Nonlinear lattice dynamics as a basis for enhancedsuperconductivity in YBa 2 Cu 3 O6.5
R. Mankowsky 1,2,3 *, A. Subedi4 *, M. Forst 1,3 , S. O. Mariager 5 , M. Chollet 6 , H. T. Lemke 6 , J. S. Robinson 6 , J. M. Glownia 6 ,M. P. Minitti 6 , A. Frano 7, M. Fechner 8 , N. A. Spaldin 8 , T. Loew 7, B. Keimer 7, A. Georges4,9,10 & A. Cavalleri1,2,3,11
Terahertz-frequencyopticalpulsescan resonantlydrive selectedvibra-tionalmodes insolidsanddeformtheir crystalstructures 1–3 .Incom-plex oxides, this method has been used to melt electronic order 4–6 ,drive insulator-to-metal transitions 7 and induce superconductivity 8 .Strikingly,coherent interlayer transportstrongly reminiscentofsuper-conductivity can be transiently induced up to room temperature(300 kelvin) in YBa 2 Cu 3 O6 1 x (refs 9, 10). Here we report the crystalstructure of this exotic non-equilibrium state, determined by fem-tosecond X-ray diffraction and ab initio density functional theory calculations.We findthatnonlinearlattice excitationin normal-stateYBa 2 Cu 3 O6 1 x atabove thetransitiontemperature of52 kelvincausesasimultaneousincreaseanddecrease intheCu–O 2 intra-bilayerand, re-spectively,inter-bilayerdistances,accompaniedbyanisotropicchangesin the in-plane O–Cu–O bond buckling. Density functional theory calculations indicate thatthese motionscause drastic changes in theelectronic structure. Among these, the enhancement in the d x 2{ y 2
character of the in-plane electronic structure is likely to favour superconductivity.
The response of a crystal lattice to strong, resonant excitation of aninfrared-activephonon mode canbedescribedbyseparatingthe crystalHamiltonian into its linear and nonlinear terms: H 5 H lin 1 H NL . Thelinear term H lin~ v 2
IR Q2IR 2 describes harmonic oscillations about the
equilibriumatomicpositions,with v IR denotingthe frequencyand QIR
thenormalcoordinate of theinfrared-active mode. In thelimit of lowest-order(cubic)couplingto othermodeswithgenericcoordinate QR ,thenon-linear term canbe written as H NL~ v 2
R Q2R 2{ a12QIR Q2
R { a21Q2IR QR .
In thisexpression, a12 and a21 areanharmoniccouplingconstants. (SeeMethods section and Extended Data Figs 1 and 2 for details on quartic
coupling.) For a centrosymmetric crystal like YBa2Cu3O6.5 , a12 QIR Q2R
is zero because QIR is odd in symmetry whereas Q2R is even and their
product therefore vanishes.Thus, the total Hamiltonian reduces to H ~ v 2
IR Q2IR =2z v 2
R Q2R =2{
a21Q2IR QR , which results in a shift in the potential energy minimum
along QR for any finite distortion QIR (Fig. 1a). Correspondingly, for aperiodically driven QIR mode, the dynamics are described by the cou-pled equations of motion
€QIR z 2c IR
_QIR z v
2IR QIR ~ f t ð Þz 2a21QIR QR
and€QR z 2c R
_QR z v2R QR ~ a21Q2
IR :
Figure 1b pictorially represents these dynamics. On resonant mid-infrared excitation of QIR , a unidirectional force is exerted along thenormalcoordinate QR , which is displaced bya magnitude proportionalto Q2
IR . This effect remains sizeable only as long asQIR oscillatescoher-ently, which occurs typically for several picoseconds.
We next discuss the specific case of YBa2Cu3O6.5 , which crystallizesin a centrosymmetric orthorhombic unit cell that has D2 h symmetry (point group), comprising bilayersof conducting CuO 2 planes separatedby an insulating layer containing Y atoms and Cu–O chains that con-trol the hole doping of the planes (Fig. 2a). The YBa2Cu3O6.5 samplecontained bothO-richandO-deficient chains,andexhibited short-range
* These authors contributed equally to this work.
1 Max Planck Institute for the Structure and Dynamics of Matter, 22761 Hamburg, Germany. 2 University of Hamburg, 22761 Hamburg, Germany. 3 Center for Free-ElectronLaser Science (CFEL), 22761Hamburg,Germany. 4 Centre de Physique The ´orique, E colePolytechnique,CNRS, 91128Palaiseau Cedex, France. 5 SwissLight Source, PaulScherrerInstitut, 5232Villigen,Switzerland. 6 Linac CoherentLight Source, SLAC National Accelerator Laboratory, Menlo Park 94025, California, USA. 7 Max Planck Institute for Solid State Research, 70569 Stuttgart, Germany. 8 Materials Theory, Eidgeno ¨ssischeTechnischeHochschuleZu ¨rich,8093Zu¨rich, Switzerland. 9 College deFrance,11 placeMarcelin Berthelot, 75005Paris,France. 10 Departementde Physiquede la Matie `re Condense´e (MaNEP),Universitede Gene ve, 1211 Gene`ve, Switzerland. 11 Department of Physics, University of Oxford, Clarendon Laboratory, Oxford OX1 3PU, UK.
15
10
5
0
–5
V R
( a . u .
)
3210–1–2QR amplitude (a.u.)
QR
Q IR
t
ba
Figure 1 | Coherent nonlinear lattice dynamics in the limit of cubiccoupling. a , A staticdistortion QIR shifts the equilibrium potential( V R ; dashedline) of all modes QR that are coupled through Q2
IR QR coupling, displacing the equilibrium position towards a new minimum (solid line). a.u., arbitrary units. b, The dynamical response of the two modes involves an oscillatory motion of the infrared mode (red line) and a directional displacement of QR
(blue line). The displacement is proportional to Q2IR and survives as long asQIR
is coherent.
Inter-bilayer
Intra-bilayer
Intra-bilayer
a b
c
c
c
a
a
a
O Cu Y Ba
c
Figure 2 | Structure of YBa 2 Cu3 O6.5 . a , Structure of orthorhombicYBa2Cu3O6.5 and motions of the optically excited B1u mode. The sketch on theleft shows the two tunnelling regions respectively within and between thebilayers. b, Cu–O chains, which are either filled (Cu on right) or empty (Cu onleft) in the ortho-II structure. c, Superconducting CuO2 planes (blue).
In our experiments, mid-infrared pump pulses of , 300 fs durationwere focused to a maximumfluence of , 4mJcm2 2 anda peak electricfield of , 3MVcm2 1 . These pulses were polarized along the c axis of YBa2Cu3O6.5 andtunedto resonancewith thesame670 cm
2 1 frequency (, 15 mm, 83meV) B1 u infrared-active mode 11 (Fig. 2a) that was previ-ouslyshown bymeansoftime-resolvedterahertzspectroscopy to enhanceinterlayer superconducting coupling 9,10 .
In analysing thenonlinear latticedynamics causedbythisexcitation,we note that the nonlinear term a21 Q2
B1uQR , where QB1u denotes the
normal coordinate of the infrared-active mode, is non-zero only if QR
isof A g symmetry, because thesquare of the irreducible representation
of B1u is A g . Thus,only A g modes cancouple to theoptically driven B1 u
motion shown in Fig. 2a. YBa2Cu3O6.5 has 72 optical phonon modes,of which 33 are Raman active. These can be further divided into 22 B g modes, which break in-plane symmetry, and 11 A g modes, which pre-serve thesymmetry of theunit cell (Extended Data Fig. 3).The geome-triesof these11 A g modes and their respectivecouplingstrengths to thedriven B1u modewerecomputedusing first-principlesdensityfunctionaltheory (DFT) calculations within the local-density approximation. Atthe3MVcm 2 1 fieldstrength ofthemid-infrared pump pulse, weexpect
a peak amplitudefor the B1u motioncorrespondingto a 2.2pm increasein thedistancebetween theapicalO atom andthe chainCuatom, whichwas used as a basis to calculate energy potentials of the A g modes for afrozen distortion of this magnitude (Fig. 3a). Only four phonon modes A g (15,21,29,74)werefoundtocouplestronglytothedriven B1 u mode,all involving a concerted distortion of the apical O atoms towards theCuO2 plane and an increase in Cu–O buckling (Fig. 3b). (The numbersin parentheses denote theindicesof thephononmodes, sortedin orderof increasing frequency.) The calculations also predict weak coupling tothree further modes A g (52, 53,61), consisting ofbreathing motion of theOatomsin theplane (Fig.3c). The remainingfourmodes A g (14,39,53,65)do not couple to the B1 u mode (Extended Data Table 1).
To experimentally determine theabsoluteamplitude of these distor-tions under the conditions relevant for enhanced superconductivity 9,10 ,
we measured time-resolved X-ray diffraction using 50 fs,6.7keVpulsesfrom the Linac Coherent Light Source free-electron laser, which wassynchronized with the optical laser that generated the mid-infraredpump pulses. Changes in diffraction intensity were recorded for fourBragg peaks at a base temperature of 100 K, which is above the equi-librium transition temperature T c 5 52 K. These peaks were observedto either increase or decrease promptly after excitation (Fig. 4) and torelax within the same timescale as the changes in the terahertz opticalproperties9,10 . For each Bragg reflection, we calculated changes in dif-fraction as functions of QB1u amplitude considering a displacement of only thefour dominant Ramanmodes A g (15, 21, 29, 74) orall 11modesin Fig. 3, taking intoaccountthe relative coupling strengths. We simul-taneously fitted the four experimental diffractioncurvesusing only twofree parameters: the amplitude of the directly driven B1 u motion and
the relative contributions of two exponential relaxation components
0.8
0.4
0.0
–0.4
E n e r g y ( m e V )
6420–2
A g displacement ( √u pm)
A g15 A g21 A g29 A g74
A g14
A g39 A g53 A g65
A g52 A g61 A g63
a b
c
Figure 3 | First-principles calculations of cubic coupling between 11 Ag
modes and the driven B 1 u mode. a , Energy potentials of all A g modes for afrozen B1u displacement of 0.14 A! u (u, atomic mass unit), corresponding toa change in apical O–Cu distance of 2.2 pm. The x axis is the amplitude of the A g eigenvector. Arrows indicate the potential minima. b, There is strong coupling to the A g (15, 21,29,74) modes,whichinvolvesa decrease in the apicalO–Cu distance and an increase in in-plane buckling. c, The A g (52, 61,63)modes are weakly coupled and govern a breathing motion of the oxygen atomsin the CuO2 plane.
0.6
0.4
0.2
0.0
–0.2
–0.4
Δ I / I ( % )
Δ I / I ( % )
Δ I / I ( % )
Δ I / I ( % )
420–2
Time (ps)
(211) 100 K
0.6
0.4
0.2
0.0
–0.2
–0.4420–2
Time (ps)
(014) 100 K
0.6
0.4
0.2
0.0
–0.2
–0.4420–2
Time (ps)
(211) 100 K
0.6
0.4
0.2
0.0
–0.2
–0.4420–2
Time (ps)
(204) 100 K
Figure 4 | Time-dependent diffracted peak intensity ( I ) for four Bragg reflections. A displacive lattice distortion is observed. The experimental dataarefitted (solidcurves)by adjustingthe B1 u amplitudeand the relative strengthof the two relaxation channels (t 1 5 1ps, t 2 5 7 ps) extracted from the optical
experiments of refs 9, 10. The relative amplitudes and signs of the curves aredetermined from the calculated structure using only the four most strongly coupled modes (green) or all A g modes (red). Error bars, 1s (67% confidenceinterval).
(t 1 5 1ps, t 2 5 7 ps) extracted from the terahertz measurements 9,10 .Very similar results were found when considering only the four dom-inant modes or all modes (Fig. 4, green and red fitting curves).
Thetransient latticestructure determined from thesefitsinvolves thefollowingelements.First, weobserve a decrease in thedistancebetweenthe apical O and the Cu atoms of the superconducting planes (Fig. 5).Thismotion is far smaller than,andis oppositein sign to,the differencein the staticapicalO positions betweenLa- andHg-basedcopperoxidesuperconductors, forwhich T c is greater at equilibrium 12 . Therefore, thetransient enhancement of superconductingtransportcannot beexplainedby this analogy. More suggestively, the Cu atoms are simultaneously drivenawayfromoneanother withinthebilayersandtowardsoneanotherbetween different bilayers. This spatially staggered motion is approxi-mately 0.63% of the equilibrium intra-bilayer distance (Fig. 5a) andqualitatively follows the decrease in intra-bilayer tunnelling and theincrease in inter-bilayer tunnelling 9,10 . Finally, an anisotropic 0.32u
increase in the in-plane O–Cu–O buckling (different along the aand b axes) is observed (Extended Data Table 2).
Although theJosephson couplingin layeredcopperoxide supercon-ductors involves many microscopic parameters that are not taken intoaccounthere13–15 , DFTcalculationsin thedistortedcrystalstructure wereused to assess the salient effects on the electronic properties (ExtendedData Figs 4–6). Our calculations predict a decrease of a few tens of millielectronvolts in theenergyof theO-deficientchain bands. Becauseat equilibrium these bandsare very close to theFermi energylevel, thissmall shiftstronglyreduces thehybridizationofthechains withtheplaneCu orbital, leading to a DFT Fermi surface with a stronger Cu d x 2{ y 2
character and higher hole doping. This effect is likely to favour super-conductivity.We also speculatethat as theDFT Fermisurface changesshapeandsize, it isquitepossiblethat thecharge-densitywaveordermay also be destabilized16–18 , which would also aid superconductivity. Thepresent calculations will serve as a starting point for a full many-body
treatment, tobecomplementedby more exhaustiveexperimental char-acterizations of the transient electronic structure.
Moregenerally, weseenonlinearphononicsasanewtoolfordynamicalmaterialsdiscovery, withoptical latticecontrolprovidinga perturbation—analogous to strain, fieldsor pressure—that caninduceexoticcollectiveelectronic behaviour.Knowledge of the non-equilibriumatomic struc-ture from ultrafast X-ray crystallography, which we provide here, isthe essential next step towards engineering such induced behaviour at
equilibrium.Online Content Methods, along with any additional Extended Data display itemsandSourceData,are available in the online versionof thepaper ; references uniqueto these sections appear only in the online paper.
Received 25 April; accepted 19 September 2014.
1. Forst, M. et al. Nonlinear phononics as anultrafast route to lattice control. NaturePhys. 7, 854–856 (2011).
3. Subedi, A., Cavalleri, A. & Georges, A. Theory of nonlinear phononics for coherentlight control of solids. Phys. Rev. B 89, 220301 (2014).
4. Tobey, R.I. et al. Ultrafast electronic phase transition inLa 1/2 Sr3/2 MnO4 bycoherent vibrational excitation: evidence for nonthermal melting of orbital order.Phys. Rev. Lett. 101, 197404 (2008).
5. Forst,M. etal. Drivingmagnetic order in a manganite by ultrafast latticeexcitation.Phys. Rev. B 84, 241104 (2011).
6. Forst,M. etal. Meltingof charge stripesin vibrationally driven La 1.875 Ba0.125 CuO4 :assessing the respective roles of electronic and lattice order in frustratedsuperconductors. Phys. Rev. Lett. 112, 157002 (2014).
7. Rini, M. et al. Control of the electronic phase of a manganite by mode-selectivevibrational excitation. Nature 449, 72–74 (2007).
8. Fausti, D. et al. Light-induced superconductivity in a stripe-ordered cuprate.Science 331, 189–191 (2011).
9. Kaiser, S. et al. Optically induced coherent transport far above T c in underdopedYBa2 Cu3 O6 1 d . Phys. Rev. B 89, 184516 (2014).
10. Hu, W. et al. Optically enhanced coherenttransport in YBa 2 Cu3 O6.5 by ultrafastredistribution of interlayer coupling. Nature Mater. 13, 705–711 (2014).
11. Homes,C. C. et al. Optical properties along the c-axis of YBa2 Cu3 O6 1 x , for x 5 0.50 R 0.95 evolution of the pseudogap. Physica C 254, 265–280 (1995).
12. Pavarini,E. etal. Band-structuretrend in hole-dopedcupratesand correlation withT c max . Phys. Rev. Lett. 87, 047003 (2001).
13. Shah, N. & Millis, A. J. Superconductivity, phase fluctuations and the c-axisconductivity in bi-layer high temperature superconductors. Phys. Rev. B 65,024506 (2001).
14. Chaloupka, J., Bernhard, C. & Munzar, D. Microscopic gauge invariant theory ofthe c-axis infrared responseof bi-layer cuprate superconductors and the originof the superconductivity-inducedabsorption bands. Phys. Rev. B 79, 184513(2009).
15. Yu, L. et al. Evidence for two separate energy gaps in underdoped high-temperature cuprate superconductors from broadband infraredellipsometry.Phys. Rev. Lett. 100, 177004 (2008).
16. Ghiringhelli, G. et al. Long-range incommensurate charge fluctuations in(Y,Nd)Ba 2 Cu3 O6 1 x . Science 337, 821–825 (2012).
17. Blanco-Canosa, S. et al. Resonant X-ray scattering study of charge density wavecorrelations in YBa 2 Cu3 O6 1 x . Phys. Rev. B 90, 054513 (2014).
18. Chang, J. et al. Direct observation of competition between superconductivity andcharge density waveorder in YBa 2 Cu3 O6.67 . Nature Phys. 8, 871–876 (2012).
Acknowledgements The research leading to these results received funding from theEuropean Research Council under the European Union’s Seventh FrameworkProgramme (FP7/2007-2013)/ERC Grant Agreement no.319286 (QMAC). Fundingfrom the priority program SFB925 of the German Science Foundation isacknowledged. Portions of this research werecarried out at the Linac CoherentLight
Source (LCLS) at the SLAC National Accelerator Laboratory. The LCLS is an Office ofScience User Facility operated for the US Department of Energy Office of Science byStanford University. This work was supported by the Swiss National SupercomputingCentre under project IDs404. Thiswork was supported by the Swiss National ScienceFoundation through its National Centre of Competences in ResearchMUST.
Author Contributions A.C. conceived this project. R.M. and M. Fo ¨rst led the diffractionexperiment,supported byS.O.M.,M.C.,H.T.L.,J.S.R., J.M.G.,M.P.M.andA.F. R.M.andA.S.analysed the data. A.S. performed the DFT calculations, with support from A.G., M.Fechner and N.A.S. The sample was grown by T.L. and B.K. R.M. and A.C. wrote themanuscript, with feedback from all co-authors.
Author Information Reprints and permissions information is available atwww.nature.com/reprints . The authors declare no competing financial interests.Readers are welcome to comment on the online version of the paper .Correspondence and requests for materials should be addressed toA.C. ([email protected]) or R.M. ([email protected]) .
–2
–1
0
1
2
d ( p m )
43210–1–2Time delay (ps)
0.60.40.20
Δ d / d
( % ) –0.1
–0.2
Intra-bilayerInter-bilayer
d
0.3
0.2
0.1
0.0
C h a n g e i n
α ( º )
1086420–2Time delay (ps)
6
4
2
0
Along a Along b
e
Δ α / α ( %
)
a b
c
c
a
c
b
a
Figure 5 | Transient lattice structure. a –c, We find a concerted displacivelattice distortion( b, c) with a decrease in theapical O–Cu distancesby 2.4pm atO-deficient sites and an increase in O–Cu–O buckling. d, The intra-bilayerdistance increases and the inter-bilayer distance decreases. Here the copperatoms of the planes at O-deficient chain sites (at left in a ) are used to define thepositions of the planes. e, The in-plane buckling angle a of the O–Cu–O bondincreases by 5% along both a and b at oxygen-deficient sites.
METHODSExperimental details. The X-ray diffraction measurements were carried out with6.7keVpulses atthe X-raypumpprobe (XPP)beamlineof theLCLS. Theenergyof the X-rays was selected using a channel-cut Si (111) monochromator with a reso-lution of 1 eV. The diffraction from each pulse was recorded individually withoutaveragingusing a diode.Shot-to-shot normalization to the intensity monitor afterthemonochromator wasused to correct thedetectedsignalsfor intensity andwave-length fluctuations of the X-ray pulses.The experiment was carried out in grazing-incidence geometrywith an angle of 5u between theX-rays and the sample surface.
TheYBa2Cu3O6.5 sample wasexcited withmid-infrared pulses of , 300fs dura-tion, generated by optical parametric down-conversion and difference-frequency generationof near-infrared pulses froma titanium–sapphire laser.These pulses weretunedto15 mm wavelengthwith2 mm bandwidth,chosento be inresonancewiththeB1u phonon mode. The measurement was carried out at a repetition rate of 120 Hz,while the repetition rate of the mid-infrared pulses was set to 60 Hz. This allowedus to measure theequilibriumand excitedstatesfor each time delay,to correctforany drifts of the free-electron laser.Structure factor calculations. To deduce theamplitudes Qi of theatomic displace-mentsalong the eigenvectors e i ofthe A g coordinates fromthe changes in scatteredintensity I / jF j
2, we calculated the corresponding modulation of the structurefactors F .Specifically,wequantifiedF ~ P j f j exp { iG :r j ,where G isthe reciprocal-
lattice vector of the corresponding diffraction peak, f j are the atomic scattering factors and r j isthe positionof the jthatomintheunitcell.Bycalculatingthestruc-ture factors for the equilibrium atomic positions r j~ r 0ð Þ j and the transient struc-ture r j
’
~ r 0ð Þ j z PiQie ji, the relativechange in diffracted intensitywas evaluated as
DI ~ F r j’
2{ F r j
2 . F r j 2.
Thechanges in signal amplitude arecalculated for A g amplitudes Qi aspredictedby DFT calculations for a certain infrared amplitude QB1u and compared with theexperimental findings to determine the quantitative values.DFT calculations. The phonon modes and the nonlinear phonon couplings wereobtainedusingDFTcalculations withplane-wave basis setsand projectoraugmented-wave pseudopotentials19,20 as implemented in the VASP software package21 . Thelocal-density approximation wasused for the exchange andcorrelations. We useda cut-off energy of 950 eV for plane-wave expansion and a 43 8 3 4 k-point gridforthe Brillouinzoneintegrationin theself-consistent cycles. We used theexperi-mental lattice parameters of the YBa2Cu3O6.5 ortho-II structure, but relaxed theinternal coordinates. The interatomic force constants were calculated using thefrozen-phonon method 22 , and the PHONOPY software package was used to cal-culate the phononfrequencies and normal modes 23 . After thenormal modes wereidentified, the total energy was calculated as a function of the QIR and QR phononmode amplitudes to obtain the energy surfaces. The nonlinear coupling betweenthe infrared and Raman modes was obtained by fitting the energy surfaces to thepolynomial
H ~12
v2IR Q
2IR z
12
v2R Q
2R { a21Q2
IR QR
Coupling strengths of the Ag to the B 1 u modes. The energy potential of an A g
Raman mode in the presence of a cubic nonlinear coupling to an infrared mode isV R ~ v 2
R Q2R 2{ a21Q2
IR QR . Atequilibrium, when theinfrared mode isnot excited,thepotentialof theRaman mode hasa minimum at QR 5 0 because thestructureisstable at equilibrium. However, when the infrared mode is excited externally, theminimum of theRaman mode shiftsby anamount a21Q2
IR v 2R . The minimaof the
A g modes energy potentials as obtained from DFT calculations for a frozen dis-placementof the B1
umodeof0.14A! u arereported in ExtendedDataTable1. The
mode displacements are given in terms of the amplitude Qi of the dimension-less eigenvectors of the mode and have units of A! u, where u is the atomic massunit. The atomic displacement due to an amplitude Qi of a mode is given by U j~ Qi. ffiffiffiffiffi m jp e ij, where U j is the displacement of the jth atom, m j is the massof this atom and e ij is the corresponding component of the normal-mode vector.We note that e ij is normalized and dimensionless.Transientcrystal structure. The static crystal structureof YBa2Cu3O6.5 (ortho-II)isgiveninExtendedDataTable 2.Thelatticeconstantsare a 5 7.6586A,b 5 3.8722 Aand c 5 11.725A, as determined by single-crystal X-ray diffraction at 100 K. Thelight-induced displacements of the atomic positions at peak change in diffractedintensity are reported in Extended Data Table 2.Changes in the electronic structure. The changes in the electronic structure dueto the light-induced distortions were studied using the generalized full-potentialmethod within the local-density approximation as implemented in the WIEN2k package24 . Muffin-tin radii of 2.35, 2.5, 1.78 and 1.53 Bohr radii (one Bohr radiusequals 0.529 A) were used for Y, Ba, Cu and O, respectively, and a 203 40 3 20
k-point grid was used for the Brillouin zone integration. The plane-wave cut-off was set such that RK max 5 7.0, where K max is the plane-wave cut-off and R is thesmallmuffin-tinradius, thatis, 1.53Bohr radii. Thedensity of states wasgeneratedwith a 323 64 3 32 k-point grid. Calculations are presented for the equilibriumstructureand the transientdisplaced structurefor three B1u amplitudes: the ampli-tude 0.3 A! u determined here, the amplitude 0.8 A ! u estimatedin refs 9, 10, and alarger amplitude of 1.2 A! u.
Our calculated electronic structure of the equilibrium YBa2Cu3O6.5 (ortho-II),shown in Fig. 2, is similar to the one calculated previously 25,26 . The bands near the
Fermi levelare derived from the Cu 3d states from both the planesand the chains.The four bands that have high dispersion along the path X–S–Y–C are due to theplanar Cu d x 2 { y 2 states. Thebroadbandthathas very littledispersionalong S–Yisdue to the Cu d z 2 states from the filled chain. The O-deficient chains that controlthe hole doping give rise to fairly flat electronic bands with dominant Cu d xz andd yz character that are very close to the Fermi level at the Y point. The electronicstructurecalculationspredictsome hybridizationbetween thesebandsandthe planarCu bands, which creates an anticrossing near Y. In the equilibrium structure, thisanticrossingis closeto the Fermi level, givingrise to pocketswithunfilled-chainCucharacter in the Fermi surface (Extended Data Figs 4 and 5).
The displacements due to the nonlinear couplings cause noticeable changes tothe electronic structure around the Fermi level. There are three main effects.
(i) The light-induced displacements reduce the width of the planar Cu bands,which leads to an increase in the planar Cu contribution to the density of states atthe Fermi level.
(ii) The atomic displacements cause a transfer of charge from the planes to theO-deficient chains. As the unfilled Cu chain bands decrease in energy and movebelow the Fermi level with increasing light-induced displacements, the planar Custates increase in energy, becoming less occupied. That is, there is an effective holedoping of the planarCu states owing to the light-induceddisplacements (ExtendedData Fig. 6).
(iii) The changes in the relativeoccupationsof the bands alsocausea topologicalchange in the Fermi surface. The light-induced displacements increase the filling ofthe unfilled-chainCu bands,which decreasesthe size of thepocketsin theFermisurface. Above a threshold QB1u amplitude of 0.8 A! u, the O-deficient chain Cubands become fullyfilled andthe Fermisurface consists solelyof two-dimensionalplanar Cu sheets and one-dimensional filled-chain Cu sheets.Quartic-order coupling. To verify that the nonlinear phonon coupling is domi-natedby the third-order contribution,as discussed in themaintext, we checkedforsignals at the next (fourth) order, described by the term Q2
IR Q2 j in the nonlinear
Hamiltonian
H NL~ 12
v 2R Q2
j { a21Q2IR Q j{ a22Q2
IR Q2 j
As noted in the text, when the directly driven infrared mode is of B1u symmetry,the only modes to which there is non-zero third-order coupling are those of A g symmetry.However, coupling to any mode Q j, in particular to in-plane B g modes,is allowed through Q2Q2 coupling.
We note first that small-amplitude B1u excitations would simply renormalizethefrequencyof a secondmode Q j. This can be directlydeducedfromthe equationof motion, wherethe driving forceis given bythe couplingterm a22Q2
IR Q j,whichislinear in Q j:
€Q jz 2c jQ j _Q jz v
2 j Q j~ 2a22Q2
IR Q j
On QIR displacement, the anharmonically coupled mode experiences a renorma-lization of its frequency: v j
’
~ v j ffiffiffiffiffiffiffiffiffiffiffiffi1{ 2a22Q2IR p .
However, above a threshold amplitude QIR , the frequency of the second mode
Q j becomes imaginary 3
andthe latticebecomes unstable. Importantly, suchinstab-ilitycan takeplace in two directions, dependingon therandom instantaneous stateof the system (mode amplitude Q j and its velocity dQ j/dt ). This manifests in achange from a parabolic to a double-well energy potential as shown in ExtendedData Fig. 1.
Hence, fourth-order effects need to be identified by analysing the diffraction of each individual X-ray pulse, whereas the unsorted average is expected to be zeroeven ifthe quartic couplingis sizeable.In theexperiment, wesorted allpositive andnegativedeviations fromthe averagesignalof all shotsto obtain the Q2Q2 responseat a specifictime delay. Averagingthem separately andsubtracting negativedevia-tions from positive then gives the intensity changes from Q2Q2 only.
Time-resolved X-ray diffraction was measured for four Bragg reflections, sens-itive to A g andto B2 g displacements. The results of theseexperiments are showninExtended Data Fig. 2.
Within our resolution, we find no evidenceof quarticcontributions.The ampli-tude of the infrared motion is below the threshold beyond which fourth-ordercoupling induces lattice displacements.
19. Blochl, P. E. Projectoraugmented-wave method. Phys. Rev. B 50, 17953(1994).
20. Kresse, G. & Joubert,D. From ultrasoft pseudopotentials to the projectoraugmented-wave method. Phys. Rev. B 59, 1758 (1999).
21. Kresse, G. & Furthmu¨ ller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B 54, 11169(1996).
23. Togo,A., Oba, F. & Tanaka, I. First-principles calculations of the ferroelastictransition betweenrutile-typeand CaCl 2 -type SiO 2 at highpressures. Phys. Rev. B78, 134106 (2008).
25. Carrington, A. & Yelland, E. A. Band-structure calculations of Fermi-surfacepockets in ortho-IIYBa 2 Cu3 O6.5 . Phys. Rev. B 76, 140508(R) (2007).
26. Elfimov, I. S.,Sawatzky, G. A. & Damascelli, A. Theory of Fermi-surface pockets andcorrelationeffectsin underdoped YBa 2 Cu3 O6.5 .Phys.Rev.B 77, 060504(R) (2008).
Extended Data Figure 1 | Nonlinear lattice dynamics in the limit of cubicand quartic coupling. Dashed lines: potential energy of a mode QR as afunction of mode amplitude. a , A static distortion QIR shifts the potential of allmodes QR that are coupled through a Q2
IR QR coupling (solid line), displacing
the equilibrium position towards a new minimum. b, Owing to quartic Q2IR Q
2 j
coupling, the energy potential of a coupled mode Q j is deformed symmetrically on static distortion QIR . The frequency of the mode first softens until it isdestabilized, which manifests in a double well potential (solid line).
Extended Data Figure 2 | Changes in diffracted intensity of specific Bragg reflections from fourth-order coupling for different time delays betweenpump and probe pulse. We find no evidence of lattice distortions originating
from fourth-order contributions to the phononcoupling. The amplitude of theinfrared mode QIR is below the threshold beyond which fourth-order effectsdestabilize coupled phonon modes. Error bars, 1 s (67% confidence interval).
Extended Data Figure 3 | Phonon modes of ortho-II YBa 2 Cu3 O6.5 . Sketches of the resonantly excited B1u mode and all 11 A g modes for which the coupling strengths (Extended Data Table 1) have been calculated.
Extended Data Figure 4 | Band structure of the equilibrium (black line)and transient crystal structure. The band structure is plotted along C(0,0,0) R X(0.5,0,0)R S(0.5, 0.5,0)R Y(0, 0.5, 0)R C(0,0,0) R
Z(0, 0, 0.5) for transient displaced structures corresponding to QB1u amplitudesof 0.8A! u (a ), which is the amplitude estimated for the geometry of refs 9, 10,and 1.2 A! u (b).
Extended Data Figure 5 | Cuts of the Fermi surface of the equilibrium(black line)and transient crystal structures (reddashedline) at k
z 5 0. Inthe
equilibrium structure, the bands of the unfilled-chain Cu atoms give rise topockets in the Fermi surface. The light-induced displacements shift thedensities of states of these bands to lower energies, increasing the filling and
reducing thepockets. Abovea thresholdof 0.8A ! u, the O-deficientchain bandsbecome fully filled, the pockets close and the Fermi surface consists solely of two-dimensional planar Cu sheets and one-dimensional filled-chain states.TheFermi surface is shownin thedisplacedstatefor QB1u amplitudes of 0.8A! u(left) and 1.2 A! u (right).
ExtendedData Figure 6 | Changes in thedensityof statesin theCuO 2 planeand the Cu–O chains. These are obtained from a projection of the density of states onto the copper muffin-tin spheres. a , b, In the light-induced state, thedensity of states of the O-deficient chain lowers in energy (a ), whereas theopposite effect is observed for the Cu in the plane below (b). This correspondsto charge transfer from theplanes to thechains. c, d, The density of statesof the
filled chain Cu is not strongly affected (c). The bands of the planar Cuatoms narrow, which leads to an increase in the density of statesnear theFermilevel both at sites with filled (d) and empty chains (b). The effect is already visible for a QB1u amplitude of 0.3 A! u (blue) but becomes more prominent forlarger displacements of 0.8 A! u (purple) and 1.2 A! u (green).
Energy potential minima of the Ag modes as obtained from DFT calculations for a frozen displacement of the B 1 u mode of 0.14A ˚ ! u, which corresponds to a change in apical O–Cu distance of 2.2 pm.
Extended Data Table 2 | Equilibrium structure of YBa 2 Cu 3 O6.5 and light-induced displacements
Equilibrium Structure (Å) Displacements (pm) 0.3Å u Displacements (pm) 0.8Å u Displacements (pm) 1.2Å uAtom x y z x y z x y z x y zY 1.922 1.936 5.863 0.111 0.000 0.000 0.769 0.000 0.000 1.835 0.000 0.000Y 5.737 1.936 5.863 -0.111 0.000 0.000 -0.769 0.000 0.000 -1.835 0.000 0.000
