state by evaluating Eq. (37. Since these states are strngly mixed and a crrect assignment with a principal quantum number n is nt pssible, we d nt use the frmula (38. We btain when using the Haken ptential r r (S y 4.39 nm 0.3 a, r (S g 4.09 nm 9.58 a, (44a (44b when using the Pllmann-Büttner ptential. We see that in bth cases the values f r fr the green S and the yellw S state are f the same size. This is expected due t the strng mixing f bth states. The resnance f the green S state with the yellw excitn series and the mixing f all even excitn states via the cubic band structure leads t an admixture f D and G states t the green S state. Hence, the three Γ + 5 states which we assigned with S g are elliptically defrmed and invariant nly under the subgrup D 4h f O h [0, 9]. The lwer symmetry f the envelpe functin allws fr a smaller mean distance between electrn and hle in a specific directin, which leads t a gain f energy due t the Culmb interactin [0]. As regards the xy-cmpnent, the symmetry axis f the accrding subgrup D 4h is the z-axis f the crystal. Since fr this state the expectatin values Ψ x Ψ and Ψ y Ψ are identical, we can calculate the semi-principal axes f the elliptically defrmed state by evaluating Ψ x Ψ = Ψ ( r z Ψ = and N L J F F t M F NLJF F t tm Ft c N L J F F t M F t c NLJF FtM Ft α Π 3 r 3 6 X( 0 Π Ψ z Ψ = N L J F F t M F NLJF F t tm Ft c N L J F F t M F t c NLJF FtM Ft α Π r Π 3 X( (45 (46 with the wave functin Ψ f Eq. (8 and the matrix elements Π X ( 0 Π and Π r Π listed in the Appendix f Ref. [4]. We btain x 6.4 a, z 9.9 a, (47 when using the Haken ptential r x 68.6 a, z 5. a, (48 r (S y 4.3 nm 0. a, r (S g 5.3 nm.5 a, (43a (43b when using the Pllmann-Büttner ptential. The significant differences in x and z shw again the strng