Superconductivity-induced optical anomaly in an iron arsenide

Abstract

One of the central tenets of conventional theories of superconductivity, including most models proposed for the recently discovered iron-pnictide superconductors, is the notion that only electronic excitations with energies comparable to the superconducting energy gap are affected by the transition. Here, we report the results of a comprehensive spectroscopic ellipsometry study of a high-quality crystal of superconducting Ba_0.68 K_0.32 Fe₂ As₂ that challenges this notion. We observe a superconductivity-induced suppression of an absorption band at an energy of 2.5 eV, two orders of magnitude above the superconducting gap energy 2Δ[approximate]20 meV. On the basis of density functional calculations, this band can be assigned to transitions from As-p to Fe-d orbitals crossing the Fermi level. We identify a related effect at the spin-density wave transition in parent compounds of the 122 family. This suggests that As-p states deep below the Fermi level contribute to the formation of the superconducting and spin-density wave states in the iron arsenides.

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Received 15 Sep 2010 | Accepted 2 Feb 2011 | Published 1 Mar 2011 DOI: 10.1038/ncomms1223

Superconductivity-induced optical anomaly in an iron arsenide

A. Charnukha1, P. Popovich1, Y. Matiks1, D. L. Sun1, C. T. Lin1, A. N. Yaresko1, B. Keimer1 & A. V. Boris1,2

One of the central tenets of conventional theories of superconductivity, including most models proposed for the recently discovered iron-pnictide superconductors, is the notion that only electronic excitations with energies comparable to the superconducting energy gap are affected by the transition. Here, we report the results of a comprehensive spectroscopic ellipsometry study of a high-quality crystal of superconducting Ba 0.68K0.32Fe2As2 that challenges this notion.

We observe a superconductivity-induced suppression of an absorption band at an energy of 2.5 eV, two orders of magnitude above the superconducting gap energy 2 20meV.Onthe basis of density functional calculations, this band can be assigned to transitions from As-p to Fe-d orbitals crossing the Fermi level. We identify a related effect at the spin-density wave transition in parent compounds of the 122 family. This suggests that As-p states deep below the Fermi level contribute to the formation of the superconducting and spin-density wave states in the iron arsenides.

1Max-Planck-Institut fr Festkrperforschung, Heisenbergstrasse 1, Stuttgart D-70569, Germany.2 Department of Physics, Loughborough University ,

Loughborough LE11 3TU , UK . Correspondence and requests for materials should be addressed to A.V.B. (email: [email protected]).

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The standard BardeenCooperSchrieer (BCS) theory of superconductivity, based solely on an eective attractive interaction between electrons mediated by phonons, does

not provide a satisfactory explanation of the properties of strongly correlated high-temperature superconductors. Theoretical proposals in earlier years suggest that electronic excitations might enhance this interaction and thus contribute to the formation of the super-conducting condensate 15.Th ese proposals appeared to gain some ground with the observation of superconductivity-induced transfer of the optical spectral weight (SW) in the cuprate high-temperature superconductors, which involves a high-energy scale extending to the visible range of the spectrum 6 . In spite of numerous studies (for a comprehensive list of references see ref. 7), no modication of interband optical transitions in the superconducting state has been directly identied in the cuprates. Instead, the observed superconductivity-induced anomalies in the optical response of highly conducting CuO 2 planes were found to be conned to the energy range corresponding to transitions within the conduction band below the plasma edge. Th ese changes are dominated by the narrowing of the broad Drude peak caused by a superconductivity-induced modication of the scattering rate 810 . A minute redistribution of the SW between the conduction band and high-energy Hubbard bands generated by Coulomb correlations may also play a role 11,12.

Current research on the recently discovered iron-pnictide superconductors 13 suggests that electronic correlations are weaker than those in the cuprates. Unlike in cuprates, the Fermi surface has been reliably determined over the entire phase diagram and shows good agreement with density functional calculations. The superconducting state of the iron-pnictides appears to t well into a BCS framework in which phonons, which in these compounds interact only weakly with electrons 14 , are replaced by spin uctuations 15.Th e ellipsometric data we present here are consistent with the hypothesis that electronic correlations result in only a modest renormalization of the electronic states. However, the superconductivity-induced optical anomalies we observed involve modication of an absorption band at an energy of 2.5 eV, two orders of magnitude greater than the superconducting gap 2 20meV. In contrast to the cuprate superconductors, this high-energy anomaly has a regular Lorentzian shape in both the real and imaginary parts of the dielectric function and is conned to energies well above the plasma edge

[planckover2pi] pl 1.6 eV. It can be explained as a consequence of non-conservation of the total number of unoccupied states involved in the corresponding optical transitions due to the opening of the superconducting gap. Th is implies that unconventional interactions beyond the BCS framework must be considered in models of the superconducting pairing mechanism.

Results

Superconductivity-induced optical anomalies in Ba 0.68K0.32Fe 2As 2 .

The measurements were carried out on a single crystal of Ba 1x K x Fe 2As 2 (BKFA) with x=0.32 and superconducting transition temperature Tc=38.5K. Specic heat measurements on the same sample conrm its high purity and the absence of secondary electronic phases 16 . We performed direct ellipsometric measurements of the in-plane complex dielectric function

(

)=

1(

)+i 2(

a

b

BKFA, Tc= 38.5 K

3 1 cm1 )

(103 1 cm1 )

[afii9846] 1

(1 cm1 )

40 K37 K10 K

0 40 80Photon energy (meV) Photon energy (eV)

2.0

0.0

5.0

0.0

5.0

2.0

[afii9829]AH

[afii9846] 1(10

1.0

0.0

2.0

3 )

40 K10 K

[afii9830] 1(10

[afii9829]AL

2.0 2.5 3.0

c d

uo = NNSuo

N SC

uo < NNSuo

SC

3.303

3 1 cm1 )

N SC

3.302

[afii9846] 1(24meV)

1.5

1.0

0.5

[afii9846] 1(2.5eV) (10

SC

Energy

EF

3.301

20 40 60

DOS

Temperature (K)

Figure 1 | Superconductivity-induced anomalies in optical conductivity. ( a) Real part of the far-infrared optical conductivity of Ba 0.68K0.32Fe2As2

and the missing area. ( b) Difference spectra of the real part of the optical conductivity (top panel) and dielectric function (bottom panel) between 40 and 10 K, with a small background shift (horizontal dashed line) detected by temperature modulation measurements. Lorentzian t to both spectra (black solid lines). ( c) Temperature scan of

1 at 2.5eV.

Contribution of the normal-state dynamics (dotted line) was estimated to determine the magnitude of the SC-induced jump (dashed lines).( d) Density of states in the normal (NS; grey dashed line), conventional superconducting state (SC; blue line), and an unconventional state with a depletion of unoccupied states (UO; red line). Filled areas of respective colours represent total number of unoccupied states.

equivalent to a London penetration depth of

p2,200. The fraction of the missing area below 12 meV not accessible to the experiment was accurately quantied from the requirement of KK consistency of the independently measured real and imaginary parts of the dielectric function.

Careful examination of the visible range uncovered a super-conductivity-induced suppression of an absorption band at 2.5 eV. Figure 1b shows dierence spectra between 40 and 10 K of the real parts of the optical conductivity and dielectric function. The suppressed band has a Lorentzian lineshape and appears abruptly across the superconducting transition, consistently in both

1 and

1, as

shown in Figure 1c for

1 (2.5eV). Th e temperature dependence of the suppression (blue open circles) coincides with that of the far-infrared optical conductivity due to the opening of the superconducting gap (red lled circles). Th us, the onset of superconductivity not only modies the low-energy quasiparticle response but also aects the overall electronic structure, including interband transitions in the visible range of the spectrum. As the SW loss

AH is not balanced in the vicinity of the absorption band ( Fig. 1b ), our data indicate a SW transfer over a wide energy range. We note that the superconductivity-induced modication of the lattice parameters only results in a minute volume change of V/V5107 , which is insuffi cient to explain the optical anomaly (C. Meingast, personal communication). A KK consistency analysis could not be carried out with suffi cient accuracy to show whether or not the SW liberated from the absorption band at the superconducting transition contributes to the response of the superconducting condensate at zero energy. We did, however, detect a minute rise in the background level of

1 (1.53.5eV)below Tc , which according to the KK rela-

over a range of photon energies extending from the far infrared (

[planckover2pi] 12meV) to the ultraviolet

)=1+4i (

)/

[planckover2pi] 6.5eV), with subsequent KramersKronig (KK) consistency analysis (see Methods). Th e far-infrared optical conductivity is dominated by the opening of a superconducting gap of magnitude 2 20meV below

Tc ( Fig. 1a ), in accordance with previous studies of optimally doped BKFA 17. Th e low-energy missing area in the optical

conductivity spectrum below Tc,

(

10

( ( ) ( )) ,

is contained within 10 and amounts to w d

pl

sc L eV

d s w s w w

AL

10 40 K K d

=

+

0

1

1

=

8 0 9

A . ,

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tion implies that the SW is transferred to energies below 1.5 eV. This eect was identied from a simultaneous t of

1 (

m m

* / ( / )

band pl

= LDA pl

w w 2 3 . Such an enhancement is consistent with de Haas-van Alphen and photoemission experiments on other compounds of the 122 family 2022 and was recently reproduced by combined LDA and dynamical mean-eld theory (DMFT) calculations for both 1111 and 122 compounds 23. Th ese calculations do not show evidence of the formation of Hubbard bands and thus indicate moderate electronelectron correlations. Th is explains the good agreement of the LDA optical conductivity above 1.5 eV with the experimental data.

Now, we turn to the physical origin of the superconductivity-suppressed absorption band. Th e same LDA calculation revealed a set of interband transitions centered at 2.5 eV, which originate or terminate in states exhibiting hole dispersion and crossing the Fermi level at the - and M-points of the Brillouin zone ( Fig. 2c ). We condently assign these states to Fe-d yz,zx andFe-d xy orbitals. Th e other states involved in these transitions belong to As-p x,y /Fe-dz2 hybridorbitals,~23eV below and above the Fermi level, giving rise to a bandwidth of E1eV, which is in remarkable agreement with the experiment. Suppression of the absorption band over its full width can be explained by redistribution of the occupation of Fe-d yz,zx and Fe-d xy states under the Fermi level below the superconducting transition. Th is mechanism is supported by LDA calculations in which the density of states within one superconducting gap energy above the Fermi level was eliminated (see Supplementary Fig. S2 and Method), leading to the observed suppression of the optical transitions shown in cyan in Figure 2b .

Spin-density wave-induced SW transfer in SrFe 2As 2 . We have further explored the validity of this scenario by repeating our ellipso-metric measurements on parent compounds of the 122 family of iron arsenide superconductors. As the spin-density wave (SDW) instability exhibited by these compounds is also believed to be induced by nesting of electronic states on dierent electronic bands, we expect an optical anomaly at the SDW transition similar to the one we observed in the superconductor. Th e magnitude of the anomaly is expected to be greater than the one in the superconductor, because the SDW transition occurs at high temperature and generates a larger energy gap. In SrFe 2As 2 (SFA), we indeed nd a strong reduction in optical absorption upon cooling below TSDW 200K.The dierence spectra of

1(

) and

) (horizontal dashed line in the bottom panel of Fig. 1b ). To estimate the background shi more accurately, temperature modulation measurements of

1 at the resonance photon energy 2.47 eV and of 1 at o-resonance photon energies of 2.12 eV and 2.82 eV were carried out. In Supplementary Figure S1 , the sample temperature was changed between 20 to 40 K with a period of 1,800 s and later averaged over 24 periods to reduce noise to

1=10 4. This conrms a minute background increase of

1 (2.47eV)=(84)104.

1 (

As the SW

AH liberated from the absorption band upon cooling below Tc comprises onl 0.5 % of the total SW

AL of the superconducting condensate, its contribution to the low-energy charge dynamics might be considered negligible. However, assuming that this additional high-energy SW contributes to the itinerant-carrier response below Tc , a simple estimate in the framework of the tight-binding nearest-neighbour approximation 18,19 shows that this would lead to a reduction in electronic kinetic energy of 0.60 meV per unit cell in the superconducting state (see Supplementary Note 1 ). Th is is close to the condensation energy F(0)=0.36meV per unit cell obtained from specic heat measurements on the same sample 16. It is thus important to establish the origin of this unusual optical anomaly.

Local density approximation calculations. We therefore compared our data with the results of ab initio electronic structure calculations in the framework of the local density approximation (LDA) ( Fig. 2a,b ). A dispersion analysis of the experimental optical conductivity in the range 0.5 6.5 eV yielded three major interband transitions in Ba 0.68K0.32Fe 2As 2 . A comparison with the LDA results

enabled us to identify the initial and nal states of these transitions. Th e lowest-energy transition is located at about 1 eV (red line) and stems from intraband Fe-d and interband As-p to Fe-d transitions. Th e major contribution to the optical response in the visible spectral range comes from transitions starting from Fe-d or As-p orbitals into strongly hybridized Fe-d to As-p or Fe-d orbitals (green line). Finally, the ultraviolet absorption comes from higher-energy transitions into Ba-d states (blue line).

Although the high-energy electronic structure of BKFA is predicted quite well by the LDA calculations, the experimental quasiparticle response due to transitions within the conduction band (or, given the multiorbital structure of iron-pnictides, a narrowly spaced set of conduction bands) shows a signicant deviation. The discrepancy can be quantied by the squared ratio of the band-structure plasma frequency wplLDA eV

) between 200 and 175 K show a double-peak structure, with maxima at 2.4 and 3.4 eV ( Fig. 3a ). A temperature scan across TSDW at the frequency of the second peak (inset of Fig.

3a) further conrms that this eect is induced by SDW formation. A direct comparison with the superconducting compound is complicated by a pronounced modication of the electronic structure due to the coincident magnetic and structural transitions. Nevertheless, certain information can be gained from the critical behav

iour of the in-plane SW

) and

2(

. (not included in Fig. 2b ) to its experimental counterpart wplexp eV

= 2 7

1 6

. , which can be obtained in practice from the residual optical response, aer the interband transitions identied using a dispersion analysis have been subtracted. This ratio approximates the quasiparticle eective-mass renormalization factor

s w w. Figure 3b shows

dierence SW of SFA between 200 and 175 K as a function of the

a

b

c

3 1 cm1 )

Experiment

FitFe-d/As-p into Fe-de Fe-d/As-p into Fe-d+As-p/Fe-de All into Ba-d

As-p into Fe-dh

3 1 cm1 )

4.0

4.0

4.0

Energy, eV

2.0

0.0

[afii9846] 1([afii9853]) (10

2.0

[afii9846] 1([afii9853]) (10

2.0

0.0

2.0

2.4 eV 3.0 eV

6.0 Photon energy (eV)

Figure 2 | Assignment of interband optical transitions. ( a) Real part of the optical conductivity of Ba 0.68K0.32Fe2As2 and contributing interband transitions determined by a dispersion analysis. ( b) Corresponding LDA calculation with a breakdown into separate orbital contributions described in a. (c) Band structure from the same LDA calculation. Colour coding of the dispersion curves corresponds to the text colour in a. Superconductivity-suppressed absorption bands (cyan arrows).

4.0

0.0

M X M

2.0 4.0 6.0 Photon energy (eV)

2.0 4.0

3

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Discussion

Th e population redistribution required to account for the superconductivity-reduced absorption in BKFA is, however, at variance with the conventional theory of superconductivity. In the framework of the standard BCS approach, opening of an energy gap in a single-band superconductor leads to bending of the quasiparticle dispersion and to expulsion of the density of states in the vicinity of the Fermi surface (grey and blue areas in Fig. 1d ) 26 , with the total number of unoccupied states below the transition conserved N N

SC

uo NS

uo

a

6.75

SFA, TSDW = 200 K

6.72

[afii9830] 2(T)

10.0

6.69

[afii9853] = 3.4 eV

6.66 180 200 220

5.0

[afii9830] 1, [afii9830] 2(102 )

2.4 eV

Temperature (K)

200 K175 K

0.0

= (blue area is equal to the grey area in Fig. 1d ). Th is can only lead to a small corrugation of an optical absorption band on the scale of one superconducting gap energy superimposed on the overall broad feature without any modication of its SW 27. The experimentally observed suppression of an absorption band on the scale of its full width necessarily requires a population imbalance N N

SC

uo NS

uo

[afii9830]2([afii9853])

[afii9830]1([afii9853])

5.0 2.0 3.0 4.0 5.0

Photon energy (eV)

< (red area unequal to the grey area in Fig. 1d ). This eect can be clearly identied as a consequence of superconductivity because the temperature dependence of the suppression mimics that of the optical conductivity in the far-infrared region due to the opening of the superconducting gap, as shown in Figure 1c .

All of the iron-pnictide superconductors are known to have multiple superconducting gaps 13 and theoretical work indicates a dominant contribution of electron pairing between dierent bands to the formation of the superconducting state 15 . Redistribution of the occupation of the dierent bands below Tc could explain the optical anomaly we observed (see Supplementary Fig. S3 ). However, even a generalization of the standard BCS theory to the multiband case 28 does not take this eect into account. It requires lowering the chemical potential of the material in the superconducting state (see Supplementary Note 2 ). In the presence of large Fe-As bond polarizability 29,30 , it can potentially enhance superconductivity in iron-pnictides.

Interactions of electrons in dierent energy bands at the Fermi level may provide a common framework for an explanation of the optical anomalies in the SDW and superconducting compounds. It is important to note that these anomalies aect only a small fraction of the interband transitions, which involve initial states of As p -orbital character deep below the Fermi level. Th is indicates that these orbitals signicantly inuence electronic instabilities in the iron arsenides, possibly due to the high polarizability of the As Fe bonds. Our study points to optical SW transfer from high energies to below 1.5 eV induced by collective electronic instabilities. In the superconductor, it occurs at energies two orders of magnitude greater than the superconducting gap energy, suggesting that electronic pairing mechanisms contribute to the formation of the super-conducting condensate.

Methods

Sample preparation. The Ba 0.68K0.32Fe 2As 2 single crystal was grown in zirconia crucibles sealed in quartz ampoules in argon atmosphere 31. Its chemical composition was determined by energy-dispersive X-ray spectrometry. Th e quality of the sample and absence of phase separation was conrmed by specic heat experiments that yielded an upper bound of 2.4 % (ref. 16 ) on its non-superconducting volume fraction. From direct-current (DC) resistivity, magnetization and specic heat measurement, we obtained Tc=38.50.2K. Th e sample surface was cleaved before every measurement.

Experimental apparatus. Th e experimental setup comprises three ellipsometers to cover the spectral range of 12meV6.5eV. For the range 12meV1eV, we used a home-built ellipsometer attached to a standard Fast-Fourier-Transform Bruker 66v/S FTIR interferometer. Th e far-infrared measurements were performed at the infrared beamline of the Angstrm Quelle Karlsruhe (ANKA) synchrotron light source at the Karlsruhe Institute of Technology. For the mid-infrared measurements, we used the conventional glow-bar light source from a Bruker 66v / S FTIR . Finally, high-energy spectra 0.7 6.5 eV were measured with a Woollam VASE (variable angle spectroscopic ellipsometer) ellipsometer equipped with an ultra high-vacuum cold-nger cryostat operated at <5109mbar chamber pressure.

Th e inherent capacity of Woollam VASE ellipsometers to measure relative changes of the dielectric function in the order of 10 2 was boosted to an

b

4.5

ExperimentSW(12 meV) = 0.015 eV2 SW(12 meV) = 0

2.0

3.0

[afii9830] 1

SW (102 eV2 )

0.0

1.5

0.25 0.50 0.75

Photon energy (eV)

0.0

200 K175 K

1.5 0.0 2.0 4.0 6.0

Photon energy (eV)

Figure 3 | SDW-induced anomaly in high-energy optical conductivity.( a) Difference spectra of the real and imaginary parts of the dielectric function of SFA between 200 and 175 K. Lorentzian t to both (solid lines). (Inset) Temperature scan of

2 at 3.4eV. (b) SW redistribution between 200 and 175 K. SW in the extrapolation region below 12 meV before (blue lled circle) and after (red lled circle) a KK consistency check. Blue and red lled areas represent regions of SW gain and loss, respectively, in the magnetic versus normal state. (Inset) Difference spectra of the real part of the dielectric function obtained experimentally (open circles) and from a KK transformation of the real part of the optical conductivity (solid lines, colours match lled circles).

upper integration limit . Th e change in the SW in the extrapolation region below 12 meV was accurately determined via a KK consistency analysis (see Methods), as illustrated in the inset of Figure 3b . With SW (12meV)=0 across the transition (blue lled circle), the KK transformation of the

2(

) (blue line) deviates signicantly from the experimentally measured

1(

). Gradually increasing this SW brings them closer until they nally coincide (red line), thus xing SW (12meV)=0.015eV2 (red lled circle). The higher-energy redistribution of the SW is broken down in Figure 3b into regions of SW gain (blue areas) and loss (red areas) in the SDW with respect to paramagnetic state. Th e SW lost because of the opening of the SDW gap 24 (the rst red region) is partly transferred to the electronic excitations across the gap (the rst blue region) and fully recovered by 1.5eV. Th ese processes are then followed by high-energy redistribution in the region 1.5 4.0 eV, involving the SW of the suppressed bands. It appears unlikely that such high-energy SW transfer could result from modication of the electronic structure due to a magnetic transition, because eects of electronic reconstruction at the SDW transition are limited by 1.5 eV. A modication of the matrix elements at the structural transitions of suffi cient strength cannot account for the observed suppression, because this would be accompanied by an even larger eect at higher energies clearly absent in Figure 3b . A redistribution of charge carriers between the SDW-coupled bands analogous to that in the superconducting compound provides a more natural explanation. Th e same physical reasons might explain the orbital polarization that breaks the degeneracy of Fe-d xz and Fe-d yz orbitals recently observed in the Ba-based parent of the same family by photoemission spectroscopy 25.

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unprecedented level of 10 4 using temperature-modulation measurements of the dielectric constant at particular photon energies.

KK consistency check . One of the strong advantages of spectroscopic ellipsometry with respect to reectometry is that independently obtained real part of the dielectric function

1 (

) and optical conductivity

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Acknowledgments

Th is project was supported by the German Science Foundation under grant BO 3537 / 1-1 within SPP 1458. We gratefully acknowledge Y.-L. Mathis for support at the infrared beamline of the synchrotron facility ANKA at the Karlsruhe Institute of Technologyand , V. Khanna for taking part in some of the measurements, and C. Meingast for communication of data prior to publication. We also thank O. V. Dolgov, L. Boeri, F. V. Kusmartsev, A. S. Alexandrov, I. I. Mazin, P. B. Littlewood for fruitful discussions.

Author contributions

A.C., P.P., Y.M. and A.V.B. carried out the measurements. A.C. and A.V.B. analysed the data. A.N.Y. carried out the LDA calculations. D.L.S. and C.T.L. synthesized the samples. A.C., B.K. and A.V.B. wrote the manuscript. B.K. and A.V.B. supervised the project.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/ naturecommunications

Competing nancial interests: Th e authors declare no competing nancial interests.

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How to cite this article: Charnukha, A. et al. Superconductivity-induced optical anomaly in an iron arsenide. Nat. Commun. 2:219 doi: 10.1038 / ncomms1223 (2011).

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1 (

) can be used in a KK consist-

ency check, in which

1 (

) and the KK transform of

1 (

) must coincide:

e w s w

1 0

1

2 2

( ) ( ) .

=

8

xx x

d

Th is additional constraint unique to ellipsometry allows one to determine with high accuracy the SW in the extrapolation region beyond the experimentally accessible spectral range, in our case below 12 meV. It drastically reduces the extrapolation uncertainty and renders the subsequent data analysis more robust. KK consistency analysis is rather insensitive to the exact shape of the optical spectrum in the extrapolated frequency range, but it does x the total SW

SW d

( ) ( ) ,

w s

w 0 0

0

1

= x x

where

0 is the experimental low-energy cutofrequency. This procedure is illustrated in Figure 3b . Taking the experimental dierence spectrum of

1 (

)

with SW (

0) = 0 (blue circle) and carrying out the KK transformation results in the deviation of the real part of the dielectric function from the measured data (blue line in the inset of Fig. 3b ). Only by increasing the SW below

0=12meV to 0.015eV2 (red circle) does one achieve complete agreement with the experiment (red line in the inset). Th e exact shape of the extrapolated

1 (

) has a minor role.

Th e maximum uncertainty introduced by the unknown shape can be calculated as the dierence of two extreme congurations: all SW (

0) at

=0 and

=

0:

(1)(1)

(2)(2)

d e w w

w

( )( ) ( ) ( )

1 =

1 0 2

0

8

SW SW

w w w

2 2

0

2

=

8 8

SW SW

( ) ( ) ,

w w

0 2

ww w

0

w w

0 2

0

w w

2

2

2 2

0

when

[greatermuch]

0. On the other hand, the accuracy with which the SW is determined at the same energy is given by

d e w d s w

x d SW

Th e relative eect of the shape change over the magnitude change in the SW in the extrapolation region is then

| ( )/ ( )| ( / )

8 8

( )( ) ( ) ( ( )) .

1 2 =

+

1

2 2

d w w

0 2

0

( ) ( )

d e w d e w w w

1

1

0

2 [greatermuch]

0. In the

present case taking

0=12meV and

1 =250meV (as in the inset of Fig. 3b), one obtains a shape uncertainty of 0.2 % . Th us, the eect is negligible already at rather low frequencies. Th e same analysis applies for the high-energy extrapolation above 6.5 eV. However, complete agreement between

1 (

) and

2 (

) within the accuracy of the experiment was found up to 6.5 eV, therefore, no experimentally discernible missing SW is present at higher energies.

References

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5. Hirsch, J. E. & Marsiglio, F. Superconducting state in an oxygen hole metal. Phys. Rev. B 39, 1151511525 (1989).

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7. Maiti, S. & Chubukov, A. V. Optical integral and sum-rule violation in high-Tc superconductors. Phys. Rev. B 81, 245111 (2010).

8. Holcomb, M. J., Perry, C. L., Collman, J. P. & Little, W. A.Thermal-dierence reectance spectroscopy of the high-temperature cuprate superconductors . Phys. Rev. B 53, 67346751 (1996).

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