PHYSICAL REVIEW B 85, 144510 (2012)

In-plane anisotropy in spin-excitation spectra originating from chain states in YBa2Cu3O6+ y

Tanmoy DasTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

(Received 5 December 2011; revised manuscript received 12 March 2012; published 9 April 2012)

We present a random-phase-approximation-based multilayer spin-susceptibility calculation for the trilayerYBa2Cu3O6+y (YBCO) system (including bilayer CuO2 planes and uniaxial CuO chain layer) in thesuperconducting state. We show that the observed in-plane anisotropy in the spin-excitation spectrum ofYBCO—which is often interpreted as evidence for the electron nematic phase—can alternatively be explainedvia incorporating the uniaxial CuO chain’s contribution. We demonstrate that the neutron spectra in YBCO isdominated by the contribution from the fourfold symmetric CuO2 plane state as in other cuprates; however, itacquires an in-plane anisotropy via finite interlayer coupling with the chain state. The result rules out the claimthat an electronic nematic phase is responsible for the pseudogap state in YBCO.

DOI: 10.1103/PhysRevB.85.144510 PACS number(s): 74.72.Kf, 71.10.Hf, 73.43.Nq, 74.40.Kb

I. INTRODUCTION

The formation of electronic liquid crystals is intimatelyrelated to two-dimensional anisotropy phenomena, which areof intense current interest in the field of strongly correlatedelectrons, semiconductors,1 two-dimensional (2D) electrongas,2 graphene,3 as well as in cold atom systems.4 Thesurge of interest has recently been extended to various fami-lies of unconventional superconductors including cuprates,5–9

pnictides,10 Sr3Ru2O7,11 and heavy fermion URu2Si2.12,13 Themicroscopic origin for the in-plane anisotropy in these su-perconducting (SC) systems which consistently accommodatetetragonal crystal symmetry is more interesting and exotic. Ageneral question that underlies all these materials is whetherthe anisotropic structure is related to an emergent nematicelectronic liquid-crystal phase14 or merely comes from thesubtle crystallographic distortion15 or something else.

In cuprates, pnictides, and heavy-fermion URu2Si2(Refs. 5–13) systems, this question has posed widespreadresearch interests with the hope to explain their unusualnormal-state phases from which the SC state is arguablyderived. In cuprates, the experimental evidence of the in-planeanisotropy has been taken seriously as definite evidence for thepresence of the electronic liquid-crystal phase,14 and also beenextended to explain the so-called 1/8 doping problem wheresuperconductivity is strongly quenched.16 Here we provide adifferent mechanism for the origin of the in-plane anisotropyin YBa2Cu3O6+y (YBCO). We demonstrate that, even in theabsence of any crystal anisotropy or “nematic” ground state,an anisotropy is introduced to the SC CuO2 plane state via theinterlayer electronic coupling with the quasi-one-dimensional(1D) CuO chain bands.

Experimental implications of anisotropy in cuprates. TheCuO2 plane state is present in all cuprates, and is arguably re-sponsible for most of the exotic low-energy phases which varydramatically within the same family. A stripe-like modulationhas been well established in La2−x(Sr/Ba)xCuO4 (LS/BCO),17

Bi2Sr2CaCuO8+δ (BSCCO),18 Ca2−xNaxCuO2Cl2,19 andHgBa2CuO4+δ (HBCO).20 On the other hand, most evidencefor the anisotropic structure has been obtained in YBCOvia transport,5 Nernst,9 and inelastic-neutron-scattering (INS)measurements,6 with some recent indication of this phasein BSCCO from scanning tunneling microscopy (STM).7,8

These material variant pseudogap properties raise an essentialquestion: Is the mechanism of pseudogap different in differentcuprates, or do such differences merely arise from the material-dependent electronic structure? By closely looking at theYBCO crystal, we indeed notice that this system hosts bothquasi-2D CuO2 layers and quasi-1D CuO chain layers, asshown in Fig. 1(a). The Fermi surfaces (FSs) measured byangle-resolved photoemission spectroscopy (ARPES) in theunderdoped [Fig. 1(b)] and overdoped [Fig. 1(c)] regionsexhibit that while a pseudogap is opened in the CuO2 bi-layerbands, the 1D CuO chain FS remains ungapped at all finitedopings.21,22 Additionally, a visual estimation to the Fermi arcseen in the underdoped case does not reveal any significantpresence of fourfold symmetry breaking, thus indicatingthat the origin of the pseudogap is not directly related tothe observed anisotropy in this system. On the other hand,YBCO being a polar compound, there exists a charge transferbetween the layers as a result of the electric fields generatedby the polar unit cell.23,24 We compute the INS spectra byrealistic multilayer BCS-RPA (random-phase approximation)calculation to demonstrate that the interlayer coupling betweenthe chain and plane layers stipulates a twofold symmetry in theelectronic states even when the isolated bulk planer states donot intrinsically involve any spontaneous fourfold symmetrybreaking.

The rest of the paper is organized as follows. In Sec. II,we introduce the tight-binding model for the trilayer YBCOsystem and the corresponding multilayer spin-excitation cal-culation. The results for the static spin susceptibility at zeroenergy and the role of two plane FSs and the chain FS are givenin Sec. III. In the SC state, the origin of the in-plane anisotropyin the dynamical spin-excitation spectrum is demonstrated inSec. IV. The doping and energy dependence of the FS nestingand the in-plane anisotropy is calculated and discussed in thecontext of the YBCO phase diagram in Sec. V. Finally weconclude in Sec. VI.

II. TIGHT-BINDING MODEL

To realistically model the low-lying electronic states ofYBCO, one requires a minimum of three band models toincorporate the bilayer splitting and the chain band, comingfrom CuO2-CuO2-CuO trilayers.25 In each isolated layer, the

144510-11098-0121/2012/85(14)/144510(5) ©2012 American Physical Society

TANMOY DAS PHYSICAL REVIEW B 85, 144510 (2012)

Cu O BaY

t

t

t'

tc

tcp

tpp

x=0.05

Experiment

Present theory

-0.5 0.5

-0.5

0.5

kx (2π/a)

k y(2

π/b)

(a) (b)

(c)

(d)

MIN

MAX

Inte

nsit

y

x=0.29

x=0.15

-0.5 0.5

FIG. 1. (Color online) (a) Crystal structure of YBCO with twoCuO chain layers residing at the top and bottom of the unit cell,and two CuO2 planes intervening between them. Arrows dictate allthe electronic hoppings considered in this calculation (except t ′′,which is not shown). (b),(c) ARPES results of FS in the extremeunderdoping and overdoping regions of the YBCO sample (Ref. 22).Here x is the hole doping, not the oxygen content. (d) ComputedFS in the paramagnetic state near the optimal doping region. Thespectral weight is most dominant on the chain band, while the bilayersplitting in the CuO2 bands is hardly visible, due to spectral weightdistribution among the trilayers.

single electron hopping between Cu dx2−y2 orbitals is mediatedby O atoms, which is captured by effective tight-bindinghopping parameters. For the plane and chain layers, it issufficient to consider up to third and next-nearest-neighborhoppings, respectively [see Fig. 1(a)], which construct theisolated layer dispersions ξ(p/c)k as26

ξpk = −2t(cx + cy) − 4t ′cxcy − 2t ′′(c2x + c2y) − μp, (1)

ξck = −2tccy − μc. (2)

Here cαx/y = cos (αkx/ya) and μc/p are the chemical potentialsfor the plane and chain bands, which encode the on-sitepotential imbalance and other crystal field effects betweenthe two layers. We consider the interlayer coupling withinsingle-electron tunneling matrix elements tpp and tcp, whichobey periodic boundary conditions along the c axis. Thereforethe final Hamiltonian in the trilayer basis is

Hk =

⎛⎜⎝

ξpk tpp tcp

tpp ξpk tcp

tcp tcp ξck

⎞⎟⎠ . (3)

Note that while isolated plane dispersion ξpk obeys fourfoldsymmetry, the total Hamiltonian in Eq. (3) breaks thissymmetry due to its coupling with the chain band via tcp.

In other words, all eigenvalues and eigenstates including thosefor the plane states are twofold symmetric. The computedspectral weight function on the 2D momentum space at theFermi level is plotted in Fig. 1(d) at a representative dopingx = 0.15 (this choice of doping represents YBCO6.45 at whichthe neutron-scattering data is compared below), which agreeswell with the corresponding FS trend obtained in ARPES data[Figs. 1(b) and 1(c)].22

Multilayer BCS susceptibility

The superconductivity is taken to be d-wave-like for allthree bands, where the value of the SC gap amplitude isconstrained by the experimental value22,27 of � = 34 meV.28

In the SC state, the layer-dependent BCS spin-susceptibilitytensor is computed from

χjj ′0ii ′ (q,iωm) = − 1

4N

∑k,n,ν,ν ′

Mν,ν ′ii ′jj ′(k,q)

×Tr[Gν(k,in)Gν ′(k + q,in + iωm)].

(4)

For each band, G is the 2 × 2 Green’s function, made of normaland anomalous parts in the Nambu-Gorkov’s notation.29

The interlayer coupling matrix element is Mν,ν ′ii ′jj ′(k,q) =

φiν(k)φj†

ν (k)φi ′ν ′(k + q)φj ′†

ν ′ (k + q), where φiν(k) is the eigen-

state of band ν or layer i. In this notation, ν (ν ′) is the initial(final) eigenstate, consists of hybridization between i and j (i ′and j ′) layers.

RPA. Finally, we incorporate the many-body interactionwithin RPA as χ(q,) = χ (q,)[1 − U χ(q,)]−1 (the sym-bol tilde represents tensor). Since we consider only the dx2−y2

orbital for all three layers, no interorbital couplings such asHund’s coupling are involved here. We include the intralayerinteraction for the plane and chain as Up and Uc, respectively,and the interlayer one between plane-plane and chain-planelayers as Upp and Ucp.30

III. ANISOTROPY IN STATIC SUSCEPTIBILITY AND THEROLE OF FS NESTING

Figure 2 shows the relevant components of the static RPAsusceptibilities in the normal state and their correspondingFS nestings. The intraplanar component in Fig. 2(a) obtains aleading incommensurate nesting at qpp

a = (π (1 − δa),π ) andqpp

b = (π,π (1 − δb)), where δa and δb measure the degree ofincommensurability along a and b bond directions. Comparingthese two nesting peaks, we immediately find that there is aprominent anisotropy present in both intensity of susceptibilityand magnitude of δa and δb. The origin of this anisotropy canbe traced back to the FS anisotropy of the plane band shownby blue lines in Fig. 2(e). This anisotropy in the plane state isintroduced via its hybridization with the chain layer (recall thatthe isolated plane dispersion ξpk itself obeys fourfold symme-try). The bilayer splitting in the plane bands is clearly visiblenear k = (0, ± π ) points while near k = (±π,0), it mixes withthe chain bands. The nesting between these split bands, qpp′

,produces a susceptibility peak in the vicinity of q ∼ �, as seenin Fig. 2(b). The intrachain and chain-plane components of thesusceptibility in Figs. 2(c) and 2(d), respectively, are naturally

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In-PLANE ANISOTROPY IN SPIN-EXCITATION . . . PHYSICAL REVIEW B 85, 144510 (2012)

0 40 0 10 0 2 0 3χpp

pp χpp'pp' χcc

cc χcpcp

0 1

0

1

-0.5 0.5

-0.5

0.5

0

1

(a) (b)

qx (2π/a)

kx (2π/a)

k y(2

π/b)

q y(2

π/b)

(c) (d)

(e) (f)

0 1

qapp

qbpp

qpp'

qccqcp

q y(2

π /b)

qx (2π/a)

FIG. 2. (Color online) (a)–(d) Various representative componentsof the static RPA susceptibility in the normal state at x = 0.15. Thedominant nesting in each component is depicted by arrows. Thearrows of same color are overlayed on the corresponding FS in (e).(e) The FS obtained from Fig. 1(d). (f) Total RPA susceptibility(summing over all components).

dominated by the quasi-1D components. Summing over itscomponents at each q and ω, we see that the full susceptibilityaccommodates the coexistence of several nestings with relativeintensities [see Fig. 2(f)]. The competition between variousinstabilities originating from the leading nesting is stronglydoping dependent, and hence on the FS topological changesas discussed below in Fig. 4.

IV. ANISOTROPY IN SPIN RESONANCEAND SUPERCONDUCTIVITY

INS directly measures the imaginary part of the spinsusceptibility as a function of q and ω. Within the standarditinerant picture of unconventional superconductivity, a spinresonance develops in the SC state due to the sign reversalof the SC order parameters at the FS nesting vectors.31,32

On the basis of this theory, only the qpp vector contributesto the BCS susceptibility, while the other nesting vectorsdisappear or become suppressed in the SC state (of course,the inelastic scattering activates other possible scatteringchannels; however, their intensities are shown earlier to becomparatively less32). The computed spin-resonance spectraplotted along the a- and b-bond directions in Figs. 3(a) and3(b), respectively, reveal that the anisotropy between δa andδb acquires energy dependence due to the d-wave nature ofthe SC gap. At zero energy, both qpp

a/b touches the nodalpoints, where δa/b attain their maximum amplitudes. Withincreasing energy, qpp

a/b approach Q, thereby reducing thevalues of δa/b. Finally, at ω = 40 eV, both q

pp

a/b march atQ = (π,π ), the intensity of χ ′′ becomes maximum, and aresonance peak appears. Although the anisotropy disappearsat the resonance in its magnitude (δa = δb = 0), it remainspresent in its intensity as can be seen from Fig. 3(e), due to theoverlap matrix-element effect between plane and chain layers[see Eq. (4)]. In fact, the anisotropy becomes reversed at andabove the resonance energy, as demonstrated in Fig. 3(c). The

Along a Along b

ω=20 meV ω=40 meV

0

40

80

ω=3 meV ω=50 meV

0 1 0 1

(a) (b) (c)

0 1

47

33

51

292420

0.3 0.70 1 0 10

01

1

q(d) (e)

(f)

(g) (h)ω

(meV

)

Inte

nsit

yIn

tens

ity

qx (2π/a) qy (2π/a)

qx (2π/a) qx (2π/a)q y

(2π/

b)q y

(2π/

b)q (2π/a)

FIG. 3. (Color online) (a),(b) Computed imaginary part of thetotal susceptibility χ ′′ in the SC state plotted along (0,π ) → (2π,π )in (a) and (π,0) → (π,2π ) in (b) (a- and b-bond directions,respectively). (c) Several representative constant energy cuts of χ ′′

are shown as a function of q along a- and b-bond directions. Eachspectrum is shifted vertically by a constant value for visualization.The peak position of each spectrum is marked by a small verticalline. (d),(e) Two representative constant energy profiles of χ ′′ shownbelow and near the resonance, respectively. (f) Two representativemomenta cuts through (e) (arrows) shown near Q, and compared withcorresponding experimental data (symbols of same color) (Ref. 6).The intensities of the theoretical and experimental data are normalizedarbitrarily to ease the comparison. The dashed line through theexperimental data is a guide to the eye. (g),(h) The constant energyexperimental profiles (Ref. 6).

results agree well with the experimental data of YBCO6.45,presented in Figs. 3(f)–3(h); the difference in the energy scalebetween Figs. 3(d) and 3(g) notwithstanding.

V. DOPING DEPENDENCE OF ANISOTROPY

Finally, we study the doping evolution of the anisotropy inYBCO, and compare it with the YBCO phase diagram in Fig. 4.In the extreme underdoped region x = 0.025, the leading staticinstability in χ ′(q,ω = 0) commences at qpp with both themagnitude and intensity of δa being larger than those of δb, asseen in Fig. 4(a). With increasing doping, we notice severalinteresting characteristic evolutions of the susceptibility: (1)Both δa and δb increase almost monotonically with doping;however, the degree of anisotropy or the difference δa − δb,does not have significant doping dependence up to x ∼ 0.15[see Fig. 4(e)]. (2) With further hole doping, the incommensu-rate peaks along the bond directions qpp

a and qpp

b begin to split.

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TANMOY DAS PHYSICAL REVIEW B 85, 144510 (2012)

0 0.1 0.2 0.30.1

0.3

δ a ,δ

b

Hole doping

(e)

0 10

1

(a) x=0.025 (b) x=0.10 (c) x=0.20 (d) x=0.30

0.4

0.2

x=0.025

x=0.30

6.0 6.5 7.0 Oxygen content

100

200

300

T(K

)

AF

M

SC0

q y(2

π/b )

qx (2π/a)

FIG. 4. (Color online) (a)–(d) Static susceptibility at ω = 0 plot-ted at four representative dopings. The evolution of the dominatingnesting from the vicinity of Q [arrow in (a)] toward the chain bandnear (π/2,0) [arrow in (d)] as a function of doping is clearly visible.(e) The degree of incommensurability along a and b axes, definedin text by δa and δb, respectively, is shown as a function of doping.The interesting jump in the value of δb at x = 0.15 is associated withthe shift of spectral weight from the plane layer to the chain layer(see text). Different color shadings represent standard YBCO phasediagrams (Refs. 21,27). Insets: Computed FSs at two extreme dopingsof YBCO.

Above x = 0.15, the splitting becomes sufficiently large sothat the maximum intensity in the vicinity of Q does not remainaligned along the bond directions, but rotates by 45◦ to thediagonal directions [see Figs. 4(c) and 4(d)]. It is noteworthythat the 45◦ of the INS spectra as a function of doping isobserved earlier in La-based materials where such rotationoccurs at the beginning of the SC dome around x = 0.05,33

while in the present case of YBCO, this crossover dopingcoincides with the optimal doping of this compound. (3) Thedoping gradually increases intensity at other small nestingvectors qpp′/cc near the � point, which all participate in nestingthe van Hove singularity (VHS) region of the quasi-2D planebands, and the quasi-1D FSs of the chain bands, respectively,as defined in Fig. 2(e). With hole doping, these flat FSs movetoward each other [compare FSs at x = 0.025 and x = 0.30 inthe insets to Fig. 4(e)], and thereby enhance nestings. SimilarVHS nesting has been theoretically predicted to be presentin the single-layer BSCCO band structure34 and observed inthe quasiparticle interference pattern of the same compoundas probed by STM35 (in YBCO, this nesting additionallybreaks fourfold symmetry). However, as mentioned earlier,

these nestings do not survive in the SC state, because theydo not involve sign reversal of the d-wave SC gap, and thuscannot be detected by INS. (4) Above the optimal doping,the maximum intensity at δb jumps suddenly from the (π,π )region (plane bands) to the chain band region at qcp [dictated byarrows in Figs. 4(c) and 4(d)] as plotted by square symbols inFig. 4(e). As the chain band is 1D, qcp only appears along onedimension, while the incommensurate peak along the a-bonddirection continues to be dominated by plane states. This iswhy the anisotropy between δa and δb becomes reversed abovethe optimal doping.

VI. DISCUSSION AND CONCLUSION

Does anisotropy cause nematic pseudogap order? Coin-cidentally, the crossover of the leading instability from theplane to the chain band occurs at the doping where evidence ismounting in YBCO that the pseudogap also disappears.20,21,27

On the basis of these results as well as strong evidence forthe presence of FS reconstruction in the pseudogap regionof YBCO, we conjecture that if pseudogap arises from a FSnesting, the planar nesting qpp ∼ (π,π ) that dominates in thisdoping region is most likely to be the one.36,37 The leadingnesting at qpp also prohibits the possibility of stabilizing acharge-density wave (CDW) in the 1D chain bands via Peierlsinstability at qcc/cp in the pseudogap state. CDW can at mostbe present as a secondary order.38 Recalling the ARPES resultspresented in Figs. 1(b) and 1(c), our results are consistent withthe observed ungapped chain bands at all finite dopings.

Conclusions. In summary, we provide a microscopicmodel for the trilayer YBCO superconductor to explain theobserved in-plane anisotropy in the spin-excitation spectrumof this compound. We calculated multilayer BCS susceptibilitywithin RPA formalism to show that while the isolatedplanar electronic states intrinsically obey fourfold rotationalsymmetry, its observables exhibit a reduced twofold symmetrydue to interlayer coupling with the quasi-1D chain bands. Thecomputed INS spectra exhibit considerable anisotropy in itsincommensurate structure which varies with energy up to theresonance energy scale, in quantitative agreement with exper-imental data at the same doping. We also present a verifiableprediction that the anisotropy becomes reversed above optimaldoping in YBCO. We conclude that an incommensurate orderwith planar nesting around (π,π ) is the leading nesting asin other cuprates, while the anisotropy is a secondary effectcoming from the chain state. Our results rule out the earlierpostulate that nematic order may be the origin of the pseudogapphase in YBCO.

ACKNOWLEDGMENTS

This work is supported by the US DOE through the Officeof Science (BES) and the LDRD Program and facilitated byNERSC computing allocation.

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15In YBCO, it is observed that a nematic phase forms around T ∼150 K (YBCO),6 which is higher than the “pseudogap” temperatureT ∗ ∼ 200 K for this compound,27 and also higher than the onsetof orthorhombic distortion or the low-temperature orthorhombicphase in LSCO17 and in pnictide; J.-H. Chu et al., Science 329, 824(2010).

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23It is known that YBCO crystals mostly cleave between the CuOchain and the BaO layers, exposing the CuO chain to the surface.21

24G. Gruner, Density Waves in Solids (Addison-Wesley, Reading,MA, 1994).

25W. A. Atkinson, Phys. Rev. B 59, 3377 (1999).26The parameters are obtained after fitting to the experimental FS25

as (t,t ′,t ′′,tc,tpp,tcp,μp,μc) = (600, − 120,150,400,400,600, −10, − 600) in meV. The doping is calculated within a standardrigid band shift approximation.

27S. Huefner, M. A. Hossain, A. Damascelli, and G. A. Sawatzky,Rep. Prog. Phys. 71, 062501 (2008).

28The widely used proximity model for YBCO suggests that thed-wave superconductivity is induced to the chain layer from theplane one due to interlayer coupling [D. K. Morr and A. V.Balatsky, Phys. Rev. Lett. 87, 247002 (2001)]. We take the gapamplitude to be the same for all bands to reduce the number ofparameters.

29The components of the BCS Green’s function areGν

11/22(k,iωn) = (iωn + ξνk)/(ω2n + E2

νk) and Gν12(k,iωn) =

Gν∗21(k,iωn) = �νk/(ω2

n + E2νk), where the SC quasiparticles are

Eνk = ±√

ξ 2νk + �2

νk. (G12 is sometimes denoted by F ). Weperform Matsubara frequency summation over the fermionic indexn, while the bosonic Matsubara frequency is treated by takinganalytical continuation to the real axis as iωm = ω + iδ (δ is smallbroadening).

30Denoting the intralayer interactions for the plane and chain as Up

and Uc, respectively, and the interlayer one between plane-plane andchain-plane layers as Upp and Ucp , the nonvanishing componentsof the U tensor are U ii

ii = Ui and Uij

ij = Uij (where i,j = p,c).The values of each component of U are constrained by theircorresponding components of χ to avoid an instability (see Fig. 2),and the obtained values are (Up,Uc,Upp,Ucp) = (3,2,0.5,0.4)t .

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