First-principles calculation of electron-phonon coupling in doped KTaO 3 Tobias Essweinand Nicola A. Spaldiny Materials Theory Department of Materials ETH Zurich Switzerland

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First-principles calculation of electron-phonon coupling in doped KTaO3

Tobias Esswein∗and Nicola A. Spaldin†

Materials Theory, Department of Materials, ETH Zurich, Switzerland

(Dated: February 21, 2023)

Motivated by the recent experimental discovery of strongly surface-plane-dependent superconduc-

tivity at surfaces of KTaO3single crystals, we calculate the electron-phonon coupling strength, λ, of

doped KTaO3along the reciprocal-space high-symmetry directions. Using the Wannier-function ap-

proach implemented in the EPW package, we calculate λacross the experimentally covered doping

range and compare its mode-resolved distribution along the [001], [110] and [111] reciprocal-space

directions. We ﬁnd that the electron-phonon coupling is strongest in the optical modes around

the Γ point, with some distribution to higher kvalues in the [001] direction. The electron-phonon

coupling strength as a function of doping has a dome-like shape in all three directions and its in-

tegrated total is largest in the [001] direction and smallest in the [111] direction, in contrast to the

experimentally measured trends in critical temperatures. This disagreement points to a non-BCS

character of the superconductivity. Instead, the strong localization of λin the soft optical modes

around Γ suggests an importance of ferroelectric soft-mode ﬂuctuations, which is supported by our

ﬁndings that the mode-resolved λvalues are strongly enhanced in polar structures. The inclusion

of spin-orbit coupling has negligible inﬂuence on our calculated mode-resolved λvalues.

I. INTRODUCTION

Perovskite-structure potassium tantalate (KTaO3,

KTO) exhibits many interesting phenomena, resulting

from its high dielectric constant [1], strong spin orbit

coupling [2] and charged ionic layers [3]. The strong

spin-orbit coupling (SOC), caused mainly by the heavy

tantalum ion, leads to a band splitting of up to 400 meV

[2,4] and possible applications in spintronic devices [5,6].

The high dielectric constant, associated with a quantum

paraelectric state [7] similar to that of SrTiO3(STO) [8],

indicates proximity to ferroelectricity, which is predicted

to yield a large strain-dependent Rashba spin splitting

[9,10]. The need to compensate the alternating charged

ionic layers at the surfaces is predicted to induce lattice

polarization in thin ﬁlms [11], and leads to the accumu-

lation of compensating charges at the surfaces of bulk

samples [3]. The origin and nature of the compensating

charge are still open questions, with reports of conducting

two-dimensional electron gases (2DEGs) [12,13], charge-

density waves with strongly-localized electron polarons

[14], and terrace-like structures of alternating termination

[15], depending on the annealing atmosphere and temper-

ature.

Perhaps the most intriguing behavior of KTO is its re-

cently discovered low-temperature superconductivity on

electron doping [16]. Superconductivity was ﬁrst achieved

using ionic liquid gating on the (001) surfaces of KTO sin-

gle crystals, for which critical temperatures (Tc) of up to

50 mK were found at 2D doping concentrations of between

2×1014 and 4 ×1014 cm−2[16,17]. Note that these val-

ues correspond to 3D doping concentrations of approxi-

mately 4.1×1020 cm−3to 1.2×1021 cm−3, considerably

higher than the ∼1.4×1020 cm−3possible using chemical

∗tobias.esswein@mat.ethz.ch

†nicola.spaldin@mat.ethz.ch

doping with barium in bulk KTO [18]. (For the conver-

sion between 2D and 3D carrier concentrations see Ref. 16

and the Appendix). A subsequent study of LaAlO3-

capped KTO (110) surfaces, with 2D doping concentra-

tions of 7 ×1013 cm−2, reached markedly higher critical

temperatures up to 0.9 K [19]; (111)-oriented KTO inter-

faces with either EuO or LaAlO3showed even higher Tcs

of up to 2.2 K at similar carrier concentrations [20]. Note

that no superconductivity was found down to 25 mK at

(001)-oriented KTO interfaces at these lower carrier con-

centrations [20]. More recently, in an ionic liquid gating

setup similar to that of Ref. 16, but at lower 2D doping

densities of around 5 ×1013 cm−2, superconductivity was

found at the (110) and (111) surfaces with Tcof around

1 K and 2 K respectively, and not at the (001) surface

down to 0.4 K [21]. The reported critical temperatures

from the literature are collected as a function of carrier

concentration in ﬁgure 1.

The mechanism underlying the superconductivity, as

well as its strong and unusual dependence on the ori-

entation of the surface or interfacial plane, are not yet

established. Indeed, even in the related quantum para-

electric STO, in which superconductivity was found more

than half a century ago [22,23], the pairing mechanism

remains a subject of heated debate (for a recent review

see Ref. 24). While the persistence to low carrier concen-

trations [23] and the anomalous isotope eﬀect [25] chal-

lenge conventional BCS theories [26,27], it is likely that

electron-phonon coupling in some form, as well as prox-

imity to ferroelectricity [28–33] play a role. Spin-orbit

coupling has also been implicated [34–37], and would be

consistent with the observed higher critical temperatures

in KTO, with its heavy tantalum ion, compared to STO

[32,33,38]. The surface-plane dependence in KTO is

captured by a model in which out-of-plane polar displace-

ments of the Ta and O ions allow a linear coupling of the

transverse optical (TO) phonon to the electrons in the t2g

(dxy, dyx and dzx) orbitals; this coupling would otherwise

go to zero as the phonon wavevector qapproached Γ [39].

arXiv:2210.14113v3 [cond-mat.mtrl-sci] 20 Feb 2023

The strong dependence of the superconducting Tcon sur-

face orientation is then explained by diﬀerent inter-orbital

hopping of electrons between adjacent tantalum sites via

the oxygen orbitals, with the highest hopping at (111)

surfaces, followed by (110) surfaces, and no hopping al-

lowed by symmetry at (001) surfaces.

FIG. 1. Superconducting critical temperatures, extracted

from studies by Ueno et al. [16], Chen et al. [19], Liu et al.

[20,39], Ren et al. [21], and Mallik et al. [40]. The (111)

surface/interface reaches the highest Tcof up to 2 K (dark

blue markers), followed by the (110) surface/interface reaching

almost 1 K (bright yellow markers). The original paper by

Ueno et al. [16] reported a Tcup to 0.05 K for the (001)

surface at high doping, but more recent publications at lower

doping found no (001) superconductivity down to 0.025 K [20]

and 0.4 K [21] (red markers at bottom).

It is clear that a thorough picture of the electron-

phonon coupling as a function of electron doping and

throughout the Brillouin Zone in KTO is an essential

step towards developing a complete theory of its super-

conductivity. While the electron-phonon coupling has

been calculated from ﬁrst principles for STO [41], to our

knowledge it is lacking for KTO, and the goal of this

work is to remedy this gap. Here we report the mode-

resolved electron-phonon coupling strengths, λ, obtained

using ﬁrst-principles calculations based on density func-

tional theory, for cubic KTaO3across the range of exper-

imentally accessible electron doping values. We extract

the mode-resolved total λas a function of carrier density,

and focus in particular on diﬀerences between the [001],

[110] and [111] high-symmetry directions, which are recip-

rocal to the corresponding experimentally measured sur-

face and interfacial planes. Additionally, for one doping

value, we compare the behavior with and without spin-

orbit coupling, and for polar and non-polar structures to

determine the eﬀect of both properties. Our main ﬁndings

are that i) the calculated total electron-phonon coupling

strengths do not follow the measured trends in supercon-

ducting Tc; ii) λis concentrated in the optical modes

around Γ and polar distortions increase λby a factor of

approximately ﬁve, suggesting a mechanism involving the

polar soft mode; iii) spin-orbit coupling has negligible in-

ﬂuence on the calculated electron-phonon coupling.

II. METHODS

To calculate the forces and total energies we use den-

sity functional theory within the generalized gradient

approximation (GGA) as implemented in the Quantum

ESPRESSO 7.0 and 7.1 codes [42–44]. We describe the

exchange and correlation using the PBEsol functional

[45], and perform the core-valence separation with the

ultrasoft GBRV [46,47] and pslibrary (to compare re-

sults with and without spin-orbit coupling) pseudopo-

tentials [48]. We use a kinetic energy cutoﬀ of 60 Ry

(816 eV) for the wavefunctions and a 24 ×24 ×24 k-point

mesh including Γ for all unit cells. Doping is achieved

in the range from 0.0001 to 0.1 electrons/formula unit

(e/fu) using the background-charge method with Gaus-

sian smearing of 1 meV width. Total energies are con-

verged to 1 µeV (7.35 ×10−8Ry) and forces to 0.1 meV/˚

(3.89 ×10−6Ry/bohr).

Both unit-cell size and shape, as well as internal coor-

dinates, are fully relaxed, resulting in a non-polar cubic

perovskite structure with a lattice constant of 3.988 ˚

which is very close to the experimental one of 3.989 ˚

[49]. Phonons are calculated on a 4 ×4×4 q-point mesh,

with convergence tests on 6 ×6×6 and 8 ×8×8 q-point

meshes showing only minor quantitative diﬀerences (see

appendix on page 8). The resulting phonon dispersion for

very low doping, using the PBEsol-relaxed unit cell, cor-

responds well with the room-temperature phonon disper-

sion calculated recently using Quantum Self-Consistent

Ab Initio Lattice Dynamics (QSCAILD), which is based

on DFT and a self-consistent sampling method to cap-

ture both thermal and quantum ﬂuctuations [50]. Note

that, since our study is for doped KTO, we neglect the

LO-TO splitting, assuming that it will be screened by the

metallicity. To our knowledge, the evolution of the LO-

TO splitting from the insulating to the metallic state as

a function of doping in transition-metal oxides has not

been determined, and this would be an important topic

for future work.

The electron-phonon coupling properties are calculated

using the EPW 5.4.1 and 5.5 codes [51,52], which are

included in the Quantum ESPRESSO package. The rel-

evant electronic bands in KTaO3are the three Ta-5dt2g

bands, which are reproduced using maximally localized

Wannier functions as implemented in the Wannier90 code

[53], used internally by EPW. The electron-phonon ma-

trix elements are ﬁrst calculated on coarse 24 ×24 ×24 k-

point and 4 ×4×4 q-point meshes and then interpo-

lated onto ﬁne grids using maximally localized Wannier

functions. We use a random ﬁne mesh with 100000000 k

points to calculate the mode-resolved electron-phonon

coupling strengths, λqν , along a path between cubic high-

symmetry points with 200 q points between each point.

Convergence test results can be found in the appendix on

page 8.

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First-principlescalculationofelectron-phononcouplingindopedKTaO3TobiasEssweinandNicolaA.SpaldinyMaterialsTheory,DepartmentofMaterials,ETHZurich,Switzerland(Dated:February21,2023)Motivatedbytherecentexperimentaldiscoveryofstronglysurface-plane-dependentsuperconduc-tivityatsurfacesofKTaO3singlecrysta...

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First-principles calculation of electron-phonon coupling in doped KTaO 3 Tobias Essweinand Nicola A. Spaldiny Materials Theory Department of Materials ETH Zurich Switzerland

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