Optical anisotropy of the Jeff = 1/2 Mott insulator Sr2IrO4
D. Pröpper, A. N. Yaresko, M. Höppner, Y. Matiks, Y.-L. Mathis, T. Takayama, A. Matsumoto, H. Takagi, B. Keimer, A. V. Boris
OOptical anisotropy of the J eff = 1/2 Mott insulator Sr IrO D. Pr¨opper, A. N. Yaresko, M. H¨oppner, Y. Matiks, Y.-L. Mathis, T. Takayama,
1, 3
A. Matsumoto,
1, 3
H. Takagi,
1, 3
B. Keimer, and A. V. Boris Max-Planck-Institut f¨ur Festk¨orperforschung, Heisenbergstraße 1, D-70569 Stuttgart, Germany Synchrotron Facility ANKA, Karlsruhe Institute of Technology, 76344 Eggenstein - Leopoldshafen, Germany Department of Physics, University of Tokyo, Hongo, Tokyo 113-0033, Japan (Dated: August 27, 2018)We report the complex dielectric function along and perpendicular to the
IrO planes in the layered perovskite Sr IrO determined by spectroscopic ellipsometry in the spectral range from 12 meV to 6 eV. Thin high qualitysingle crystals were stacked to measure the c -axis optical conductivity. In the phonon response we identified10 infrared-active modes polarized within the basal plane and only four modes polarized along the c -axis, infull agreement with first-principle lattice dynamics calculations. We also observed a strong optical anisotropyin the near-infrared spectra arising from direct transitions between Ir 5d t J eff =1/2 and J eff =3/2 bands, whichtransition probability is highly suppressed for light polarized along the c -axis. The spectra are analyzed anddiscussed in terms of relativistic LSDA+U band structure calculations. I. INTRODUCTION
A rich variety of electronic ground states of transition metaloxides (TMOs) emerges from strong electron correlations andcooperative phenomena with competing interactions, includ-ing the on-site Coulomb repulsion U , crystal-electric field(CEF), and spin-orbit coupling (SOC). The transition from el-ements with 3d via 4d to 5d valence orbitals progressively re-sults in larger single particle band width W , reduced U , andenhanced SOC. TMOs of the type (La , Sr) MO , where M =Cu (3d), Ru (4d), or Ir (5d), allow one to consider the mag-nitudes of these interactions as variable parameters which cansignificantly influence the electronic structure within the samelayered perovskite ’214’ structure. La CuO is particularlywell known as the parent compound of a hole-doped high- T c superconductor family in close proximity to a Mott insu-lator ground state with antiferromagnetic ordering . Whilst Sr RuO has the same crystal symmetry, its exotic low- T c superconducting state emerges from a Fermi-liquid metallicstate . Its 5d counterpart Sr IrO represents, in turn, a proto-type spin-orbit Mott insulator . In the presence of strongSOC, the five t states of the Ir ions with 5d electronconfiguration form bands described by the effective quantumnumbers J eff =3/2 and J eff =1/2. The latter, which is half-filled,is split already by moderate U into a lower and upper Hubbardband, opening the spin-orbit Mott gap.The magnetic interactions in Sr IrO also bear a resem-blance to those in La CuO and can be described within anantiferromagnetic Heisenberg model with an effective spin 1/2on a quasi two-dimensional square lattice . The discovery ofa low temperature d -wave gap and a splitting of the Fermisurface into so-called separated Fermi-arcs in electron doped Sr IrO , which are hallmarks of the doped cuprates , have re-cently been reported. These findings underscore the similarityof the the low-energy effective physics of Sr IrO and thatof the superconducting cuprates, and they encourage furtherresearch to elucidate the relationship between Mott physicsand superconductivity and to search for new routes to high- T c superconductivity.Infrared and optical spectroscopies provide valuable infor- mation about the low-energy excitations, charge dynamics,and electron correlations in this class of materials. The in-plane conductivity spectra of single crystals of Sr IrO havebeen systematically studied and show evidence of the co-operative electron correlation and SOC effects in the pres-ence of the orbital-dependent electron-phonon interaction. Tomake further inferences about the electronic structure and un-derlying interactions in the layered iridate Sr IrO , the in-terplane response needs to be carefully examined; likewisein the ruthenates and cuprates , where valuable informa-tion about the interplane coupling, phonon anomalies, thepseudo-gap phase and precursor Cooper-pair formation hasbeen drawn from studies of the interlayer electrodynamics.Furthermore, c -axis optical conductivity data, along withthe in-plane spectra, can be used to significantly constrain themodel parameters for band structure calculations, such as theon-site Coulomb interaction U .A thorough and reliable study of the interplane responseis impeded by the small size of the currently available crys-tals along the c -axis, orthogonal to the IrO planes. Theinterplane optical conductivity measured on a -axis-oriented Sr IrO epitaxial films is obscured by the substrate contri-bution and by the distorted electronic structure caused by theanisotropic biaxial strains . Instead, we have used an ar-ray of high-quality and well-aligned Sr IrO single crystalsstacked along the c -axis.In this paper, we report a comprehensive ellipsometricstudy of the dielectric function anisotropy of Sr IrO over awide range of photon energies, extending from the far-infrared(far-IR) into the ultraviolet (UV), and its interpretation basedon band-structure and lattice dynamics calculations. The pa-per is organized as follows. Section II describes experimentaland computational details. In Section III, the far-IR in- andout-of-plane phonon spectra are reported, followed by grouptheory analysis of the zone-center phonons and first-principleslattice dynamics calculations. The optical anisotropy of theinterband transitions is discussed in Section IV. In Section Vrelativistic calculations, which use the local spin density ap-proximation +U (LSDA+U) approach to account for the on-site Coulomb interaction U simultaneously with strong spin-orbit coupling, are reported in order to explain the observed a r X i v : . [ c ond - m a t . s t r- e l ] F e b anomalies and the anisotropy of the optical response. Finally,our conclusions are summarized in Section VI. II. EXPERIMENTAL AND COMPUTATIONAL DETAILS
High quality single crystals of Sr IrO were grown bya self flux method following Ref. 24. They crystallize inthe K NiF structure with lattice parameters a ≈ . ˚Aand c ≈ . ˚A. The plate-like crystals mechanically ex-tracted from the crucible had lateral dimensions of about . × mm in the ab -plane and thicknesses less than µmin the c -direction. In order to gain a sample thickness alongthe c -axis that is sufficient for optical spectroscopy we pre-pared stacks of about 10 to 15 individual single crystals gluedon top of each other by a minimal amount of GE-varnish. Thecrystals were co-aligned according to their in-plane crystallo-graphic axes using Laue x-ray back scattering. Subsequentlyone ac -face of the stack was polished with dry polishing pa-per. The in-plane optical data were obtained from individualas-grown plate-like crystals.We report spectroscopic ellipsometric data in a wide en-ergy range from meV to eV over temperatures T = Kto
K. For the IR range, we used home-built ellipsome-ters in combination with a Bruker IFS 66v/S and Vertex 80vFourier Transform IR spectrometers. Some of the experimentswere performed at the infrared beam line IR1 of the ANKAsynchrotron light source at Karlsruhe Institute of Technol-ogy, Germany. Spectra in the visible and UV range weremeasured with a Woollam VASE variable angle spectroscopicellipsometer equipped with an ultra-high vacuum cold-fingercryostat.Spectroscopic ellipsometry determines the complex re-flectance ratio ρ = r pp r ss = tan Ψ e i ∆ (1)where r pp and r ss denote the reflectance of p and s polarizedlight and are given by the Fresnel equations, from which thefull dielectric response ε ( ω ) is extracted.For the calculations we used the low temperature exper-imental structural data according to Ref. 25. To calculatethe phonon spectrum, we employed scalar relativistic densityfunctional perturbation theory as implemented in quantumespresso . We used ultrasoft pseudopotentials , the gen-eralized gradient approximation and set the wave function(charge density) plane wave cutoff to 80 Ry (960 Ry), respec-tively. The initial structure was optimized to have a stressbelow 0.1 kbar and residual forces per atom smaller than0.1 mRy/Bohr prior to the lattice dynamics calculation.The relativistic band structure calculations were performedwithin the local spin density approximation (LSDA) us-ing the linear muffin-tin orbital (LMTO) method . TheCoulomb interaction of Ir
5d electrons in the presence ofstrong SOC was taken into account using the rotationally in-variant LSDA+U method . The on-site Coulomb repulsion U was varied in the range from 1.15 eV to 2.15 eV and Hundscoupling J H = 0 . eV was fixed to the value estimated from LSDA. Thus, the parameter U eff = U − J H , which crudelydetermines the splitting between the lower and upper Hub-bard bands, varied between 0.5 and 1.5 eV. Since the calcu-lated optical conductivity does not show any significant de-pendence on the actual spin orientation within the antiferro-magnetic phase, we assumed collinear antiferromagnetic or-der in ab -plane with Ir moments aligned along the c -axis. III. PHONON SPECTRUM
Panels (a) and (b) of Fig. 1 show the ellipsometric an-gles tan Ψ( ω, θ, T ) and cos ∆( ω, θ, T ) obtained from the ab -plane in the far-IR and mid-IR spectral range up to meV( cm − ). Panels (c) and (d) exhibit the corresponding ac -plane response, where the c -axis was co-aligned with theplane of light incidence as sketched in panel (a) and (c). Far-IR (open symbols) and mid-IR (closed symbols) data weretaken at different angles of incidence and overlap in the spec-tral range from cm − to cm − . While there is cer- FIG. 1. Ellipsometric angles tan Ψ and cos ∆ in the far-infraredspectral range measured on (a,b) the ab -plane and (c,d) the ac -plane,with the c -axis aligned in the plane of incidence at T = 10 K (grey)and T = 300 K (red). Solid lines are the results of fits to model cal-culations involving of multiple harmonic oscillators with Lorentzianlineshapes.
FIG. 2. Fitted far infrared phonon spectra. Real parts of the opticalconductivity σ and permittivity ε for (a,b) the ab -plane and (c,d) c -axis response, respectively, at selected temperatures. tainly a finite spread in the alignment of individual single crys-tals of the stacks prepared for the c -axis measurements thereis no significant “contamination” from the in-plane responseinto the out-of-plane response as already evident from the rawdata. For instance, there is no significant absorption in the ac -plane at the position of the highest energy ab -plane featurearound cm − . This underlines the validity of our stackingapproach.In the case of a system with uniaxial anisotropy the dielec-tric tensor ε ( ω ) has two complex eigenvalues – ε ab ( ω ) and ε c ( ω ) . In principle, two independent ellipsometric data sets on ab - and ac -faces, as presented here, allow for wavelength-by-wavelength numerical inversion of the corresponding Fresnelequations. However, due to numerical instability of the in-version process wherever the ellipsometric angles tan Ψ and cos ∆ approach their extreme values of [0 , and ± , respec-tively, which happens especially around sharp spectral fea-tures, we have fitted the full anisotropic data set at the sametime by two sets of harmonic oscillators with Lorentzian line- shapes: ε ab,c ( ω ) = ε ab,c ( ω ) + iε ab,c ( ω )= ε ab,c ∞ + (cid:88) j ∈ ab,c ∆ (cid:15) j Ω j Ω j − ω − iω Γ j , (2)where ∆ ε j , Ω j and Γ j denote the oscillator strength, centerfrequency and line width of the j -th phonon resonance, re-spectively, and ε ab,c ∞ the effectively constant contribution ofthe high energy interband transitions to the real part of thedielectric function in the far-IR range. For the highest en-ergy c -axis phonon at cm − we have to use a Voigt profile(that is a Lorentzian profile broadened by convolution with aGaussian with width Γ Gauss ) to account for the anomalouslylarge line width. This additional broadening might be causedby the stacking approach and polishing of the crystal stack.The results of the corresponding fits are shown as solid linesin Fig. 1 for temperatures T = 10 K and
K, respectively.Accordingly, the fitted complex dielectric response is shownin Fig. 2 for both the ab -plane and the c -axis also for interme-diate temperatures. The best fit parameters for T = 10 K aresummarized in Table I along with the ab -plane results fromMoon et al. , where we find good agreement.We unambiguously distinguish eight phonon resonances inthe ab -plane and four in the c -axis dielectric response in con-trast to six in-plane modes reported so far . The doublepeak structure located at cm − and cm − developsat low temperatures only and might be related to two addi-tional phonons. It can be clearly seen in the spectra presentedby Moon et al. but is not discussed there. The dip feature at cm − clearly seen in the raw data of the in-plane responseis a result from the ellipsometric measurement scheme andcorrectly modeled and reproduced by the lowest energy c -axisphonon at cm − . While the ab -plane phonons exhibit asmall line width Γ j , indicating a high single crystal quality, allfour c -axis resonances are considerably broader. This mightbe attributed to the stacking procedure of many single crystalswith enhanced contribution of the near-surface regions intothe c -axis optical response. Mechanical polishing as appliedhere can also induce strain effects in the surface layer whichmight significantly decrease the phonon lifetime , althoughthe large penetration depth of far-IR radiation (of the order ofseveral micrometers) increases significantly the bulk sensitiv-ity, hence averaging out pure surface effects.We find for both in- and out-of-plane response a nonzerobackground of absorption, which can be attributed to oxygendeficiency in the crystals . This might indeed have occurredunder the growth conditions applied here. This background ismodeled by a broad Lorentzian and shown as a shaded areafor the T = 10 K case in Fig. 2. It shows only moderatetemperature dependence between T = 200 K and
K.The temperature dependence of the resonant frequencies Ω j and corresponding line widths Γ j for a representative set of in-plane and all out-of-plane modes is shown in Fig. 3(a-d). Theresonance frequencies of the in-plane modes reproduce the re-ported behavior and the out-of-plane modes show qualita-tively similar characteristics: a regular anharmonic softeningby 1.5% at K of the modes at cm − and cm − , TABLE I. Best fit results for the phonon resonances in the far-IR spectral range of Sr IrO at T = 10 K with Ω j , ∆ ε j , Γ j being thecontribution to the static permittivity, resonance frequency and line width, respectively, according to the Lorentz oscillator model. AndIR active optical zone center phonon resonance frequencies Ω calc j from first-principle lattice dynamics calculations. The degree of c -axispolarization of the eigenvectors is given in the range [0 , .experiment calculation Ω j (cm − ) ∆ ε j Γ j (cm − ) Ω j a (cm − ) Ω calc j (cm − ) symmetry polarization ab -plane E u . . E u . .
33 4 . E u . .
31 2 . E u . .
07 2 . E u . .
17 4 . E u . . .
93 3 . E u . .
13 13 322 251 E u . .
17 16 338 298 E u . .
57 8 . E u . .
43 8 . E u . A u . c -axis
192 0 .
46 5 . A u . .
11 17 323 E u . .
53 13 374 A u . . b A u . a Ref. 11 b Γ Gauss = 24 cm − whereas the other two exhibit a small hardening upon heating.For the c -axis modes as expected from the already enhancedline width at low temperatures the increase of Γ j with ris-ing temperature is only moderate compared to their in-planecounterparts. FIG. 3. Temperature dependent phonon parameters of Sr IrO nor-malized to T = 10 K. (a,c) Normalized resonance frequencies Ω j and (b,d) line width Γ j for ab -plane and c -axis response, respec-tively. The gray lines in panel (a) are reproduced from Ref. 11. A space group analysis helps us to crosscheck our findings.The space group I /acd with the Wyckoff positions for Ir (8a), Sr (16d) and the 2 different oxygen sites (apical 16d andbasal 16f) allows four A u and twelve doubly degenerate E u infrared active optical phonons . All modes belonging tothe same irreducible representation (here either A u or E u )are allowed to mix in order to form the eigenmodes of the ionlattice excited by infrared photons. If an irreducible represen-tation embodies both in- and out-of-plane polarizations thisintermixing can lead to allowed phonons both in the in-planeand out-of-plane response. The large unit cell of four for-mula units, which adds up to 28 atoms per unit cell, and thereduced crystal symmetry makes a phonon calculation com-putationally expensive but indispensable for further insightson the lattice dynamics. Therefore we compare our experi-mentally extracted phonon parameters with results from lat-tice dynamics calculations. The set of calculated zone cen-ter phonon frequencies is summarized in Table I. We quantifythe degree of c -axis polarization of each phonon eigenmodeby looking at the projection p of the normalized eigenvec-tor (cid:126)e onto the ab -plane, e ab , and c -axis, e c , respectively, with p = 0 . . e c − e ab ) in the range [0,1].According to the calculated eigenvectors, we find that theset of four A u modes actually consists of three with pure c -axis polarization and one polarized in the ab -plane. Therespective mode patterns are depicted in Fig 4. The c -axis modes A u (1) to A u (3) lead to octahedron bendingand oscillations of the IrO octahedra against the Sr ions,well known from the high symmetry structure I /mmm . FIG. 4. Calculated eigenvectors of the infrared active optical phononmodes with A u symmetry at the Γ -point of I /acd structure. A u (4) , however, involves the planar oxygen atoms only andleads to a quasi-quadrupolar mode, which stems from thebackfolding of zone boundary modes (from the M point ofthe high symmetry structure I /mmm ) due to the rotation ofthe octahedra and concomitant enlargement of the unit cell.Thus for A u , there is actually no intermixing of the in- andout-of-plane contributions.In the group of the doubly degenerate E u modes, however,there is also one irreducible representation, which originatesfrom the planar oxygen atoms, that generates a displacementin the c -direction. This is an IR-inactive zone boundary modeof I /mmm folded back to the Γ point that tilts the oxygenoctahedra. It mixes with other E u in-plane modes which couldgive rise to a non-vanishing dipole moment along the c -axisfor several eigenmodes. In our experimental data, however,we find in total only four out-of-plane resonances. This sug-gests only a weak or even absent out-of-plane dipole momentin all but one of the E u phonon modes.With the highest phonon frequency at cm − the totalfrequency range of the calculation matches quite well the ex-perimental one. In qualitative agreement with the experimen-tal results, we also find the highest c -axis phonon at consider-ably lower frequency ( cm − ). The considerable numeri-cal discrepancy between the calculated and measured phononfrequencies is probably due to the fact that the material ismetallic in this calculation. The phonon frequency and os-cillator strength are therefore affected by charge screening,which is not present in the Mott-insulating compound.Following this analysis and in contrast to previous work we assign the experimental c -axis modes at , and cm − to be the one of A u symmetry and the one at cm − of E u type. The in-plane modes are of E u sym-metry, except of the highest energy one, that could be eitherrelated to the A u quasi-quadrupolar mode or the E u mode,which we find close by in energy in the calculation.In summary, the far-infrared phonon spectrum is consistentwith group symmetry considerations and lattice dynamics cal-culations. Distinct in- and out-of-plane spectra prove that thesample stack can be considered as a quasi-single domain interms of its optical response, which allows us to examine theanisotropic dielectric tensor also at higher photon energies. IV. OPTICAL ANISOTROPY
Figure 5 shows the real parts of the dielectric function andoptical conductivity in-plane ( ε a, and σ a, ) as well as out-of-plane ( ε c, and σ c, ) in the photon energy range from . eVto . eV. Since tan Ψ and cos ∆ are far from their extremawe apply the numerical inversion to correct for the anisotropy.In this energy range only moderate corrections to the absolutevalues of ε are introduced.First, we will focus on the in-plane response at photon en-ergies up to . eV. The measured spectra agree very wellwith the spectra previously reported in the spectral range upto eV by Moon et al. and Sohn et al. . We assign the lowenergy interband transitions accordingly as α and β . Above eV we find strong absorption setting in due to interband tran-sitions with a plateau-like feature around eV and a further in-crease at higher photon energies with another shoulder around . eV.Following Fermi’s golden rule one finds the following fre-quency dependence of the imaginary part of the dielectricfunction ε ( ω ) for photon energies just above the direct op-tical gap ∆ dir : ε ( ω ) ∝ ω − [ (cid:126) ω − ∆ dir ] / . There-fore we plot ( ε · ω ) in the inset of Fig. 5(a) and obtain ∆ dir = 0 . eV at T = 10 K of the linear fit. Both the ampli-tude and the temperature dependence of the gap are in goodagreement with the reported values .The out-of-plane response, however, shows remarkably dif-ferent behavior. While at high photon energies the opti-cal conductivity is almost identical to the in-plane responsewith very similar characteristic shoulder features around and eV, the two bands α and β at lower photon energies arestrongly suppressed with a remaining broad, hump-like back-ground extending down to low frequencies. Surprisingly,the temperature dependence of the c -axis response is ratherweak up to room temperature. This low-energy c -axis opti-cal response can be understood in terms of an indirect gap.Phonon assisted absorption across an indirect optical gap in-volves essentially two processes. When the photon is ab-sorbed an additional phonon can either be absorbed or emittedin order to fulfill energy and momentum conservation con-ditions. While the former strongly depends on the phonondensity and is therefore strongly suppressed at low temper-atures, the latter is stronger and weakly temperature depen-dent. For an indirect gap of two parabolic bands one expects ε ( ω ) ∝ ω − ( (cid:126) ω ± (cid:126) Ω ph − ∆ indir ) . As depicted in the insetof Fig. 5(b), this expression provides an excellent descriptionof the experimental data. In this way we estimate a low tem-perature indirect optical gap ∆ indir of about . eV. But theabsence of enhanced absorption at higher temperatures sug-gest additional absorption mechanisms, such as impurities, tobe at play.A similar trend has recently been reported on thin-filmsof Sr IrO epitaxially grown along the (cid:104) (cid:105) direction on LaSrGaO (100) substrates which also provides access tothe c -axis response by normal incidence transmission . Al-though lattice mismatch and inevitable bi-axial strain effectslead to a significant orthorhombic distortion and hence a shiftin the α and β bands to lower energies, the main features FIG. 5. Temperature dependence of the real parts of the optical conductivity σ and dielectric permittivity ε of Sr IrO in the spectralrange of 0.01 to . eV. (a,b) ab -plane and (c,d) c -axis optical response. α and β denote the low energy interband transitions between.Insets: (a) In-plane direct optical gap ∆ dir = 0 . eV extracted from ( ε ω ) , (b) c -axis optical response ( ε ω ) / and indirect optical gap ∆ indir ≈ . eV. – strong suppression of α and β followed by an upturn andshoulder around eV – are quite similar in both sets of mate-rials.In an ionic picture the origin of the optical anisotropy withrespect to the α and β interband transitions might be attributedto inter-site hopping of the excited electrons since on-site d - d transitions are forbidden by the dipole selection rules , butthis picture neglects the spatially extended nature and hy-bridization of the d valence states as evident from the rel-atively large electronic bandwidths W realized in d TMOs.To elucidate the complex pattern of interband transitions andoptical gaps we compare our experimental results with the op-tical conductivity from electronic band structure calculations.
V. BAND STRUCTURE CALCULATIONS
The electronic band structure in the energy range of − to eV and an expansion of the region around the Fermi energy[ − . eV, eV] are depicted in Fig. 6(a) and (b), respectively.In panel (b) the bands are decorated with circles proportionalin size to their orbital character projected onto the basis set ofthe Ir d / (blue) and d / (orange) states. As expected, wefind the upper Hubbard band with pure d / , i.e. J eff = 1 / ,character well separated from the lower J eff = 1 / Hubbardband and the J eff = 3 / states below the Fermi level [redand purple lines in Fig. 6(a)]. We adjusted U eff to match theposition of the low energy α and β transitions [inset of Fig. 7].Hence, the direct optical band gap is naturally found to beabout . eV in our calculations.Before we discuss the high energy features we will focuson the analysis of the low energy α and β double peak struc- ture of the in-plane response. The α band indeed stems fromtransitions between initial and final bands formed by pure J eff = 1 / states but only from restricted sections of the k space near zone boundaries, e.g., around the X -point or P - N high symmetry line [solid red arrows and gray shaded areasin Fig. 6(b)]. AFM order of Ir moments within ab -plane sta-bilized by the on-site Coulomb repulsion U causes openingof a gap near the zone boundary between two pairs of bandswhich are nearly degenerate in non-spin-polarized relativisticLDA calculations. These two pairs of bands show nearly par-allel dispersions which insures a high joint density of statesfor interband transitions responsible for the α band. This isin line with both previous theoretical dynamical mean-field and experimental photoemission results. Both find the high-est occupied states with J eff = 1 / character around the X -point, too.The β band located around eV, however, is more intri-cate. The occupied J eff = 1 / bands exhibit rather strongdispersion with the total width of about . eV. They cross J eff = 3 / bands so that near the Brillouin zone center along Γ – T – M line the bottom of the J eff = 1 / bands is buriedwell below the top of the J eff = 3 / ones. In agreement withprevious results we found the β band to have a dominantcontribution from transitions with J eff = 3 / initial states[green arrows]. However, in contrast to the previous inter-pretation based on a simplified band picture transitions from J eff = 1 / initial bands also contribute to the optical con-ductivity at ∼ . eV [dashed red arrows]. While the DMFTresults lead to the same conclusions, Kim et al. favor aninterpretation in terms of a Fano-type interference of the broad J eff = 1 / electron-hole continuum with an optically inac-tive so-called SO-exciton, i.e., a magnetically active mode FIG. 6. (a) Electronic band structure and respective dominant or-bital character from LSDA+U calculations with U eff = 1 . eV. Col-ors represent dominant orbital character as in the legend. (b) En-largement of the band structure in (a). The size of the blue andorange circles is proportional to the weight of the orbital charac-ter when projected onto d / and d / states, respectively. Thered and green arrows indicate J eff = 1 / → J eff = 1 / and J eff = 3 / → J eff = 1 / , respectively. (c) Brillouin zone of I /acd . (d) Sketch of the main contributions to the low energy α , β double-peak structure. FIG. 7. Calculated (a) out-of-plane and (b) in-plane optical conduc-tivity σ , zz and σ , xx , respectively, along with the experimental spec-tra at T = 300 K. Inset: Near-infrared in-plane double peak structurefor different values of on-site repulsion U eff . U eff = 1 . eV is chosento match the position of the peaks. found as a broad peak around . eV in resonant inelastic x-ray experiments. This depletes the optical excitation spec-trum in that energy range which leaves the two-peak struc-ture. Our calculations give the lower α -band about twice asstrong as the higher β -band while in the experimental spectrathe strength is approximately the same for both. Kim et al. observed a similar trend in their microscopic model calcula-tions when considering clusters of 4 Ir ions. They relate thisto interband mixing of J eff = 3 / and J eff = 1 / states whichreflects the itinerancy of the system, i.e., the hybridization of Ir d states via neighboring oxygen 2 p states.Furthermore, our calculations allow us to analyze the fullanisotropic optical response up to high energies. Fig. 7(a,b)presents the real parts of the calculated optical conductivity σ , zz (out-of-plane) and σ , xx (in-plane) in the spectral rangeup to eV along with the experimental data taken at T =300 K.Most noticeable is the large optical gap of about . eValong the c -axis [Fig. 7 (a)] followed by a weak band centeredaround eV and several stronger bands around 5.5 and eVin rather good overall agreement with the experimentally ob-tained c -axis response. For α and β we find the dipole matrixelements for c -axis polarization below eV either completelyvanishing as for the J eff = 1 / → J eff = 1 / transitionsor for J eff = 3 / → J eff = 1 / to be strongly suppressedin strength by roughly two orders of magnitude compared tothe in-plane response. Above eV interband transitions fromthe low lying oxygen p states into the unoccupied J eff = 1 / states set in. This matches the absorption edge we find in theout-of-plane response around eV.In the in-plane response [Fig. 7(b)] we identify three majorfeatures. Beside the discussed double peak structure the nextset of interband transitions sets in around . eV concurrentwith the c -axis response but at somewhat lower frequency thanin the experimental spectra. From there a plateau reaches outto about eV followed by a further rise peaking at eV withapproximately half the strength of the c -axis response. Theweak feature around . eV seen in the experiment might findits counterpart at eV in the calculation.Along the Brillouin zone boundaries, for example betweenthe P and N points (see Fig. 6), we find an indirect optical gapof about . eV. This could indeed enable phonon mediatedindirect transitions, which as second order processes are be-yond our calculations. These transitions may contribute to thehump in the c -axis response as well as the strongly temper-ature dependent absorption edge tail and far-IR background seen in the ab -plane. The latter has also been shown to be rel-evant for Sr Ir O , the narrow band gap bilayer analogue . VI. CONCLUSIONS
We demonstrate the feasibility of a stacking approach ofsingle crystals to extract the c -axis optical conductivity of Sr IrO as proven by a distinct phonon spectrum, which isin full accordance with lattice dynamics calculations. The ob-served uniaxial anisotropy in the infrared excitation spectrumis consistent with the suggested novel J eff = 1 / ground statewithin the LSDA+U band structure calculations. The absenceof the characteristic IR double peak structure in the out-of-plane response is explained by vanishing dipole matrix ele-ments. The additional information from the c -axis responseseverely constrains the parameter space for calculations. Ourcomprehensive investigation of the optical response of a pro-totypical spin-orbit Mott insulator thus provides an excellentbasis for experiments on doped layered iridates, which arepromising candidates for exotic ground states including un-conventional superconductivity. H. Takagi, Nat Mater , 179 (2007). A. P. Mackenzie and Y. 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