Polarization-dependent infrared reflectivity study of Sr 2.5 Ca 11.5 Cu 24 O 41 under pressure: Charge dynamics, charge distribution, and anisotropy
S. Frank, A. Huber, U. Ammerahl, M. Huecker, C. A. Kuntscher
aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Polarization-dependent infrared reflectivity study of Sr . Ca . Cu O under pressure:Charge dynamics, charge distribution, and anisotropy S. Frank , A. Huber , U. Ammerahl , M. Huecker , and C. A. Kuntscher , ∗ Experimentalphysik 2, Universit¨at Augsburg, D-86195 Augsburg, Germany Laboratoire de Physico-Chimie de L ´Etat Solide, ICMMO,UMR 8182, Universit´e Paris-Sud, 91405 Orsay, Cedex, France and Condensed Matter Physics and Materials Science Department,Brookhaven National Laboratory, Upton, New York 11973, USA (Dated: October 26, 2018)We present a polarization-dependent infrared reflectivity study of the spin-ladder compoundSr . Ca . Cu O under pressure. The optical response is strongly anisotropic, with the high-est reflectivity along the ladders/chains ( E k c) revealing a metallic character. For the polarizationdirection perpendicular to the ladder plane, an insulating behavior is observed. With increasingpressure the optical conductivity for E k c shows a strong increase, which is most pronounced below2000 cm − . According to the spectral weight analysis of the E k c optical conductivity the holeconcentration in the ladders increases with increasing pressure and tends to saturate at high pres-sure. At ∼ δ =0.09( ± . Ca . Cu O remains electronically highly anisotropic up to high pressure, also at lowtemperatures. PACS numbers: 78.20.-e,78.30.-j,62.50.-p,71.45.Lr
I. INTRODUCTION
The quasi-one-dimensional spin ladder compoundsSr − x Ca x Cu O have been studied extensively dueto the emergence of superconductivity for high Cacontent and high pressure. The theoretically pre-dicted superconducting state was first observed inSr . Ca . Cu O . below T c =12 K and pressures ≥ The crystal structure of Sr − x Ca x Cu O consists of two types of copper oxide layers that are par-allel to the crystallographic a-c plane and alternate alongthe b axis: the Cu O planes which contain the two-legladders, and CuO planes containing chains with edge-shared CuO plaquettes. At ambient conditions the par-ent compound Sr Cu O has an intrinsic hole dopingof six holes per formula unit, which results in a aver-age Cu valence of +2.25. Although the substitution ofSr by isovalent Ca does not change the intrinsic chargeconcentration in the system, the physical properties alterdrastically, which has been attributed to a redistributionof hole carriers among the ladder and chain subsystems. Among the key issues for understanding the mecha-nism of superconductivity in Sr − x Ca x Cu O are thedistribution of charge carriers among the ladders andchains, and the dimensionality of the system. Both as-pects can be addressed by polarization-dependent in-frared spectroscopy, which is also applied in the presentstudy. According to optical studies for the undoped par-ent compound Sr Cu O , at ambient conditions onehole resides on the ladders and five holes on the chains. These values are close to those found by XAS and NMRexperiments.
Ca doping changes the carrier distribu-tion as was recently discussed and summarized in Ref.10:Despite some discrepancies in the absolute number of holes, it is now generally accepted that Ca doping trig-gers a chemical pressure induced transfer of holes fromthe chains to the ladders.Besides a high Ca content, high pressure is needed toinduce superconductivity in Sr − x Ca x Cu O . The pressure-dependent charge distribution inSr − x Ca x Cu O has been studied by NMR mea-surements for x =0 and x =12 for pressures up to3.2 GPa. According to this study the hole concentrationon the ladders increases by ∆ δ ≈ Ca Cu O , when a pressure of 3.2 GPa is applied.Furthermore, pressure-dependent electrical transportmeasurements on Sr . Ca . Cu O suggest that thesuperconducting state has a quasi-two-dimensionalcharacter. In the present study we investigate the charge distri-bution in Sr . Ca . Cu O up to a high pressure of7.5 GPa by infrared spectroscopy. Additionally, we ad-dress the dimensionality of the system under pressureby presenting polarization-dependent infrared spectra atroom temperature and at low temperature. II. EXPERIMENT
The studied Sr . Ca . Cu O single crystal wasgrown by the traveling-solvent floating-zone. Thepolarization-dependent room-temperature reflectivity asa function of pressure was measured over a broad fre-quency range (300 - 12000 cm − ) using a Bruker IFs66v/S Fourier transform infrared spectrometer. Twodiamond-anvil pressure cells were employed for pres-sure generation: A clamp diamond-anvil cell (DiacellcryoDAC-Mega) and a Syassen-Holzapfel diamond-anvil R e f l e c t i v i t y R s - d E IIc E IIa E IIb E IIc O p t i c a l c ondu c t i v i t y s ( W - c m - ) E IIa E IIb
400 5000.00.20.4 R e f l e c t i v i t y Energy (cm -1 )
400 50004080 O p t i c a l c ondu c t i v i t y Energy (cm -1 ) Energy (eV)
Energy (eV) (a)(b)(c) (d)(e)(f) Energy (cm -1 )
300 10 Energy (cm -1 ) (c) (f) FIG. 1: (a)-(c): Reflectivity spectra R s − d of Sr . Ca . Cu O for the polarization E of the radiation along the three crystalaxes at room temperature. (d)-(f): Real part of the optical conductivity σ obtained from the Drude-Lorentz fitting of thereflectivity spectra. cell (DAC). Finely ground CsI powder served as quasi-hydrostatic pressure transmission medium to ensure di-rect contact of the sample with the diamond anvil. Thesample size was 80 x 80 µ m for the high-frequencyrange and about 200 x 200 µ m for frequencies below700 cm − to avoid diffraction effects. To focus the in-frared beam onto the small sample in the pressure cell,a Bruker IR Scope II infrared microscope with a 15 × magnification objective was used. Spectra taken at theinner diamond-air interface of the empty cell served asthe reference for normalization of the sample spectra(see Ref.16 for an illustration of the measurement geom-etry). The absolute reflectivity at the sample-diamondinterface, denoted as R s − d , was calculated according to R s − d ( ω ) = R dia × I s ( ω ) /I d ( ω ), where I s ( ω ) denotes theintensity spectrum reflected from the sample-diamond in- terface and I d ( ω ) the reference spectrum of the diamond-air interface. For R dia a value of 0.167 was calculatedfrom the refractive index of diamond n dia , which is as-sumed to be independent of pressure.The polarization-dependent reflectivity measurementsfor frequencies 780 - 6000 cm − at low temperature andhigh pressure were carried out using a home-built in-frared microscope coupled to the FTIR spectrometer andmaintained at the same vacuum conditions, in order toavoid absorption lines of H O and CO molecules. De-tails about the home-built infrared microscope can befound in Ref.17. A Syassen-Holzapfel DAC for thepressure generation was mounted in a continuous-flow he-lium cryostat (Cryo Vac KONTI cryostat). More detailsabout the geometry of the reflectivity measurements canbe found in our earlier publications. As reference,
Energy (cm -1 ) O p t i c a l c ondu c t i v i t y s ( W - c m - ) Energy (eV) E IIc
P=1 GPa
FIG. 2: Real part of the optical conductivity σ for E k c atthe lowest applied pressure at room temperature and the var-ious contributions (Drude term, MIR band, phonon modes)as obtained from the Drude-Lorentz fits. M I R - band po s i t i on ( c m - ) Pressure (GPa)
Energy (cm -1 ) O p t i c a l c ondu c t i v i t y s ( W - c m - ) MIR-Band
Energy (eV) E IIc
FIG. 3: MIR band as a function of pressure at room temper-ature, as obtained from the Drude-Lorentz fits of the reflec-tivity spectra. Inset: Position of the MIR band as a functionof pressure. we used the intensity reflected from the silver coatedsteel gasket inside the DAC. Correspondingly, the ab-solute reflectivity at the sample-diamond interface R s − d was calculated according to R s − d ( ω ) = I s ( ω ) /I Ag ( ω ),where I s ( ω ) denotes the intensity spectrum reflected fromthe sample-diamond interface and I Ag ( ω ) the referencespectrum of the diamond-silver interface. All reflectivityspectra shown in this paper refer to the absolute reflec-tivity at the sample-diamond interface R s − d . The pres-sure in the DAC was determined in situ by the standardruby-fluorescence technique. III. RESULTS AND ANALYSIS
The pressure-dependent reflectivity spectra ofSr . Ca . Cu O at room temperature are depictedin Fig. 1(a)-(c) for the polarization direction alongthe ladders/chains ( E k c), along the rungs ( E k a), andperpendicular to the ladder plane ( E k b), respectively.Features in the frequency range 1700 - 2700 cm − areartifacts originating from multiphonon absorptions inthe diamond anvils, which are not fully corrected bythe normalization procedure, and are not considered inthe following analysis of the spectra. The polarization-dependent reflectivity spectra at the lowest appliedpressure clearly reveal the electronic anisotropy ofSr . Ca . Cu O , consistent with earlier results :For E k c, i.e., along the ladders and chains, the overallreflectivity is high, whereas for E k a it is considerablylower, but still reveals a metallic behavior (see alsothe corresponding optical conductivity as describedbelow). For E k b a typical insulating behavior is found,with an overall low reflectivity and strong phononexcitations in the low-frequency range. As comparedto the results from ambient-pressure measurements onSr . Ca . Cu O the observed phonon modes havea lower intensity and are broadened. According tothe pressure-dependent reflectivity data the electronicanistropy of the sample is preserved up to the highestapplied pressure ( ≈ E k c the reflectivityincreases monotonically with increasing pressure. Alsofor E k a a gradual increase is found, which is howevermuch lower as compared to the E k c direction. Alongthe insulating E k b direction the reflectivity does notchange with pressure, except the alterations related tothe phonon mode excitations.The real part of the optical conductivity σ was ob-tained by fitting the reflectivity spectra with the Drude-Lorentz model combined with the normal-incidenceFresnel equation R s − d = (cid:12)(cid:12)(cid:12)(cid:12) n dia − √ ǫ s n dia + √ ǫ s (cid:12)(cid:12)(cid:12)(cid:12) , ǫ s = ǫ ∞ + iσǫ ω , (1)where n dia is the refractive index of diamond and ǫ s thecomplex dielectric function of the sample. ǫ ∞ is the op-tical dielectric constant. The so-obtained optical con-ductivity spectra are shown in Fig. 1(d)-(f) for the threepolarization directions. The optical conductivity is high-est for E k c and the metallic character is clearly revealedby a Drude contribution. The various contributions tothe optical conductivity are illustrated in Fig. 2: Besidesthe Drude term a pronounced mid-infrared absorptionband (MIR band) centered at around 2000 cm − andphonon excitations are found. For E k a the conductivityat the lowest frequencies is finite indicating a metalliccharacter. At around 5000 cm − an absorption bandis observed followed by the onset of higher-frequency ex-citations. For E k b the low-energy optical conductivitycontains phonon excitations besides higher-frequency ex- F r equen cy ( c m - ) Pressure (GPa) E IIa E IIb F r equen cy ( c m - ) Pressure (GPa) (a) (b)
FIG. 4: Frequencies of the phonon modes as a function of pressure at room temperature, as obtained from the Drude-Lorentzfits of the reflectivity spectra, for (a) E k a and (b) E k b. Energy (eV) E IIc R e f l e c t i v i t y E IIa R e f l e c t i v i t y (a)(b) Energy (cm -1 ) (b) FIG. 5: Reflectivity spectra of Sr . Ca . Cu O for (a) E k cand (b) E k a up to 15 GPa at room temperature. citations with the main spectral weight above the mea-sured frequency range.With increasing pressure a strong increase in the E k coptical conductivity is observed, which is most pro-nounced below 2000 cm − [see Fig. 1(d)]. Based on theDrude-Lorentz fit of the pressure-dependent reflectivityspectra, the MIR band was extracted and is depicted inFig. 3. From the maxima we have estimated the energy position of the MIR band, which we plot in the inset ofFig. 3 as a function of pressure. One can see that with in-creasing pressure the MIR band shows a red shift. Simul-taneously, its spectral weight increases. For the polariza-tion E k a the overall optical conductivity increases withincreasing pressure [see Fig. 1(e)]. The frequencies of thephonon modes were obtained from the Drude-Lorentz fitsand are depicted in Fig. 4(a). The phonon modes show aslight hardening with increasing pressure. The mode at530 cm − splits into two modes above 4.5 GPa. For E k bthe overall optical conductivity is pressure-independent,except for the low-frequency range, where the phononmodes are observed [see Fig. 1(f) and its inset]. Thepressure dependence of the phonon frequencies from theDrude-Lorentz fits are plotted in Fig. 4(b). The phononmodes harden with increasing pressure. Above 2 GPaadditional, but weak phonon modes appear, and above4.5 GPa the phonon mode close to 400 cm − splits. Ingeneral, a pressure-induced symmetry change of the crys-tal structure will modify the phonon spectrum - with typ-ical signatures being anomalies in the pressure-inducedfrequency shifts, mode intensity or width, or splitting ofmodes. In the case of Sr . Ca . Cu O the new highpressure modes appear as shoulders of the strong modespresent at ambient conditions. These additional modesmight already be present at ambient conditions and theirintensity just increases upon pressure application due tocharge redistribution, like transfer of charges from thechains to the ladders; therefore, we hesitate to inter-pret them in terms of pressure-induced crystal symme-try changes. Also the non-hydrostatic components of thepressure in the DAC might play a role. Furthermore, it isdifficult to attribute the observed phonon modes to Cu–O vibrations of either chains or ladders, since the bondlengths are similar. In conclusion, the pressure depen-dence of the infrared-active phonon modes support thefindings of x-ray diffraction measurements, which did notfind indications for a pressure-induced structural phasetransition up to 9 GPa. Additional information on the pressure dependence ofthe electronic properties of Sr . Ca . Cu O has beenobtained by reflectivity measurements within the ladderplane for higher pressures, i.e., up to 15 GPa [see Fig.5]. Along both E k c and E k a directions the reflectiv-ity spectra barely change for pressures above ∼ IV. DISCUSSIONA. Pressure-induced hole transfer onto the ladders
The high conductivity in Sr − x Ca x Cu O along the c direction has been attributed to the charge carriers ofthe ladder subunits. Optical studies of Sr Cu O at ambient conditions found an intrinsic hole doping ofsix holes per formula unit, with one hole in the Cu O ladders and five holes in the CuO chains at ambientconditions. This distribution of holes among laddersand chains is close to the one found by XAS and NMRexperiments.
With increasing Ca doping the totalcarrier concentration in the material is conserved, butthe physical properties change drastically, which was at-tributed to a redistribution of charge carriers from thechains to the ladders with increasing Ca content.
Because the ionic radius of Ca is smaller than that ofSr, the lattice parameter b decreases for increasing Cacontent. Since a similar effect occurs when externalpressure is applied, the substitution of Sr by Ca canbe interpreted in terms of a chemical pressure. Thus onemay ask, whether the application of external pressurecauses a similar charge carrier redistribution.The real part of the optical conductivity ofSr . Ca . Cu O for E k c (see Fig. 2) consists of aDrude term and an MIR band. Upon pressure applica-tion an increase in the optical conductivity is observed,which is most pronounced for the frequency range belowabout 2000 cm − . Obviously, this increase suggests aredistribution of spectral weight from high to low fre-quencies. According to the sum rule (see below) thespectral weight is a measure of the effective charge car-rier concentration N eff . The excitations below 1.2 eV( ≈ − ) can be attributed to the ladders, whereasthose above 1.2 eV are related to the chains. Therefore,the increase in the low-frequency optical conductivity isrelated to a transfer of holes from the chain to the laddersubsystem.By applying the sum rule the number of charge car-riers in the ladders can be calculated from the opticalconductivity spectrum according to N eff ( ω ) = 2 m ∗ Vπe Z ω ω σ ( ω ′ ) dω ′ , (2)where m ∗ is the effective mass of the charge carri-ers, V the pressure-dependent volume of the unit cell, V o l u m e ( Å ) Pressure (GPa)
FIG. 6: Estimated unit cell volume of Sr . Ca . Cu O as a function of pressure at room temperature. and ω =300 cm − , i.e., the lowest measured frequency.Hereby, we assume that only the holes on the ladder sub-units are mobile, i.e., the spectral weight for the conduct-ing E k c direction in the frequency range up to ω =1.2 eVis related to the holes of the ladders. Hence the cal-culated N eff gives the effective carrier concentration inthe ladders per Cu atom δ according to N eff = A δ , where A is a constant determined according to literature data(see details given below).To be able to calculate the effective charge car-rier concentration N eff the pressure-dependent vol-ume of the unit cell must be known. Since corre-sponding data are not available for the studied com-pound, we made use of the linear pressure coeffi-cients for the lattice parameters of the closely re-lated compound Sr . Ca . Cu O , for which val-ues of α = 0,0093 ˚A/GPa, β = 0,0823 ˚A/GPa, and γ = 0,0074 ˚A/GPa, respectively, can be found in theliterature. These values were applied to the lattice pa-rameters of Sr . Ca . Cu O . The so-obtained volumeof the unit cell is depicted in Fig. 6.To obtain the effective carrier concentration in the lad-ders per Cu atom, δ , from the spectral weight accord-ing to N eff = A δ , the coefficient A needs to be deter-mined. This parameter was obtained based on publishedresults for Sr Cu O , for which NEXAFS and opticalmeasurements found one hole per formula unit on theladders. With 14 Cu atoms on the ladder units perunit cell, this gives δ = = 0.07 (corresponding ladderCu Valence of +2.07). From the spectral weight anal-ysis of the optical data of Osafune et al. we obtainN eff ≈ N eff δ = 2.12.For consistency check, the above-described proce-dure has been applied to our lowest-pressure data ofSr . Ca . Cu O , which should deviate only slightlyfrom the ambient-pressure data. We find V = 549,26 ˚A and the spectral weight 8.93 · Ω − cm − at ω =1.2 eV,which gives N eff ( ω ) = 0.523 according to Equ. (2).From this value the number of holes in the ladders perCu atom is calculated according to δ = N eff A = 0.25 ± (a) (b) N e ff ( pe r l adde r- C u ) Energy (eV) E IIc
Energy (cm -1 ) C u v a l en c e ( l adde r s ) Pressure (GPa) D d FIG. 7: (a) Effective charge concentration N eff per Cu atom obtained according to Equ. (2). (b) Cu valence in the ladders asa function of pressure. At high pressure the increase of holes in the ladders saturates. Inset: Relative increase of the numberof holes per Cu atoms in the ladders. sult is in agreement with Ca doping dependent infraredmeasurements, where for Sr Ca Cu O a hole con-centration of 0.2 holes per Cu atom on the ladders (cor-responding ladder Cu valence: +2.20) was found, keepingin mind the slightly higher Ca content and the pressureoffset of 1 GPa in our measurement.The resulting effective charge concentration N eff perCu atom and the Cu valence of the ladder subunits asa function of applied pressure are depicted in Fig. 7 (a)and (b). With increasing pressure the Cu valence andthe related number of holes in the ladders increases andtends to saturate at high pressure. At P ∼ ± Ca Cu O observed a pressure-induced increase of∆ δ ≈ ∼ which compares well with thevalue ∆ δ = 0.04 ± > E k c and E k a up to ∼ c direction and theassociated concentration of charge carriers on the laddersare approximately constant above ∼ B. Nature of the MIR band
The optical conductivity spectrum ofSr . Ca . Cu O for the polarization E k c con-sists of a pronounced MIR band, whose origin willbe discussed in the following based on its pressuredependence. Photoemission experiments at 130 K foundan electronic band located at ∼ − ) forSr Cu O , which shifts toward the Fermi energy withincreasing Ca content. The band was interpretedin terms of strong electronic correlations. Within thesingle-band Hubbard model for correlated electronsystems characteristic contributions are expected in theoptical conductivity spectrum.
For the metallicsolution of the Hubbard model a Drude term and anexcitation band in the mid-infrared frequency range areobserved in the optical conductivity spectra, consistentwith experimental findings.
These contributionsare also present in the optical conductivity spectrumof Sr . Ca . Cu O (see Fig. 2). With increasingpressure the MIR band shifts to smaller frequencies,which is consistent with a pressure-induced bandwidthincrease. Although the spectral weight of the MIRband is large compared to that of the Drude term, an interpretation of the MIR band within the Hubbardmodel appears reasonable.On the other hand, the importance of electron-phononinteraction in Sr − x Ca x Cu O has been discussedearlier. In fact, MIR absorption features are finger-prints for the excitation of polaronic quasiparticles. Aninterpretation of the MIR band in Sr − x Ca x Cu O interms of the dissociation of bipolarons composed of holeson one rung was suggested in Refs.. Generally, po- R e f l e c t i v i t y a t c m - Pressure (GPa) E IIc E IIa E IIb
FIG. 8: Room-Temperature reflectivity ofSr . Ca . Cu O at 1000 cm − for E k c, E k a, and E k b as a function of pressure. laronic excitations cause a characteristic MIR band inthe optical conductivity spectrum, whose energy posi-tion is related to the polaron binding energy and whichis expected to shift to lower frequencies with increasingpressure. The pressure-induced shift of the MIRband in Sr . Ca . Cu O to lower frequencies is thusconsistent with the polaron picture.Furthermore, it has been shown theoretically that inSr − x Ca x Cu O a pressure-induced phase transitionfrom single polarons to bipolarons can occur. Duringthe formation of bipolarons two polarons are localizedwithin the same potential well. The absorption bandof small bipolarons generally appears at higher energiescompared to single small polarons, with the frequencyposition ω p = 4 E b − U , where E b is the polaron bindingenergy and U the Coulomb interaction between the twopolarons. Hence, a pressure-induced phase transition ofsingle polarons to bipolarons would result in a shift of theMIR band to higher frequencies. This is not observed inour data, and therefore such a phase transition seems tobe unlikely. C. Pressure-induced dimensional crossover?
In Sr . Ca . Cu O superconductivity is observedat 4 GPa and 6 K. Owing to the similarity in crystalstructure with the high-temperature copper-oxide super-conductors the question at issue is, whether the supercon-ductivity in Sr − x Ca x Cu O is a one-dimensional or– like in the high-temperature superconductors – a two-dimensional phenomenon. It was suggested earlier thatthe superconductivity in the spin ladder compounds isof two-dimensional nature. Pressure-dependent in-frared data for various polarization directions can giveinsight into the anisotropy of a material.According to Figs. 1(a) - (c) only for the direction E k cthe reflectivity – and concomitant the optical conduc-tivity – significantly increases under pressure. For il- E IIa E IIc R e f l e c t i v i t y R s - d Energy (cm -1 ) 1 GPa 4 GPaT = 10 K FIG. 9: Reflectivity spectra R s − d of Sr . Ca . Cu O at10 K for the lowest ( ∼ ∼ E k c and E k a within theladder/chain plane. lustration we plot in Fig. 8 the reflectivity level alongthe three polarization directions at 1000 cm − , i.e., out-side the phonon mode region, which is representative forthe overall behavior of the reflectivity spectra. For E k bthe reflectivity is unchanged under pressure, and for E k aonly a small increase of the reflectivity with increasingpressure is observed. Hence, no strong increase of thereflectivity level for the directions perpendicular to E k cis observed. Therefore, a pressure-induced dimensionalcrossover towards a more two-dimensional character atroom temperature seems to be unlikely according to ourdata.An electrical transport study suggested that pressur-ized Sr . Ca . Cu O becomes two-dimensional at lowtemperature. To test the dimensionality of the systemwithin the ladder plane at low temperature, we carriedout additional reflectivity measurements at various pres-sures. Fig. 9 depicts the reflectivity spectra at 10 Kfor the polarization direction along the two crystal axeswithin the ladder/chain plane, i.e., along the c and a axis, at the lowest ( ∼ ∼ . Ca . Cu O is not ob-served up to 4 GPa within the studied frequency range. V. CONCLUSIONS
In conclusion, our polarization-dependent infrared re-flectivity study shows that Sr . Ca . Cu O is elec-tronically highly anisotropic with a conducting behavioralong the ladders and an insulating behavior perpendicu-lar to the ladder plane. Along the in-plane direction per-pendicular to ladders the metallic character is lower ascompared to the ladder direction. The pressure-inducedincrease in the reflectivity and the corresponding opticalconductivity is strongest for the polarization directionalong the ladders, E k c. The pronounced MIR band inthe E k c optical conductivity spectrum red shifts underpressure, which can be attributed to strong electroniccorrelations or to the excitation of polaronic quasiparti-cles. The hole concentration in the ladders increases withincreasing pressure and tends to saturate at high pres-sure, resulting in a Cu valence in the ladders of +2.33at P ∼ ± . Ca . Cu O remainshighly anisotropic up to high pressure and at low tem- peratures within the studied frequency range. VI. ACKNOWLEDGMENT
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