Critical role of magnetic moments on lattice dynamics in YBa 2Cu3O6 Jinliang Ning1Christopher Lane2 3Yubo Zhang4Matthew Matzelle5Bahadur Singh6 Bernardo Barbiellini7 5Robert S. Markiewicz5Arun Bansil5and Jianwei Sun1

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Critical role of magnetic moments on lattice dynamics in YBa2Cu3O6

Jinliang Ning,1Christopher Lane,2, 3 Yubo Zhang,4Matthew Matzelle,5Bahadur Singh,6

Bernardo Barbiellini,7, 5 Robert S. Markiewicz,5Arun Bansil,5and Jianwei Sun1, ∗

1Department of Physics and Engineering Physics,

Tulane University, New Orleans, Louisiana 70118, United States

2Theoretical Division and Center for Integrated Nanotechnologies,

Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

3Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA

4MinJiang Collaborative Center for Theoretical Physics,

College of Physics and Electronic Information Engineering, Minjiang University, Fuzhou 350108, China

5Department of Physics, Northeastern University, Boston, MA 02115, USA

6Department of Condensed Matter Physics and Material Science,

Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India

7Department of Physics, School of Engineering Science,

Lappeenranta University of Technology, FI-53851 Lappeenranta, Finland

(Dated: December 22, 2022)

The role of lattice dynamics in unconventional high-temperature superconductivity is still vigor-

ously debated. Theoretical insights into this problem have long been prevented by the absence of

an accurate ﬁrst-principles description of the combined electronic, magnetic, and lattice degrees of

freedom. Utilizing the recently constructed r2SCAN density functional that stabilizes the antiferro-

magnetic (AFM) state of the pristine oxide YBa2Cu3O6, we faithfully reproduce the experimental

dispersion of key phonon modes. We further ﬁnd signiﬁcant magnetoelastic coupling in numerous

high energy Cu-O bond stretching optical branches, where the AFM results improve over the soft

non-magnetic phonon bands.

I. INTRODUCTION

Despite the discovery of unconventional high-

temperature superconductivity in the cuprates a little

over thirty years ago, there is still no consensus on

the underlying microscopic mechanism[1–6]. Early

theoretical works [7–9] suggested that conventional

electron-phonon coupling does not play an important

role in driving superconductivity in the cuprate family

of materials. However, recent experiments ﬁnd a more

nuanced picture[10–16]. A strong anomaly in the Cu-O

bond-stretching phonon beyond conventional theory

is observed near optimal doping and is associated

with charge inhomogeneity in the system[11]. Optical

spectroscopy reports ﬁnd that antiferromagnetic spin

ﬂuctuations are the main mediators for the formation

of Cooper pairs, but that the electron-phonon coupling

gives a contribution to the bosonic glue of at least

10%[17]. Moreover, recent ARPES observations suggest

that the electronic interactions and the electron-phonon

coupling reinforce each other in a positive-feedback loop,

which in turn drives a stronger superconductivity[16].

One reason why the role of phonons was dismissed by

the theoretical community is, in part, based on the failure

of density functional theory (DFT) calculations to ﬁnd

any appreciable electron-phonon coupling at both the lo-

cal density approximation (LDA) and generalized gradi-

ent approximation (GGA) levels[8]. This issue was fur-

ther compounded by these density functional approxima-

∗https://www.matcomp.org/; jsun@tulane.edu

tion’s inability to stabilize the correct antiferromagnetic

ground state in the parent phase, let alone its evolution

with doping[18, 19]. While corrections such as the ad-

dition of a Hubbard U [20–23] stabilize an antiferromag-

netic (AFM) ground state [24], they can spoil the good

agreement of PBE with experimental low-temperature

equilibrium volumes[25]. Therefore, it is evident that a

holistic ab initio treatment is required to simultaneously

satisfy both the lattice and magnetic degrees of freedom.

Recent advances in the construction of density func-

tionals gives new hope in addressing the electronic struc-

tures of correlated materials at a ﬁrst-principles level.

Speciﬁcally, the strongly-constrained and appropriately-

normed (SCAN) meta-GGA density functional [26, 27],

which satisﬁes 17 exact constraints, has demonstrated

excellent performance across a diverse range of bonding

environments. In particular, SCAN accurately predicts

the correct half-ﬁlled AFM ground state and the transi-

tion from the insulating to the metallic state with doping

observed in the cuprates [18, 19]. Moreover, SCAN pro-

vides improved estimates of lattice constants, across cor-

related and transition metal compounds [18, 19, 26–33].

Thus, by accurately capturing the electronic and mag-

netic ground state SCAN has the potential to provide

a good description of lattice dynamics and associated

electron-phonon coupling. Unfortunately, SCAN suf-

fers numerical problems that are exacerbated in phonon

calculations, making reliably obtaining accurate phonon

spectra from SCAN calculations a challenging task. Re-

cently, we have shown that a revised version of SCAN,

called r2SCAN [34], solves the numerical instability prob-

lem and delivers accurate, transferable, and reliable lat-

arXiv:2210.06569v2 [cond-mat.supr-con] 20 Dec 2022

tice dynamics for various systems with diﬀerent bonding

characteristics[35].

In this article, we examine the role magnetoelastic ef-

fects play in explaining the experimental phonon disper-

sion of pristine YBa2Cu3O6by taking advantage of the

numerically stable r2SCAN functional. We ﬁnd speciﬁc

branches of the phonon band structure to be sensitive

to the ground state magnetic order. Moreover, these

phonons correspond to breathing modes within the CuO2

plane, suggesting a sensitive dependence on magnetoe-

lastic coupling, which may facilitate a positive-feedback

loop between electronic, magnetic, and lattice degrees of

freedom.

II. METHODS

Ab initio calculations were performed using the pseu-

dopotential projector-augmented wave method[36, 37]

implemented in the Vienna ab initio simulation pack-

age (VASP) [38, 39] with an energy cutoﬀ of 600 eV

for the plane-wave basis set. Exchange-correlation ef-

fects were treated using the r2SCAN [34, 35] meta-GGA

scheme. The calculations are performed with a Gamma-

centered mesh having a spacing threshold of 0.15 ˚

A−1

for the k-space sampling. We used the experimental

low-temperature P4/mmm crystal structure to initialize

our computations[40]. All atomic sites in the unit cell

along with the cell dimensions were relaxed using a con-

jugate gradient algorithm to minimize the energy with

an atomic force tolerance of 0.001 eV/˚

A and a total en-

ergy tolerance of 10−7eV. The harmonic force constants

were extracted from VASP using the ﬁnite displacement

method (with displacement 0.015˚

A) as implemented in

the Phonopy code [41]. In some calculations, to give bet-

ter agreement with the experimental volume an eﬀective

U was added to r2SCAN[34].

III. RESULTS

Table I compares various properties calculated with the

non-magnetic (NM) and antiferromagnetic (AFM) states

to available experimental values for YBa2Cu3O6. In the

NM state, the lattice parameters and corresponding vol-

ume are the furthest away from the experimental values.

Speciﬁcally, the in-plane lattice parameters slightly un-

derestimate those from neutron scattering, whereas the

predicted c-height is signiﬁcantly larger, consistent with

previous studies employing PBE and SCAN[18, 19].

When the majority and minority spins are allowed to

self-consistently relax, we stabilize the experimentally

observed G-type AFM order across the planar copper

atomic sites. The predicted value of the magnetic mo-

ment on copper sites is 0.45 µB, which is in good accord

with the corresponding experimental value of 0.55 µB[45].

Due to the localization of electrons on the copper sites

the ab-plane expands with a concomitant shrinking of the

c-axis, bringing the equilibrium crystal geometry in line

with the experimental values. Since the phonon disper-

sion is determined by the inter-atomic forces, which de-

pends sensitively on the ground state electronic structure

and equilibrium atomic positions, the excellent perfor-

mance of r2SCAN in predicting the equilibrium ground

state bodes well for an accurate prediction of the lattice

dynamics.

Figure 1 (a) compares the theoretically predicted

phonon dispersion of YBa2Cu3O6in the AFM phase

with the experimental bands obtained by inelastic neu-

tron scattering [42–44]. For convenience we plotted the

phonon spectra in the NM Brillouin zone of Fig. 1 (b).

Overall, r2SCAN yields phonon frequencies and their dis-

persion along all three directions in momentum space in

excellent accord with experiment.

To analyze the sensitivity of the phonon bands to

magnetoelastic coupling we compare the NM and AFM

phonon dispersion in Fig. 2 with the experimentally re-

ported bands overlaid. By inspection, it is clear that

a majority of the phonon branches are not signiﬁcantly

aﬀected by the change in electronic environment [NM

vs. AFM], which is consistent with the small diﬀerence

in lattice constants given in Table I. However, for some

branches, the AFM order results in a hardening of the af-

fected phonon branches, i.e. a shift to higher frequency,

which improves agreement with experiment.

While overall agreement between experiment and the

AFM phonon dispersions is quite good, for several

branches there remains an underestimation of the hard-

ness of the atomic bonds. Interestingly, many of these

branches seem to be of relevance for the electronic prop-

erties of the cuprates. In Fig. 2, we highlight seven

branches, A-D along Γ-X, and E-G along Γ-M, for special

discussion, illustrating the associated atomic displace-

ments in Fig. 2(b) and listing properties of the A, B,

D, and G modes in Table II. Not only do these branches

mostly show strong eﬀects of magnetic order, but most

are also of experimental interest, having strong doping

dependence or changes at the superconducting transi-

tion. These branches are all related to the deformation

(stretching or buckling) of Cu-O bonds. Branches A and

B feature Cu-O bond out-of-plane buckling vibrations,

while branch F features buckling in the CuO2plane.

Branches C, D, E and G are all related to Cu-O bond

stretching, whereas branches C, D and G also have a no-

table admixture of apical Cu-O bond stretching. This

admixture has been noted in previous studies[46]. While

branch D is usually called a half-breathing mode con-

sidering only the motion of the in-plane oxygens, there

is a strong motion of the apical oxygen so we term this

branch as an ac-plane full breathing mode. Branch G,

normally called the full-breathing mode, is denoted as

the 3D full-breathing mode. Mode E is normally called

quadrupolar mode. Mode F is called the scissor mode

in terms of the in-plane Cu-O bond buckling behav-

ior. The Cu-O bond stretching modes are experimen-

tally interesting due to their softening upon doping [47],

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CriticalroleofmagneticmomentsonlatticedynamicsinYBa2Cu3O6JinliangNing,1ChristopherLane,2,3YuboZhang,4MatthewMatzelle,5BahadurSingh,6BernardoBarbiellini,7,5RobertS.Markiewicz,5ArunBansil,5andJianweiSun1,1DepartmentofPhysicsandEngineeringPhysics,TulaneUniversity,NewOrleans,Louisiana70118,UnitedStates...

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Critical role of magnetic moments on lattice dynamics in YBa 2Cu3O6 Jinliang Ning1Christopher Lane2 3Yubo Zhang4Matthew Matzelle5Bahadur Singh6 Bernardo Barbiellini7 5Robert S. Markiewicz5Arun Bansil5and Jianwei Sun1

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