Quantitative Determination of Eliashberg Function and Evidence of Strong Electron Coupling with Multiple Phonon Modes in Heavily Over doped (Bi,Pb)2Sr2CuO6

0.36 MB·English

by  Lin Zhao

Quantitative Determination of Eliashberg Function and Evidence of Strong Electron
Coupling with Multiple Phonon Modes in Heavily Over-doped (Bi,Pb)2Sr2CuO6+δ
Lin Zhao1, Jing Wang2, Junren Shi2,∗ Wentao Zhang1, Haiyun Liu1, Jianqiao Meng1, Guodong
Liu1, Xiaoli Dong1, Wei Lu1, Guiling Wang3, Yong Zhu3, Xiaoyang Wang3, Qinjun Peng3,
Zhimin Wang3, Shenjin Zhang3, Feng Yang3, Chuangtian Chen3, Zuyan Xu3, and X. J. Zhou1,†
1National Lab for Superconductivity, Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Science, Beijing 100190, China
0 2Beijing National Laboratory for Condensed Matter Physics,
1 Institute of Physics, Chinese Academy of Science, Beijing 100190, China
0 3Technical Institute of Physics and Chemistry, Chinese Academy of Sciences, Beijing 100190, China
2 (Dated: January 31, 2010)

n Super-high resolution laser-based angle-resolved photoemission spectroscopy measurements have
a been carried out on a heavily overdoped (Bi,Pb)2Sr2CuO6+δ (Tc ∼ 5 K) superconductor. Taking
J
advantageofthehigh-precisiondataonthesubtlechangeofthequasi-particledispersionatdifferent

1 temperatures,wedevelopageneralproceduretodeterminethebarebanddispersionandextractthe
3 bosonicspectral function quantitatively. Ourresultsshow unambiguouslythat the∼70meV nodal
kinkisduetotheelectroncouplingwiththemultiplephononmodes,withalargemassenhancement
] factor λ∼0.42 even in the heavily over-doped regime.
n
o
PACSnumbers: 74.25.Jb,71.38.-k,74.72.Gh,79.60.-i
c
-
r
p
The interaction of electrons with phonons and other Coming specifically to high-T cuprate superconduc-
u c
collective excitations (bosons) in solids dictates funda- tors, although there is a general consensus on the
s
. mental physical properties of materials [1]. In a Fermi electron-boson coupling as the origin of the universally
t
a liquid picture, such interaction can be described by the observed 50∼80 meV dispersion kink along the (0,0)-
m Eliashberg spectral function α2F(ω), which accounts for (π,π) nodal direction [14–20], it remains under debate
- the bosonic modes involved in the coupling and their on the nature of the boson(s) involved, mainly between
d
strengths. In the conventional superconductors, extrac- phonons [15] and magnetic origins [19, 21, 22]. Theoret-
n
o tion of the Eliashberg function played a key role in pin- ical calculations suggest that the electron-phonon cou-
c ningdownthephononasthegluefortheelectronpairing pling in cuprates is too weak to account for the nodal
[
thatgivesrisetotheBCSsuperconductivity[2]. Forhigh kink[23,24]. Toeventuallyresolvethe issue,itiscrucial

1 temperaturecupratesuperconductorsandothercomplex toextractthebosonicspectralfunctionquantitativelyso
v compounds,oneexpectsthattheextractionoftheEliash- that a direct comparison between the experiments and
0 berg function will also be important in understanding the calculations could be possible [6–8].
2
theexoticphysicalpropertiesandthemechanismofhigh
1 In this paper, we introduce a new approachwhich can
temperature superconductivity [3–8].
0 unambiguously determine the bare quasi-particle disper-
2. The electron-boson coupling gives rise to the band sion, and hence enables the quantitative determination
0 ofthebosonicspectralfunction. Themethodutilizesthe
renormalizationandthechangeofthequasiparticlescat-
0 subtle changeofthe quasi-particledispersioninducedby
tering rate, which can be described by the real part
1
the temperature. This is made possible by the super-
: (ReΣ(k,ω)) and the imaginary part (ImΣ(k,ω)) of the
v highresolutionlaser-basedARPESmeasurements,which
electron self-energy Σ(k,ω), respectively. With the dra-
i present data of the significantly higher quality feasible
X matic improvement of resolution, angle-resolvedphotoe-
for a quantitative analysis of a modified maximum en-
r mission spectroscopy (ARPES) has emerged as a pow-
a tropy method [6]. By carrying out such a procedure
erful tool in probing such many-body effects [5]. Under
the sudden approximation, it measures the single par- onaheavilyover-doped(Bi,Pb)2Sr2CuO6+δ (Pb-Bi2201,
T ∼5K),weareabletodeterminethebarebandandex-
ticle spectral function A(k,ω) = (1/π)ImΣ(k,ω)/{[ω − c
ǫ0 − ReΣ(k,ω)]2 + [ImΣ(k,ω)]2}, which is directly re- tract the bosonic spectral function quantitatively along
k
the nodal direction. Our results present a conclusive ev-
lated to the electron self-energy [5]. ARPES has been
idence that the electron-bosoncouplingis strong(with a
extensively employed in investigating many-body effects
mass enhancement factor λ ≈ 0.42) even in the heavily
in conventional metals [9, 10], graphene [11] and com-
over-doped cuprate compounds, and it originates from
plex materials [5, 12]. However, a long-standing issue
the electron coupling to the multiple phonon modes.
withtheARPEStechniqueistheambiguous“bareband”
ǫ0[13],whichhinders aquantitativedeterminationofthe Moreover, our approach is general enough to be appli-
k
electron self-energy ReΣ(k,ǫ)=ǫ-ǫ0 and the underlying cable for many other materials.
k
bosonic spectral function α2F(ω) [6–8]. The angle-resolved photoemission measurements have
2
E - E (eV)F---000...0321a G HLoigwhEDC Intensity (Arb. Units)bX5 kF E - E (eV)F---000...3210 c -1MDC Width (A)00.0.-1050.3E- 0- .E2F -(0e.V1) Energy Difference (meV)Energy Difference (meV)54321-432100000000.3ed-0.25-0.2-0.15-0.1-0.050 lbabMsFoetyeewroTre∼vtmrnheehe7ndioeer3evmemousrepnlgnoroery,mdoVtphaedfee,ereunc∼lacycEtlthee4uiuDanad1mrngrmCegisdnb(seekaiVos(pFst.se,eon)∼aWva.netenri4erc7gda13ynsl∼mpodmit1eneiecs7edVttVmetrrhpaiaeb(anelVFuntdftiudatgi∼noehr.nncee1tt1srci7e(ooumbmnbtr)hevu)eαaresVescst2uea),Farnsnfne(ieeωgmdbaan)etrehiunficaotrtaehvbesdnees-.
0.3Mom0.e4ntu0m.5 k (p0/.6a) -0.3 -E0.2 - E-F0 .(1eV)0 -0.09 -0k.0 6- kF (A-0-.01)3 0 -0.06E --0 .E04F (e-0V.0)2 0 be extracted from the measuredrealpart of the electron
self-energy [6]. The real part of self-energy is related to
α2F(ω) by:
FIG. 1: Fine structure in the nodal dispersion of a heavily
tooveemr-disospioendiPmba-gBei2t2a0k1en(Taclo∼ng5tKh)em(0e,a0s)u-(rπe,dπa)tn1o5daKl.d(iar)ecPthioon- ReΣ(ǫ;T)= ∞dωK ǫ , ω α2F(ω), (1)
Z0 (cid:16)kT kT(cid:17)
with the location of the momentum cut shown in the inset.
(b)Correspondingphotoemissionspectra(energydistribution whereK(y,y′)= ∞ dxf(x−y)2y′/(x2−y′2)withf(x)
curves, EDCs). The EDC at the Fermi momentum kF (red −∞
curve)showsadipfeatureat∼73meV,asindicatedbyanar- being the Fermi dRistributionfunction [1]. In the ARPES
measurements, the real part of the electron self-energy
row in the expanded spectrum. (c) Quasi-particle dispersion
obtainedbyfittingMDCs(momentumdistributioncurves)at is determined from the measured MDC dispersion ǫk by
different binding energies. The inset shows the correspond- ReΣ(ǫk)=ǫk−ǫ0k,whereǫ0k isthebarequasi-particledis-
ingMDCwidth(FWHM).(d)Energydifferencebetweenthe persionforanad-hocsystemwiththeelectron-bosoncou-
measureddispersionandastraightlineconnecting0(EF)and plingturnedoff[26]. Attempts havebeenmadetoinvert
−0.3eV.Apeakat∼73meVisclearlyobservedasindicated the“effective”bosonicspectralfunctionα2F(ω)fromthe
byanarrow. (e)Energydifferencebetweenthemeasureddis-
persion and a straight line connecting 0 and −0.07 eV. Two measuredMDCdispersionbyassuminganempiricalbare
features can be identified at ∼41 meV and ∼17 meV. bandofaquadraticformǫ0k =a(k−kF)2+b(k−kF)2 [6–
8]. It was found that the qualitative features of the
bosonic spectral function do not sensitively depend on
the choice of the bare band. However, the arbitrariness
been carried out on our newly developed vacuum ultra-
violet (VUV) laser-based ARPES system [25]. The pho-
ton energy of the laser is 6.994 eV with a bandwidth of 0
-0.06
0.26 meV. The overall instrumental energy resolution is
setat1meV,whichissignificantlyimprovedfrom10∼20 -0.07
mlareVreosoflpurteiovniouiss ∼me0a.s3urdeemgerneet,s c[6o–r8re,s1p4o–n2d0i]n.gTthoeaanmguo-- 2450KK -0.08E (eV)F 2450KK
6m.9en94tuemV,remsoolruetitohnan∼0tw.0i0c4e Aim−p1roavtetdhefropmhot0o.0n0e9nAer−g1yaotf 11 05680000KKKK -0.09E - 1 068000KKK
a usual photon energy of 21.2 eV for the same angular V) -0.036 -0.032 -0.028 -0.10 150K
(rPesbo-lButi2io2n0.1)TshineglheecarvyilsytaolsvewridtohpaedTc(B∼i,P5bK)2wSre2rCeugOro6w+nδ (eF-0.1 k - kF (A-1) -0.2 -0.15E - E-0F. 1(eV-)0.05
by the traveling-solventfloating zone method. The sam- E 0.08
-
ples are cleaved in situ in vacuum with a base pressure E
better than 5 × 10−11 Torr. -1h (A) 0.06 1 0205KK
Figure 1(a) shows the raw data of photoemission im- dt 150K
Wi
age taken on a heavily overdoped Pb-Bi2201 (Tc∼5 K) C
sample along the (0,0)-(π,π) nodal direction at 15 K. MD 0.04
Clear kink is observed near 70 meV in the dispersion
extractedfromMDC(momentumdistributioncurve)fit- 0.02
ting (Fig. 1(c)), with a corresponding drop in the MDC -0.2
-0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 0
width (inset of Fig. 1(c)). In order to revealthe possible -1
k - k (A )
F
finestructuresinthedispersion,weplottheenergydiffer-
ence between the measured dispersion and a featureless
straight line connecting 0 (EF) and −0.3 eV on the dis- FIG. 2: Temperature dependence of the nodal dispersion
persion (Fig. 1(d)). A prominent peak shows up near measuredon aheavilyover-dopedPb-Bi2201 (Tc∼5K).The
∼73 meV. In addition, in the energy difference between dispersions near the kink region are expanded in the up-left
the measured dispersion and another straight line con- inset. Thebottom-rightinsetshowsMDCwidthmeasuredat
necting 0 and −0.07 eV (Fig. 1e), one can also identify 25 K, 100 K and 150K.
3
ipspsnperrleoecfv-caheteconrnhoaetslrtsigonfyuagcnictrchqhctuaeuionamanngpvetpi[et6arna–tott8pdit]vr.iheiffaiesHtdrepeeeprnrtoeaetbrrtwmaleememmineppatbreteoyriroapsnmtofuosareorkeftisanh.tgeghWebeuneaseerbareosaosbslfuoaatnmnhpidece- Energy eaDReS0(e) ReS1(eR)eS2(e) DReS(e) Self-Energy Difference SSRe(25K)-Re(100K) (meV)-13210 c ∆∆∆RRReeeΣΣΣ012
thatα2F(ω)andthe barebandhavenegligibletempera- Dispersion 1 0.4
Dispersion 2 n d
turedependenceoverarelativelysmalltemperaturewin- Momentum nctio
dthoews,ealfn-dentehregyEdliiaffsehrbeenrcgefautndctiffioenrecnatntebmepeexrtaratucrteeds bfryo:m 0 bMMeeaassuurreedd,,TT==2150K0K Spectral Fu0.2 SSSFFF012
ReΣ(ǫ;T1)×−(cid:20)RKeΣ(cid:18)(kǫ;TǫT12,)k=ωT1Z(cid:19)0∞−dKω(cid:18)α2kFTǫ(2ω,)kωT2(cid:19)(cid:21) (2) E - E (eV)F-0.1 BBBaaarrreeebbbaaannnddd012 SRe (meV)2211055050 e RRReeeΣΣΣ012
-0.2 0
The successful application of the approach relies crit- -0k.0 -6 k (A-0-.10)3 0 0 0.05 Energ0y. 1w (eV) 0.15 0.2
ically on the high quality of data, as it utilizes a F
small temperature-induced change between two disper-
sions subjected to noises. Such a stringent requirement FIG.3: ExtractionprocedureofthebarebandandtheEliash-
onthedataqualitycannowbemetbythesuper-highres- bergspectralfunction. (a)Schematicplotforthetwodisper-
olution laser-ARPES measurements(Fig. 2). In this pa- sions and a bare band to show the real parts of self-energies
per, we test our new approach on the heavily overdoped andtheirdifferenceforagiven energyǫ. (b)Measured nodal
dispersionsofheavilyoverdopedPb-Bi2201at25Kand100K
Pb-Bi2201, in which the electron-electron correlation is
andtheextractedbarebandsfordifferentiterations. (c)Self-
relatively weak, and Eq. (2) is believed to be applicable.
energy difference between the 25K and 100K dispersions for
Figure 3 shows the detailed procedure to extract the
different iterations. (d) Extracted Eliashberg spectral func-
bare band dispersion and the bosonic spectral function. tion for different iterations. (e) Real part of electron self-
To obtain the bosonic spectral function α2F(ω) from energy calculated from the extracted spectral function at 25
Eq.(2), one needsto knowthe difference ofthe realpart K for different iterations.
ofself-energy∆ReΣ(ǫ)≡ReΣ(ǫ;T1)−ReΣ(ǫ;T2) attwo
different temperatures, as schematically shown in Fig.
needed for this analysis.
3(a). Here,weusetwodispersionsmeasuredat25Kand
100 K. Note that ∆ReΣ(ǫ) depends on the selection of Fig. 4 summarizes the final results of the bare band
the bare band dispersion. As the result, an iteration ap- (Fig. 4(a)),Eliashbergspectralfunction(Fig. 4(b)),and
proachis needed, as detailed in the following: (1) Deter- electronself-energy(insetofFig. 4(a)). Althoughthere-
mine ∆ReΣ(ǫ)basedonthe currentselectionofthe bare sultsareobtainedbyusingthetworepresentativedisper-
band dispersion. The initial value (∆ReΣ0(ǫ)) is taken sions at 25K and 100K, the self-energy calculated from
as the direct difference between the two dispersions, as the extracted Eliashberg function (Fig. 4(b)) at other
schematically shown in Fig. 3(a). The result is shown temperatures(hereweshow60Kdata)matchesverywell
in Fig. 3(c); (2) Extract the “bosonic spectral function” with the measured values, as shown in the inset of Fig.
SF from ∆ReΣ(ǫ) by Eq. (2), using a maximum entropy 4(a),indicatingtheinternalconsistencyofourapproach.
method (MEM) similar to that in Ref. [6] but with the The obtained bare band is close but less steeper than
modifiedintegralkernel. TheresultisshowninFig.3(d). the one given by the LDA calculations [27] (Fig. 4(a)),
Note that the significant change of the real part of the and is significantly different from the empirical one used
self-energy is limited to the energy up to ∼ 100 meV, previously (green line in Fig. 4a) [8].
above which the self-energy difference fluctuates around Sharp features are clearly identified in the extracted
zero and is most likely due to the data noise, as shown Eliashberg spectral function (Fig. 4(b)). Five peaks at
in Fig. 3(c). Therefore, we assume a maximum bosonic (14-17),(30-32),(41-44),(54-58) and (70-74) meV can be
mode energy 100 meV when doing the MEM analysis; identifiedbelow100meV,andtheyarereproduciblefrom
(3)Calculatetherealpartofself-energyat25Kfromthe several independent measurements. In fact, the features
extractedbosonicspectralfunctionusingEq.(1). There- (14-17),(41-44)and(70-74)meVareclearlyvisibleinthe
sult is shown in Fig. 3(d); (4) Determine the bare band raw data (Fig. 1). These sharp features show reason-
dispersionbytakingthedifferencebetweenthemeasured ablygoodagreementwiththephononmodesobservedin
dispersionandthe calculatedrealpartofthe self-energy, the same materialby infraredreflectance(Fig. 4(c)) [28]
both at 25K. The result is shown in Fig. 3(b). These and Raman scattering (Fig. 4(d)) [29]. Note that these
steps iterate until the result converges. We find that the optical measurements only see phonon modes with zero
procedureconvergesquicklyandonlythreeiterationsare wavevector,andthefullbandofphononswillexhibitdis-
4
0 a n0.4 b nique to be quantitative in probing many-body effects.
nctio Our results on the heavily overdopedPb-Bi2201 demon-
u
MB BaDarreCebb aDannidsd,p ,L eOrDslidAon pectral F0.2 sktirnakteisudnuaemtboigeuleocutsrloynthcoautptlhineg∼t7o0tmheeVmnuoltdipalledpishpoenrosinosn.
Bareband, New S 0 In particular, we present a conclusive evidence that the
c
E - E (eV)F-0.1 40 Energy ω 1(0e0VK)(+20) Infrared x-Axis Loss Function00..21 OOpDt ethleecthreoanv-iplyhoonvoenr-dcoopuepdlinrgegiismest.rong in cuprates even in
V)30 0d
ΣRe (me210000 0. 05C0al.c1u6l2 a05tKeK0d(.+1150) Raman Intensity(Arb. Units) XZZX ∗† CCoorrrreessppoonnddiinngg aauutthhoorr:: jXrsJhZih@[email protected]
-0.2 [1] G.Grimvall, The Electron-Phonon Interaction in Metals,
0.4M4omentum (p/a)0.48 0 0.02En0e.r0g4y 0w.0 (6eV0).08 0.1 editedbyE.Wohlfarth(North-Holland,NewYork,1981)
[2] J. M. Rowell et al., Phys. Rev. Lett.10, 334 (1963); D.
J. Scalapino et al.,Phys.Rev. 148 263 (1966)
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Eliashbergspectralfunctionofheavilyover-dopedPb-Bi2201. (2006)
(a) Measured nodal dispersion (empty blue circles) at 25 K [4] S.V.Dordevicetal.,Phys.Rev.B.71,104529(2005);E.
andtheextractedbareband(thickredline). Forcomparison, vanHeumen et al.,Phys. Rev.B.79, 184512 (2009)
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grey line)[27]and thebarebandusedbypreviousmaximum Phys. 75, 473 (2003); J.C.Campuzano et al., in The
entropy method (thin green line) [27] are also shown. The Physics of Superconductors, edited by K.H.Bennemann
bottom-rightinsetshowsthemeasuredelectronself-energyat and J.B.Ketterson(Springer New York, 2004),Vol.2;
different temperatures and their comparison with those cal- X.J.Zhou et al.,in Handbook of High-Temperature su-
culated from the extracted Eliashberg spectral function. (b) perconductivity: Theory and Experiment, edited by
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from infrared measurements on optimally-doped (red line) [6] J. R.Shi et al.,Phys. Rev.Lett.92, 186401(2004)
and over-doped (blue line) Bi-2201 [28]. (d) Raman spectra [7] X. J. Zhou et al.,Phys. Rev.Lett.95, 117001(2005)
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[12] N. Mannella et al., Nature(London)438, 474 (2005)
energy phonon near 70∼80 meV [30]. Considering the
[13] A. A.Kordyuk et al.,Phys. Rev.B.71, 214513 (2005)
ratherweakmagneticsignal[31],andtheabsenceofmag- [14] P. V. Bogdanov et al., Phys.Rev.Lett.85, 2581 (2000)
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the ∼70meVdispersionkinkisduetoelectroncoupling [17] A. Kaminski et al.,Phys. Rev.Lett.86, 1070 (2001)
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[20] W. T. Zhanget al.,Phys.Rev.Lett.100, 107002(2008)
to determine quantitatively the strength of the electron-
[21] T. Dahm et al., NaturePhys.5, 217(2009).
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even in the heavily over-doped cuprate samples. It is ous because the retarded effects of the electron-electron
interaction couldalsoactastheeffectivebosonicmodes.
muchstrongerthanthevaluesuggestedbythefirstprin-
Therefore, the “bare band” actually depends on the en-
ciples calculations (0.14∼0.20) [23]. Our present work
ergy scale we are looking at. For phonon modes which
thus asksfor a re-examinationoftheoreticalcalculations
usually have much smaller energy, the so called ”bare
onthestrengthofelectron-phononcouplinginthesecom- band” actually already includes the renormalization ef-
pounds. fects from the otherhigher energy collective excitations.
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to many other materials, thus promoting ARPES tech-