Direct Determination of the Electron-Phonon Coupling Matrix Element in a Correlated System

Huajun Qin,1 Junren Shi,2,1 Yanwei Cao,1 Kehui Wu,1 Jiandi Zhang,3 E.W. Plummer,3 J. Wen,4 Z. J. Xu,4

G.D. Gu,4 and Jiandong Guo1

1Beijing National Laboratory for Condensed-Matter Physics and Institute of Physics, Chinese Academy of Sciences,Beijing 100190, China

2International Center for Quantum Materials, Peking University, Beijing 100871, China3Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70808, USA

4Brookhaven National Laboratory, Upton, New York 11973, USA(Received 23 April 2010; published 15 December 2010)

High-resolution electron energy loss spectroscopy measurements have been carried out on an optimally

doped cuprate Bi2Sr2CaCu2O8þ�. The momentum-dependent energy and linewidth of an A1 optical

phonon were obtained. Based on these data as well as detailed knowledge of the electronic structure, we

developed a scheme to determine the electron-phonon coupling (EPC) matrix element related to a specific

phonon mode. Such an approach is general and applicable to elucidating the full structure of EPC in a

system with anisotropic electronic structure.

DOI: 10.1103/PhysRevLett.105.256402 PACS numbers: 71.27.+a, 71.38.�k, 74.72.�h, 79.20.Uv

The interaction between electrons and various collectiveexcitations (bosons) is the central ingredient for under-standing many of the novel physical properties incondensed-matter systems [1–3]. For simple systemswith isotropic electronic structures, such an interaction[electron-boson coupling (EBC)] can be characterized bythe Eliashberg spectral function �2Fð!Þ [4], which de-scribes the energy of the bosonic modes involved as wellas their coupling strengths. Thus it is important to directlydetermine the Eliashberg function. For instance, in Pb, itplayed a crucial role in establishing the BCS theory ofsuperconductivity [5]. Experimentally, there exist severaltechniques for determining the Eliashberg function, such asthe McMillan-Rowell inversion method applied to thetunneling data of the conventional superconductors [5,6],and more recently, the maximum entropy method inanalyzing the quasiparticle dispersion kink observed inangle-resolved photoemission spectroscopy (ARPES)measurements [7–9]. These approaches attempt to eluci-date the EBC by probing its effects on the electrons. On theother hand, effects of the EBC on bosonic modes are notcommonly investigated.

For more complex systems, neither the isotropic�2Fð!Þnor the anisotropic �2Fð!; kÞ determined from ARPES issufficient for fully characterizing EBC. A notable exampleis the high-Tc cuprate, which has highly anisotropic elec-tronic structure and unconventional dx2�y2 superconductiv-

ity. By probing the electrons alone, one can at best

determine �2Fð!; kÞ with dependence on the direction kalone using ARPES and the aforementioned maximumentropy method analysis [10]. However, unlike the caseof the conventional s-wave superconductors, the so-determined Eliashberg function cannot be directly relatedto the strength of the unconventional dx2�y2 pairing. This is

because the Eliashberg function and the dx2�y2 pairing

strength belong to different symmetries [11]. For thesesystems, it is necessary to resolve the full structure of theEBC [i.e., the matrix element gðk;k0Þ], which characterizesthe probability amplitude for the electron transition from kto k0 induced by interaction with bosons [12,13]. Becausemeasurements such as ARPES only contain informationabout the electron final state k0, with the contributions fromthe different initial states k integrated, it is in generalimpossible to determine the full structure of gðk;k0Þ byprobing electrons alone. Manifestation of the EBC on thebosonic modes also needs to be probed.In this Letter, we developed a method to resolve the full

structure of EBC by probing the bosonic modes, in thiscase phonons, with high-resolution electron energy lossspectroscopy (HREELS) [14] for the phonons and by usingexisting ARPES data for the electrons. The momentum-dependent linewidth, as well as the dispersion of an A1

optical phonon, were measured as a function of the direc-tion relative to the crystalline orientation. The initial andfinal states involved in the coupling and the correspondingstrength are simultaneously determined. The developedscheme is general and particularly applicable to manysystems with anisotropic electronic structures.We used the high-Tc cuprate Bi2Sr2CaCu2O8þ� (re-

ferred to as BSCCO in the following) [15,16] as a testsystem, whose electronic band structure, Fermi surface,and surface properties have been well characterized [17].Many experiments have revealed a signature of EBC.ARPES studies showed the electron self-energy renormal-ization in the form of kinks in the dispersion at energyscales of �70 meV below the Fermi energy in the nodalregion [18–21] and �40 meV near the antinodal region[13,22]. Direct measurements of phonon spectra have beencarried out with Raman spectroscopy [23], infrared spec-troscopy [24], HREELS [25], inelastic neutron scattering

PRL 105, 256402 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending

17 DECEMBER 2010

0031-9007=10=105(25)=256402(4) 256402-1 � 2010 The American Physical Society

[26], and inelastic x-ray scattering [27] for BSCCO andother high-Tc cuprates. However, a systematic study ofthe momentum-dependent phonon renormalization, whichis important for elucidating the details of EBC, is stillmissing.

The experiments were carried out in a commercial sys-tem containing variable-temperature scanning tunnelingmicroscope (STM) and HREELS (LK-5000). The basepressure was better than 1� 10�10 mbar. The supercon-ducting property of the optimally doped BSCCO singlecrystal was carefully characterized with a transition tem-perature at Tc ¼ 91 K. The sample was cleaved in situ invacuum. Atomically flat surface was obtained exposing theBi-O plane with the (1� 5) superstructure as characterizedwith STM and low-energy electron diffraction, which alsoindicates the relative angle of the crystalline orientation tothe measurement geometry. The HREELS measurementswere taken with an electron incident angle of 65�, andexcitation energy ranging from 3.5 to 50 eV. All the ob-served spectral features showed no excitation energy de-pendence. Therefore we employed an excitation energy of50 eV that offered a wide momentum transfer range. TheHREELS sample stage was coupled to a constant-flowliquid helium cryostat, and the measurements were takenat different temperatures ranging from 60 to 300 K, acrossthe superconducting Tc.

Angle-resolved HREELS of the optimally dopedBSCCO sample was measured along different directionsrelative to the crystalline orientation, as shown in Fig. 1. Atthe Brillouin zone (BZ) center (the in-plane momentum

transfer q ¼ 0), two main features are resolved near 50(F1) and 80 meV (F2), respectively. The feature F1 corre-sponds to the out-of-plane vibration of oxygen atoms in theBi-O plane [23–25], with two shoulders at both sides withenergies of �41 and �62 meV that were reported earlier[25]. The feature F2, with a major focus in the following,appears as a main peak at �80 meV with a broad tailextending to high energies. The vibration along the c axisof the apical oxygen atoms in the Cu-O semioctahedra isresponsible for the main peak [23–25], while the high-energy tail might be related to the imperfections. Also,considering the selection rule of HREELS, we were able torule out other energy loss mechanisms that fall into thisenergy range, such as the phonons of the in-plane breathingmodes of oxygen in the Cu-O2 plane because they are notdipole active along the surface normal and therefore un-detectable by HREELS at the BZ center. Along any mea-surement direction, we detected no obvious temperaturedependence across superconducting Tc.As shown in Fig. 1(c), the most prominent characteristic

of the momentum-resolved HREELS is observed along thenodal direction (’ ¼ 45�) where dispersion of the 80-meVfeature F2 softens significantly from the BZ center toward

the boundary (half of the BZ length �=a0 ¼ 0:82 �A�1).The momentum-dependent phonon energy (�q) and the

FWHM [�ðqÞ] along different directions relative to thecrystalline orientation are shown in Fig. 2. An abruptphonon softening occurs (from 81 to 74 meV) with an

onset at qk � 0:45 �A�1. With the same onset as indicated

by the arrows in Fig. 2, �ðqÞ increases suddenly from

Energy Loss (meV)

q//

(Å-1)

0.0

0.31

0.43

0.56

0.65

0.76

0.8840 60 80 100

1

2

10

25

25

x25

25

0.0

0.15

0.35

0.55

0.63

0.68

1

4

10

15

25

25

20

Inte

nsity

(ar

b. u

nits

)

F1

F2

0.77

q//

(Å-1)

20

40 60 80 100

F1

F2

200 nm 200 nm

(d)(c)(a)

(b)

A B

0.51

1.5

0

nm

BA

[110]

ee

65o

[010][100]

[001]

= 0o= 45o

FIG. 1 (color online). (a) STM image of the cleaved BSCCOsurface. The left inset shows the 1� 5 superstructure with the‘‘�5’’ direction aligning to [110] orientation. The right insetshows the low-energy electron diffraction patterns taken withbeam energy of 61 eV at room temperature. The line profilealong AB is displayed in the lower panel, which indicates a stepof 1.5 nm corresponding to the interspacing between two cleav-able BiO planes. (b) Schematic drawing of the scatteringgeometry for the angle-resolved HREELS measurements.(c),(d) The HREELS spectra measured with ’ ¼ 45� and 0�,respectively, with an excitation energy of 50 eVat 60 K. The bars(red) are guides to the eye for the phonon energy shift.

0.0 0.4 0.8

75

78

81

84 4530100

-45

q// (Å-1)

8

12

16

20

24

01030

-45

45

0.0 0.4 0.8

q// (Å-1)

FIG. 2 (color online). (a) �q and (b) �ðqÞ of the apical oxygenphonon at 60 K. The upper insets illustrate the azimuth angles ofqk with corresponding symbols in k space relative to the Fermi

surface, while the lower insets show the data taken at differenttemperatures across superconducting Tc. The dotted lines areguides to the eye. Because of the existence of the (1� 5)superstructure, there are two irreducible nodal directions (’ ¼�45�). Our measurements show identical momentum depen-dence along both directions.

PRL 105, 256402 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending

17 DECEMBER 2010

256402-2

8 meV at qk � 0:45 �A�1 to 18 meV at qk � 0:8 �A�1. This

behavior is a signature of strong electron-phonon coupling(EPC).

Both �q and �ðqÞ show a dramatic dependence on the

crystalline orientation ’. As ’ decreases from 45� (thenodal region), the anomalous behavior persists until ’ getsclose to the antinodal region (’ � 10�), when neither theenergy nor the linewidth shows a prominent momentumdependence at temperatures above or below Tc [also seeFig. 1(d)]. Therefore,�q and �ðqÞmeasured with different

’ values can be divided into two groups, respectively. Nearthe nodal region, �q decreases and �ðqÞ increases sud-

denly with the same onset. In contrast, near the antinodalregion, �ðqÞ increases slowly by less than 3 meV while�q

is almost momentum independent. It is worth noting thatthe HREELS spectra show a weak shoulder at �81 meV[see high-qk spectra in Fig. 1(c)], which can be deconvo-

luted from the main feature in all spectra by fitting withtwo Fano functions. The deconvolution indicates that thishigh-energy shoulder is weak and completely qk indepen-dent, consistent with the characteristics of the vibrationsconnected to the distorted or nonideal structure of thesample or to the extra oxygen atoms in the Bi-O layers[23,28]. Furthermore, the influence of the shoulder, as wellas that of the electron-hole pair excitations at high-energyside of F2 on the analyses of �ðqÞ, can be eliminated bymeasuring �ðqÞ from the left side of the peak only.

The dramatic behavior observed in the phonon spectra isundoubtedly related to the electronic structure and specifi-cally to EPC. The following is a formalism that allowsus to decouple the anisotropic electron properties from theEPC matrix element. In general, the momentum-dependentphonon broadening �EPCðqÞ induced by EPC can be writtenas [4]

�EPCðqÞ ¼ �2jgðqÞj2 Im½�ðq;�qÞ�; (1)

where �ðq;�qÞ is the Lindhard response function and

jgðqÞj2 is the EPC matrix element in a surface Brillouinzone. We have assumed the EPC matrix element to begðk; k0Þ ¼ gðk0 � kÞ gðqÞ, which is a good approxima-tion for the particular out-of-plane apical oxygen phononmode. The imaginary part of �ðq;�qÞ is given by

Im ½�ðq; !Þ� ¼ZBZ

dk

2�½fkþq � fk��ð@!þ �k � �kþqÞ;

(2)

where fk fð�kÞ is the Fermi distribution function and�k is the energy dispersion of the quasielectrons. Thequasiparticle dispersion �k of BSCCO has been measuredpreviously (ARPES) and fitted in a tight-binding phenome-nological model [17,29]. Im½�ðq;�qÞ� can then be calcu-

lated numerically, as shown in Fig. 3(a), which displaysrather complicated features originating from the highlyanisotropic electronic structure.

The experimentally observed �ðqÞ also include amomentum-independent background �0, which mostlyoriginates from the surface roughness of the cleaved sam-ple and the instrumentation broadening:

�expðqÞ ¼ �0 þ �EPCðqÞ: (3)

We estimate such a background using the value of theexperimental data near q ¼ 0: �0 ¼ �expðq ! 0Þ, since

there should have been no EPC-induced broadening atq ¼ 0. With the background subtracted, the EPC matrixelement can then be determined straightforwardly byjgðqÞj2 ¼ ��EPCðqÞ=2 Im½�ðq;�qÞ�. We note that the de-

termined jgðqÞj2 does not show a strong dependence on themeasurement direction ’, indicating that the dramaticallydifferent behaviors for nodal and antinodal directionsobserved in the phonon linewidth are mainly due to theanisotropy in the electronic structure.The determined EPC matrix element jgðqÞj2 is shown in

Fig. 4(a). Using the least-squares method, the data can bewell fitted by

jgqj2 ¼ fð2aþ bÞ � a½cosðqxa0Þ þ cosðqya0Þ�� b cosðqxa0Þ cosðqya0Þg2; (4)

which is consistent with an A1 phononmode that couples toelectrons in the lattice withC4v symmetry. Here a and b areconstants, representing the coupling strength between anatom displacement and its nearest and next-nearest neigh-boring sites, respectively.Only the electron transition satisfying both energy

and momentum conservations, i.e., k0 � k ¼ q and �k0 ��k ¼ @�q, contributes to the EPC-induced broadening.

This imposes a stringent constraint on the possible initial

0.0 0.4 0.8-4

-2

0

2

(a)

-Im

[(q

)] (

meV

-1Å

-2)

q// (Å-1)

Re[

(q)]

( m

eV-1

Å-2

)

= 0o

10o

30o

45o

(b)0

2

4

kx (Å-1)

k y(Å

-1)

0(EF)- 80 meV

q+80 meV

(c)

0

0

0.82

0.82-0.82-0.82

FIG. 3 (color online). (a),(b) The imaginary and real parts of�ðq;�qÞ calculated with Eqs. (2) and (6), respectively, at 60 K

for different ’. (c) The energy contour plot (dotted lines) of thequasiparticle band near EF of BSCCO. The initial states con-tributing to the EPC-induced broadening for qk � 0:47 �A�1 at

’ ¼ 45� can be determined by superimposing the density plot ofthe integrating function in Eq. (2) [dark (brown) areas] onto (c).Connecting by q (indicated by the arrows), the final states [light(pink) areas] correspond to an energy gain that equals @�q.

PRL 105, 256402 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending

17 DECEMBER 2010

256402-3

and final states of electron for a given phonon momentumq. Therefore, the determined jgðqÞj2 should be consideredas a subset of the more general jgðk; k0Þj2. Figure 3(c)depicts the electronic excitations near EF with a fixedmomentum transfer q contributing to the EPC-inducedbroadening.

To further test our analysis, we calculate the phononsoftening induced by EPC [4]:

�ðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�bareðqÞ2 þ 2�bareðqÞjgqj2Re½�ðq;�qÞ�

q; (5)

Re ½�ðq; !Þ� ¼ 2ZBZ

dk

ð2�Þ2fk � fkþq

�k � �kþq þ @!; (6)

where Re½�ðq; !Þ� is the real part of the Lindhard responsefunction, as plotted in Fig. 3(b), and �bareðqÞ is the ficti-tious bare phonon dispersion in the absence of EPC.Figure 4(b) shows the fittings to the experimental data ofthe phonon dispersion, as well as the assumed�bareðqÞ. It isimportant to note that, in contrast with �ðqÞ, �bareðqÞ arenearly isotropic, indicating that the large anisotropy ofthe phonon softening observed in the experimentoriginates from EPC.

In conclusion, we demonstrate that the EPC matrixelement jgðqÞj2 for a specific phonon mode can be deter-mined directly by measuring the phonon structure withHREELS, on the basis of a detailed knowledge of the

electronic structure of the system. Such an approach iscompletely general and could be applicable for other com-plex systems with highly anisotropic electronic structures.Our study also highlights the necessity of probing bosonsfor revealing the full structure of EBC in these complexsystems.This work is supported by Chinese NSF-10704084,

Chinese MOST (2006CB921300 and 2007CB936800),and NSF (DMR-0346826, DMR-0451163, DMS & E).J. Z. and J. G. gratefully acknowledge the support ofK. C. Wong Education Foundation, Hong Kong.

[1] J. P. Carbotte, Rev. Mod. Phys. 62, 1027 (1990).[2] M. L. Kulic, Phys. Rep. 338, 1 (2000).[3] O. Gunnarsson and O. Rosch, J. Phys. Condens. Matter 20,

043201 (2008).[4] Goran Grimvall, The Electron-Phonon Interaction In

Metals (North-Holland, Amsterdam, 1981), p. 196.[5] J.M. Rowell et al., Phys. Rev. Lett. 10, 334 (1963); D. J.

Scalapino et al., Phys. Rev. 148, 263 (1966).[6] W. McMillan and J. Rowell, in Superconductivity, edited

by R.D. Park (Marcel Dekker, New York, 1969), Vol 1.[7] J. R. Shi et al., Phys. Rev. Lett. 92, 186401 (2004).[8] X. J. Zhou et al., Phys. Rev. Lett. 95, 117001 (2005).[9] W. Meevasana et al., Phys. Rev. Lett. 96, 157003 (2006).[10] T. Chien et al., Phys. Rev. B 80, 241416(R) (2009).[11] N. Bulut and D. J. Scalapino, Phys. Rev. B 54, 14 971

(1996).[12] T. Devereaux et al., Phys. Rev. Lett. 93, 117004 (2004).[13] T. Cuk et al., Phys. Rev. Lett. 93, 117003 (2004).[14] H. Ibach and D. L. Mills, Electron Energy Loss

Spectroscopy and Surface Vibrations (Academic, NewYork, 1982).

[15] J. C. Campuzano et al., Phys. Rev. Lett. 83, 3709(1999).

[16] T. Cuk et al., Phys. Status Solidi B 242, 11 (2005).[17] A. Damascelli et al., Rev. Mod. Phys. 75, 473 (2003).[18] A. Lanzara et al., Nature (London) 412, 510 (2001).[19] P. V. Bogdanov et al., Phys. Rev. Lett. 85, 2581 (2000).[20] P. D. Johnson et al., Phys. Rev. Lett. 87, 177007 (2001).[21] J. D. Koralek et al., Phys. Rev. Lett. 96, 017005 (2006).[22] A. D. Gromko et al., Phys. Rev. B 68, 174520 (2003).[23] M. Kakihana et al., Phys. Rev. B 53, 11 796 (1996).[24] N. N. Kovaleva et al., Phys. Rev. B 69, 054511 (2004).[25] R. Phelps et al., Phys. Rev. B 48, 12 936 (1993).[26] L. Pintschovius et al., Phys. Status Solidi B 242, 30

(2005).[27] J. Graf et al., Phys. Rev. Lett. 100, 227002 (2008).[28] M. Osada et al., Phys. Rev. B 56, 2847 (1997).[29] M. R. Norman et al., Phys. Rev. B 52, 615 (1995).

= 0o

10o

30o

45o

= 0o

10o

30o

45o

FIG. 4 (color online). (a) jgqj2 determined by Eqs. (1)–(3) andthe fitting (solid lines) to Eq. (4) with a ¼ 0:47 meV �A andb ¼ 0:29 meV �A. The inset shows the measured �ðqÞ as wellas the data calculated with jgqj2 for different ’. (b) Measured

�ðqÞ with data calculated with Eq. (5). Here we assume a barephonon dispersion �bare¼83:5�1:16½cosðqxa0Þþcosðqya0Þ��1:11cosðqxa0Þcosðqya0Þþ0:32½cosð2qxa0Þþcosð2qya0Þ� ðmeVÞ,as plotted in broken lines.

PRL 105, 256402 (2010) P HY S I CA L R EV I EW LE T T E R Sweek ending

17 DECEMBER 2010

256402-4