The high-energy anomaly in ARPES spectra of the cuprates-many body or matrix element effect?
E. D. L. Rienks, M. Ärrälä, M. Lindroos, F. Roth, W. Tabis, G. Yu, M. Greven, J. Fink
aa r X i v : . [ c ond - m a t . s up r- c on ] D ec The high-energy anomaly in ARPES spectra of the cuprates—many body or matrixelement effect?
E. D. L. Rienks, M. Ärrälä, M. Lindroos, F. Roth, W. Tabis, , G. Yu, M. Greven, and J. Fink Helmholtz-Zentrum Berlin, Albert-Einstein-Strasse 15,D-12489 Berlin, Germany Department of Physics,Tampere University of Technology,PO Box 692, FIN-33101 Tampere, Finland Center for Free-Electron Laser Science / DESY,Notkestrasse 85, D-22607 Hamburg, Germany School of Physics and Astronomy,University of Minnesota,Minneapolis, Minnesota 55455, USA AGH University of Science and Technology,Faculty of Physics and Applied Computer Science,30-059 Krakow, Poland Leibniz-Institute for Solid State and Materials Research Dresden,P.O.Box 270116, D-01171 Dresden, Germany (Dated: August 14, 2018)We used polarization-dependent angle-resolved photoemission spectroscopy (ARPES) to studythe high-energy anomaly (HEA) in the dispersion of Nd − x Ce x CuO , ( x = 0 . ). We have foundthat at particular photon energies the anomalous, waterfalllike dispersion gives way to a broad,continuous band. This suggests that the HEA is a matrix element effect: it arises due to a suppressionof the intensity of the broadened quasi-particle band in a narrow momentum range. We confirmthis interpretation experimentally, by showing that the HEA appears when the matrix element issuppressed deliberately by changing the light polarization. Calculations of the matrix element usingatomic wave functions and simulation of the ARPES intensity with one-step model calculationsprovide further proof for this scenario. The possibility to detect the full quasi-particle dispersionfurther allows us to extract the high-energy self-energy function near the center and at the edge ofthe Brillouin zone. PACS numbers: 74.25.Jb, 74.72.Ek, 79.60.âĹŠi
One of the unique assets of angle-resolved photoemis-sion spectroscopy (ARPES) is the ability to determinethe spectral function A ( ω, k ) in energy and momentumspace. The finite width and deviation of the dispersionfrom that calculated in an independent particle model areinterpreted in the majority of cases in terms of many-body effects [1]. In the cuprate high- T c superconduc-tors, various kinks in the dispersion have been discov-ered which were analyzed in terms of a coupling of thecharge carriers to bosonic excitations possibly mediat-ing high- T c superconductivity in these materials. Be-sides the kinks in the low binding energy ( E B ) region( E B ≤ E B = E H ' . eV the band appearsto bend sharply and seems to proceed almost verticallytowards the valence bands. This phenomenon has beentermed "waterfall" or high-energy anomaly (HEA) [2].The HEA has been observed in undoped cuprates [3] aswell as in their hole-doped [2, 4–13], and electron-dopedderivatives [4, 13–15]. In the latter two systems E H showsa d -wave momentum dependence being larger along thenodal direction and smaller near the antinodal point op-posite to the momentum dependence of the d -wave su-perconducting gap [6, 12, 14]. The values of E H exhibit a difference of ≈ I ( ω, k ) ∝ | M ( ω, k ) | A ( ω, k ) (1)where the matrix element M ( ω, k ) = h f | er | i i (2)is determined by the final state h f | , the initial state | i i ,and the dipole operator er ( e is the unit vector along thepolarization direction of the photons).Some ARPES studies pointed out that extrinsic ef-fects due to matrix element effects may explain theHEA [8, 10, 12] since changes of the waterfalllike disper-sion to a Y-shaped dispersion have been observed uponphoton energy variation or by changing the Brillouin zone(BZ). Thereupon a combination of extrinsic and intrinsiceffects have been invoked to explain the ARPES resultsof cuprates at high energies [11, 13, 33].In this Letter we address the controversy of the ex-planation of the HEA in terms of extrinsic or intrinsiceffects. We present a polarization and photon energy de-pendent ARPES study on the electron-doped cuprate [34] Nd − x Ce x CuO ( x = 0 . ) in several BZs. We find thatthe waterfalllike dispersion transforms into a normal, dis-persive band at certain photon energies. In addition, awaterfalllike dispersion can be induced in the intact bandby changing the polarization of the incoming photons.The results can be explained in terms of a wipe-out of theintensity of the broadened quasi-particle band in a par-ticular momentum range. Thus we give strong evidencethat the HEA is not caused by intrinsic many-body ef-fects but rather by extrinsic matrix element effects. Fur-thermore, the newfound ability to observe the dispersionin the entire BZ allows us to determine the mass renor-malization of the quasi-particle band relative to densityfunctional theory (DFT) calculations.The Nd − x Ce x CuO ( x = 0 . ) single crystal wasgrown in about 4 atm of oxygen using the traveling-solvent floating-zone technique, annealed for 10 h in ar-gon at 970 ◦ C followed by 20 h in oxygen at 500 ◦ C [35].The sample was antiferromagnetic with a Néel tempera-ture of about T N = 82 K [36]. ARPES measurementswere carried out at the synchrotron radiation facilityBESSY II using the UE112-PGM2a variable photon po-larization beam line and the " "-ARPES end stationequipped with a Scienta R8000 analyzer. All measure-ments were performed in the normal state at T = 50 K.The total energy resolution was set between 10 and 15meV, while the angular resolution was ≈ . ◦ . We pointout that the ARPES experiments were performed at rel-atively high photon energies (h ν = 50 to eV), a rangein which the cross section for Cu d state excitations is5 to 7 times larger than that for O p states [37]. Thecrystal was mounted on a 6-axis cryomanipulator allow-ing polar, azimuthal, and tilt rotation of the sample inultrahigh vacuum with a precision of . ◦ . The experi-mental geometry is shown in Fig. 1(a). The mirror plane (0 , , π ) − Γ − ( π, , b =( x, , z ) was turned into the scat-tering plane. In this way cuts shown in Fig. 1(b) parallelto the Γ − (0 , π ) direction for various k x values could be recorded by changing the polar angle from nearly normalincidence to more grazing incidence. In the chosen sam-ple orientation, the Cu 3 d x − y conduction band stateshave an even symmetry with respect to the scatteringplane as shown in Fig. 1(a). For non-zero photoemis-sion intensity on the mirror plane, the final state mustbe even with respect to this plane and the same holds forthe product of the dipole operator and the initial state.Therefore, for this sample orientation and for p -polarizedlight ( e parallel to the mirror plane, dipole operator even)the matrix element should be finite near the mirror plane,while for s -polarized light ( e perpendicular to the mirrorplane, dipole operator odd) the matrix element shouldvanish (see Eq. 2). In a study of the origin of the shadowFermi surface in cuprates it was shown that the intensityvanishes in a narrow range of ± ◦ when the transition issymmetry forbidden [38].ARPES intensity calculations were done fully rela-tivistically based on the Dirac equation [39]. One-stepmodel [40] and multiple scattering theory have been uti-lized, the latter being used also for the final states. Thepotential for Nd CuO has been calculated by usingself consistent electronic structure calculations with theKorringa-Kohn-Rostoker method [41, 42]. In the calcu-lations, many-body correlations were taken into accountby the complex self-energy function Σ . For the initialstate we used the experimental self-energy function de-rived from the ARPES experiments (see below). For thefinal state a constant value ℑ Σ f = 2 eV is used.In Fig. 1(c) and (d) we present the sum of ARPES in-tensity plots along the edges of the BZ recorded withright ( c + ) and left ( c − ) circularly polarized photons hav-ing two different photon energies. For h ν = 120 eV [seeFig. 1(c)] and k x = 5 π [cut E H = 0 . eV is detected. At this energy the normaldispersion transforms into a vertical waterfalllike disper-sion accompanied by a reduction of the intensity near k y = 0 . On the other hand, changing the photon en-ergy to h ν = 94 eV [see Fig. 1(d)] and k x = π [cut inFig. 1(b)] transforms the dispersive feature into a bandhaving a normal dispersion with a width at constant en-ergy which increases continuously with increasing bind-ing energy. Since we observe similar changes as a func-tion of photon energy for k x = nπ n = 1 , , and 5 (notshown) we conclude that the change of the spectra shownin Fig. 1(c) and (d) is caused by the variation of thephoton energy but not by the change of k x . Further-more, Nd − x Ce x CuO is essentially a two-dimensionalelectronic system [43] and therefore upon variation thephoton energy k ⊥ changes, but this should not changethe spectral function. Thus the drastic change of the in-tensity plots presented in Fig. 1(c) and (d) signals thatthe HEA is caused by matrix element effects but not bychanges of the spectral function.In Fig. 1 (e)-(g) we present similar energy distributionmaps but now recorded with the wave vectors k x = π x y (a) (b) detector45º h s p scattering plane k x k y k y ( π / a)k y ( π / a)–0.5 0.0 0.5–2 –1 0–0.4–0.20.0 –0.5 0.0 0.5 E – E F ( e V ) E – E F ( e V ) –2 –1 00.0–0.4–0.80.0–0.4–0.80.0–0.4–0.8–0.4–0.20.0 (c)(d)(e)(f)(g) (h)(i)(j)(k)(l) s
110 eV p
110 eV p
100 eV + + c –
94 eV + + c –
120 eV
Figure 1. (Color online) (a) Experimental geometry. (b) Cutsin the reciprocal space used in the present investigation. (c)Experimental ARPES intensity distribution maps recordedwith parameters given in the labels. (h)-(l) calculated ARPESintensities using the parameters of (c)-(g), respectively. [cut ν = 110 eV and p -polarization, near k y = 0 a normal dispersion is observed[see Fig. 1(f)]. Compared to the antinodal point, the bot-tom of the band has moved to higher E B , which is ex-pected from band structure calculations. The spectralweight of the band extends into the region of the non-bonding oxygen valence bands. On the other hand, near k y = − π , a HEA is observed. At the same photon en-ergy (h ν = 110 eV) but for s-polarization, we know thatthe intensity must vanish at the mirror plane ( k y = 0 ).In Fig. 1(e) we see that, when the matrix element is in-tentionally suppressed in this way, a HEA appears at E H = 0 . eV. Thus the existence of the waterfalllikedispersion can be unambiguously attributed to a van-ishing matrix element near k y = 0 . In the second BZwe detect a more normal dispersion. Furthermore, forthe photon energy h ν = 100 eV but for p -polarization,a HEA is observed in the first and the second BZ [seeFig. 1(g)]. Comparing the spectra presented in Fig. 1(f)and (g) which were measured both with the same ( p ) po-larization, the difference near k y = 0 cannot be explainedby an extinction of the matrix element due to a specificphoton polarization but by an extinction of the matrixelement near the ( k x , k y = 0) line for specific photon en-ergies. Comparing the intensities near k y = 0 presentedin Fig. 1(e) and (g), it is remarkable that the distancesbetween the waterfalllike dispersions in momentum spacealong k y is in the former about twice as large as in thelatter. We attribute the differences in the energies E H (changing from 0.4 eV to 0.55 eV) to the different extinc-tion ranges of the matrix elements. In Fig. 2(a) we plotthe bottom of the band along the Γ − ( π, direction,derived from fits of energy distribution curves at k y = 0 using spectra showing no HEA and compare this resultwith our density function theory (DFT) calculations.The interpretation of the ARPES results on the HEAin terms of matrix element effects is supported by a sim-ple calculation of the matrix element for the selected ge-ometry and sample orientation using atomic wave func-tions for the | d i initial state and h p | / h f | final states [44].As expected from the discussion above, these calcula-tions yield for s -polarization a vanishing matrix elementat the mirror plane, but for photons emitted at a finiteangle relative to the mirror plane the matrix element in-creases linearly with that angle. This means that forsmall angles the intensity should increase proportionalto k y which is in perfect agreement with a momentumdistribution curve at E B = 0 . eV of the ARPES datashown in Fig. 1(e) [see Fig. 2(b))]. One explanation toaccount for the observed hν -dependence would be thatfor particular photon energies only final states which areodd with respect to the mirror plane can be reached. Thiswould explain why for p -polarization the intensity is zeroin the mirror plane. The calculation predicts for finiteemission angles relative to the mirror plane a finite in-tensity proportional to k y due to even final states whichcontribute to the matrix element linearly with increasingangle. This would explain why for certain energies evenfor p -polarization a waterfalllike dispersion is observed[see Fig. 1(g)].The interpretation of the HEA in terms of extrinsicmatrix element effects is furthermore strongly supportedby the ARPES intensity calculations which are in quali-tative agreement with the experimental ARPES data [seeFig. 1(h)-(l))]. Part of the remaining differences are re-lated to the fact that for the non-bonding O 2 p bands,the same self-energy function as for the Cu d band hasbeen used which leads to an unphysical broadening of the –0.5 0.0E – E F (eV)anti-nodalnodal–0.4 –0.3 –0.2 –0.1 (d) Γ ( e V ) F (eV) I m Ʃ ( e V ) (c) anti-nodalnodal –0.3 0 0.3k y ( π / a) (b) PE I n t en s i t y ( a r b . un i t s ) (a) DFTTB0.0–0.4–0.8–1.2 E – E F ( e V ) x ( π / a) Figure 2. (Color online) (a) Bottom of the band along the(- π ,0)-( π ,0) direction between the center of the BZ and theantinodal point. Circles: ARPES data. Dashed line: tightbinding (TB) fit. Solid line: present density functional the-ory (DFT) calculations. (b) Momentum distribution curve ofthe data shown in Fig. 1(e) at E B = 0 . eV, symmetrizedrelative to k y = 0 . The blue line shows a quadratic momen-tum dependence. (c) Imaginary part of the self-energy ℑ Σ as a function of the binding energy near the nodal and theantinodal point. The lines are fits to the data (see text). (d)Width Γ of the spectral weight at the bottom of the bandnear the antinodal point [ k k = ( π, ] and at the cut throughthe nodal point [ k k = ( π , ]. Red region indicates the rangeof the existence of quasi-particles (Γ < E B ) . former bands.The presented ARPES data together with the support-ing calculations provide strong evidence that the HEA isnot related to an anomalous spectral function, i.e., tospecific many-body effects. Rather, it is caused by awipe-out of a broad intensity distribution of the spectralweight near particular high-symmetry lines due to theextinction of the matrix elements (see also Fig. 2 in thesupplement [44]), i.e., by extrinsic effects. We point outthat the formation of a waterfalllike dispersion requiresboth a vanishing matrix element and a strong broadeningof the spectral weight at higher binding energies. Thisexplains why the HEA has only been found in stronglycorrelated systems in which a strong broadening of the"bands" at higher binding energies due to high scatteringrates is present.The present work can explain various results presentedin the literature which were not understood previously.For example, the momentum dependence of E H , whengoing from the nodal to the antinodal point, previouslyobserved in La − x Sr x CuO [6] and Nd − x Ce x CuO [14]can be readily explained in terms of a wipe-out of the intensity of a normal quasi-particle band, in which thebottom of the band near Γ is deeper than near the antin-odal point. Furthermore, the difference of the E H valuesof ≈ p - and n -doped cuprates, which waslinked to the difference of the chemical potential [14] canbe now understood by a wipe-out of quasi-particle bands,ranging to different energies below the Fermi level.Since we can now follow the dispersion to the bottomof the band, we can derive reliable results for the massenhancement compared to DFT calculations. Using thedata presented in Fig. 2(a), we obtain a high-energy massrenormalization of 2.1 near the antinodal point while at Γ a value of 2.3 derived. We remark that these valuesare connected with large errors of about 30% since thereis a considerable scattering of the energy position of theCu-O conduction band relative to the Fermi level in thepublished DFT calculations. Moreover, we have analyzedthe energy-dependence of the imaginary part of the self-energy, extracted from momentum distribution curves,using the relation ℑ Σ = − A − BE α [see Fig 2(c)]. Nearthe antinodal point we obtain A = 0 . ± . eV, B = 0 . ± . eV − α and α = 1 . ± . , while nearthe nodal point A = 0 . ± . eV, B = 1 . ± . eV − α , and α = 2 . ± . . The observation of anearly quadratic increase as a function of energy at thenodal point indicates a Fermi liquid behavior and ex-plains the quadratic temperature dependence of the in-plane resistivity above the superconducting transitiontemperature [45]. The nearly linear increase at the antin-odal point signals the proximity to a marginal Fermi liq-uid [46]. Near the center of the BZ, at k k = ( π , thetotal life-time broadening amounts to 65 % of E B [seeFig. 2(d)]. This indicates that even at the bottom of theband, spectral weight of quasi-particles is observed andthat at these E B s we are not in the incoherent range assuggested previously in the literature [5, 18, 21, 26, 32].In addition, the lack of a high-energy kink and the con-tinuous increase of the width as a function of E B do notsupport any scenario of a strong coupling of the chargecarriers to discrete high-energy bosonic excitations suchas magnetic excitations with an energy of J ≈ . eV (Jis the exchange energy) [34] which could mediate high- T c superconductivity [9, 17, 19, 29, 33]. Rather, the resultspresented here indicate a strong coupling of the chargecarriers to low-energy electronic excitations, e.g. spin ex-citations between regions near the antinodal points lead-ing there to a marginal Fermi liquid behavior as describedby [47, 48]. Finally, we postulate that the present resultscan be generalized to all cuprates since in the p -dopedcompounds very similar ARPES results, e.g. energy de-pendence of E H , have been obtained [6].To summarize, our ARPES results on an n -dopedcuprate together with a calculation of the ARPES inten-sity in a one-step model clearly show that the HEA is notrelated to an intrinsic anomalous dispersion of the spec-tral weight. Rather, it is exclusively caused by a combina-tion of a wipe-out due to matrix element effects and highscattering rates at high energies. By selecting suitablephoton energies we are able to obtain important infor-mation on the many-body properties of doped cuprateswhich places strong constraints on theories of high- T c superconductivity in these systems. 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E. D. L. Rienks, M. Ärrälä, M. Lindroos, F. Roth, W. Tabis,
4, 5
G. Yu, M. Greven, and J. Fink Helmholtz-Zentrum Berlin, Albert-Einstein-Strasse 15, D-12489 Berlin, Germany Department of Physics, Tampere University of Technology, PO Box 692, FIN-33101 Tampere, Finland Center for Free-Electron Laser Science / DESY, Notkestrasse 85, D-22607 Hamburg, Germany School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA University of Science and Technology, Faculty of Physics and Applied Computer Science, 30-059 Krakow, Poland Leibniz-Institute for Solid State and Materials Research Dresden, P.O.Box 270116, D-01171 Dresden, Germany
In this supplement, we provide angle-dependent calculations of the matrix element for photoe-mission experiments described in the main paper using atomic orbitals for the Cu 3 d initial and thedipole allowed Cu 2 p or 4 f final states. When the matrix element is symmetry-forbidden on the mir-ror plane, it is found to increase linearly with increasing emission angle relative to this plane. Thusthe ARPES intensity should increase quadratically with the distance k y from the mirror plane. Thisexplains the wipe-out near the x − z plane for s -polarization and thus the waterfalllike dispersion. PACS numbers: 74.25.Jb, 74.72.Ek, 79.60.i
In ARPES the measured photocurrent is given by I ( ω, k ) ∝ | M ( ω, k ) | A ( ω, k ) (1)where A ( ω, k ) is the spectral function and the matrixelement M ( ω, k ) = h f | er | i i (2)is determined by the final state h f | , the initial state | i i ,and the dipole operator er ( e is the unit vector along thepolarization direction of the photons) [1]. In the follow-ing we calculate the matrix elements for the geometryand sample orientation described in the main paper us-ing atomic initial and final states. The dipole operatorfor s - and p -polarization is proportional to a linear com-bination of Θ and ϕ dependent spherical harmonics by Y − Y − ∝ sin (Θ) and Y − Y − ∝ cos (Θ) , respectively. Θ denotes the angle in the z − y plane perpendicular tothe mirror plane, while ϕ is the angle around the z axis.The Θ -dependence of the matrix element is given by therelation M ( f, i, Θ) ∝ π Z Y m f l f ( Y ± Y − ) Y m i l i sin θdθ (3)The initial state is a Cu 3 d x − y state, hence the ini-tial state is determined by l = 2 , m = ± and therefore h i | ∝ Y ± ∝ sin ( θ ) . According to the dipole selectionrules ∆ l = ± and ∆ m = 0 , ± excitations into h p | and h f | final states can occur. In the case of p -polarization al-lowed final states are Y ± ∝ sin ( θ ) which is even with re-spect to the mirror plane. For p -polarization (even dipoleoperator) the matrix element is finite since both the ini-tial and the final states are even. On the other hand,for s -polarization (odd dipole operator) the matrix ele-ment vanishes since both the initial state and the finalstate are even. In this case when the emitted photons Figure 1. Calculated squared matrix element M versus thewave vector k y perpendicular to the mirror plane. have a small angle δ versus the mirror plane, the finalstates are proportional to sin (Θ + δ ) ≈ sin (Θ)+ δcos (Θ) .The second term now adds an odd contribution to the fi-nal state and we will have a finite matrix element whichis proportional to δ . Thus for small δ the intensity is I ( δ ) ∝ M ∝ δ ∝ k y . In Fig.1 we show the calculatedsquared matrix element M versus k y . A similar calcula-tion for transitions into h f | final states results in a matrixelement which is also proportional to k y .In Fig. 2(d) of the main paper we show the intensityalong k y from a momentum distribution curve at a bind-ing energy of 0.5 eV of the ARPES intensity distributionmap shown in Fig. 1(e) of the main paper. At small k y the intensity agrees perfectly with a quadratic depen-dence on k y indicating that the appearance of the water-falllike dispersion can be explained in terms of a simplecalculation of the matrix element using atomic initial andfinal state wave functions.In Fig. 2(a) we show the calculated intensity derivedfrom a spectral function using for convenience a parabolicband and a complex self-energy function [which shows no -0.4 -0.2 0.0 0.2 0.4 Wave vector ( π /a) -0.6-0.5-0.4-0.3-0.2-0.10.0 E ne r g y ( e V ) (a) -0.4 -0.2 0.0 0.2 0.4 Wave vector ( π /a) -0.6-0.5-0.4-0.3-0.2-0.10.0 E ne r g y ( e V ) (b) Figure 2. (Color online) (a) Calculated ARPES intensity dis-tribution map using a parabolic band together with a self-energy without a high-energy anomaly derived from the ex-periment. (b) The same intensity distribution map as in (a)but multiplied by a matrix element which is quadratic in thewave vector. A waterfalllike dispersion is produced. high-energy anomaly (HEA)] taken from the experiment at the nodal point which was mentioned in the main pa-per. A Shirley background [2] was added. Multiplyingthe spectral function with a matrix element which is pro-portional to k y we obtain the intensity distribution mapshown in Fig. 2(b) which shows a HEA. This representsa further indication that the waterfall like dispersion iscaused by matrix element effects.Finally we discuss the energy dependence of theARPES intensity for p -polarized photons using thepresent calculation of the matrix element [see Fig. 1(f)and (g) in the main paper]. If the final state is evenand thus proportional to sin Θ the three terms of in theintegral over θ are even and thus the matrix element isfinite yielding a normal dispersion without HEA. Uponchanging the photon energy, at particular energies pre-dominantly odd final states could be reached which areproportional to cos Θ . One explanation to account forthe observed hν dependence would be that for particularphoton energies only odd final states are reached. Thenfor p -polarized photons and photoelectrons emitted inthe mirror plane the matrix element is zero. Detectingelectrons which are emitted at a small angle δ relativeto the mirror plane, the final state is now proportionalto cos (Θ + δ ) ≈ cos θ + δsinθ . The second term yieldsan even contribution to the final state proportional to δ .Thus the finite matrix element increases proportional to δ and the intensity proportional to δ causing a HEA sim-ilar to the case for s -polarized photons. In this way it ispossible to explain the energy dependence of the ARPESintensity, shown in Fig. 1(f) and (g) in the main paper,by an energy dependent change of the parity of the finalstates. [1] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod.Phys. , 473 (2003).[2] D. A. Shirley, Phys. Rev. B5