Generalized Rashba electron-phonon coupling and superconductivity in strontium titanate Maria N. Gastiasoro1 2Maria Eleonora Temperini3Paolo Barone4and Jos e Lorenzana1y

2025-04-27 0 0 1.26MB 18 页 10玖币

侵权投诉

Generalized Rashba electron-phonon coupling and superconductivity in strontium

titanate

Maria N. Gastiasoro,1, 2, ∗Maria Eleonora Temperini,3Paolo Barone,4and Jos´e Lorenzana1, †

1ISC-CNR and Department of Physics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185, Rome, Italy

2Donostia International Physics Center, 20018 Donostia-San Sebastian, Spain

3ISC-CNR Institute for Complex Systems, via dei Taurini 19, 00185 Rome, Italy

4SPIN-CNR Institute for Superconducting and other Innovative Materials and Devices,

Area della Ricerca di Tor Vergata, Via del Fosso del Cavaliere 100, 00133 Rome, Italy

(Dated: April 24, 2023)

SrTiO3is known for its proximity to a ferroelectric phase and for showing an “optimal” doping for

superconductivity with a characteristic dome-like behaviour resembling systems close to a quantum

critical point. Several mechanisms have been proposed to link these phenomena, but the abundance

of undetermined parameters prevents a deﬁnite assessment. Here, we use ab initio computations

supplemented with a microscopic model to study the linear coupling between conduction electrons

and the ferroelectric soft transverse modes allowed in the presence of spin-orbit coupling. We ﬁnd

a robust Rashba-like coupling, which can become surprisingly strong for particular forms of the

polar eigenvector. We characterize this sensitivity for general eigenvectors and, for the particular

form deduced by hyper-Raman scattering experiments, we ﬁnd a BCS pairing coupling constant of

the right order of magnitude to support superconductivity. The ab initio computations enable us

to go beyond the linear-in-momentum conventional Rashba-like interaction and naturally explain

the dome behaviour including a characteristic asymmetry. The dome is attributed to a momentum

dependent quenching of the angular momentum due to a competition between spin-orbit and hopping

energies. The optimum density for having maximum Tcresults in rather good agreement with

experiments without free parameters. These results make the generalized Rashba dynamic coupling

to the ferroelectric soft mode a compelling pairing mechanism to understand bulk superconductivity

in doped SrTiO3.

I. INTRODUCTION

A research surge in recent years has uncovered novel

behavior involving the interplay between ferroelectricity

(FE) and superconductivity (SC) in SrTiO3(STO) [1,2].

Noteworthy examples include strain enhanced supercon-

ductivity [3,4] in samples with polar nanodomains [5,6]

and self-organized dislocations with enhanced ferroelec-

tric ﬂuctuations [7]. Alternative methods for tuning

ferroelectricity such as Ca or 18O isotope substitution

also present enhanced superconducting critical temper-

atures Tc[8–12]. In doped samples with a global po-

lar transition (and thus global broken inversion symme-

try) signatures of mixed-parity superconductivity have

been reported [13]. Theoretically, these doped polar sam-

ples have also been recently proposed as a platform for

the emergence of exotic phases such as Majorana-Weyl

superconductivity [14] and odd-frequency pair correla-

tions [15].

Despite having experimentally established a qualita-

tive connection between the superconducting and ferro-

electric phases in STO, and while there is some indication

that the dominant mode responsible for pairing might be

the ferroelectric soft transverse optical (TO) mode [16],

there is still no consensus about the pairing mechanism

in this system [17]. One the of the prominent theoretical

∗maria.ngastiasoro@dipc.org

†jose.lorenzana@cnr.it

challenges is its very low density of states and Fermi en-

ergy due to low carrier densities, which places supercon-

ductivity in STO outside of the standard BCS paradigm.

Proposed pairing theories include the dynamical

screening of the Coulomb interaction due to longitudi-

nal modes [18–21] recently challenged in Ref. [22], bipo-

laron formation [23], and diverse approaches to linear

coupling [1,24–32] or quadratic coupling to the FE

mode [33–37]. The last two proposals have the advantage

that, coupling electrons directly to the FE soft mode, pro-

vide a natural explanation to the sensitivity to the FE

instability.

In the more general context of polar or nearly po-

lar metals, the coupling between electrons and the

soft FE modes has received attention only very re-

cently [26,27,30,32,37–40]. The reason probably be-

ing that, as already mentioned, the FE soft modes in

these systems have a predominantly transverse polariza-

tion. Within the conventional electron-phonon interac-

tion scheme, this implies a decoupling of the soft modes

from the electronic density to linear order [41]. One

promising alternative route involves going to next order

by coupling the electrons to pairs of TO modes, i.e. the

quadratic coupling mentioned above [33–36,42]. Another

possibility, and subject of the present article, is the linear

vector coupling to the electrons, allowed in the presence

of spin-orbit coupling (SOC) [15,26,27,39,43–46].

In a recent work [30] we derived a vector coupling

based on a Rashba-like interaction within a minimal mi-

croscopic model and ab initio frozen phonon computa-

arXiv:2210.05753v2 [cond-mat.supr-con] 21 Apr 2023

tions. The interaction originates from a combination of

inter-orbital coupling to the inversion breaking polariza-

tion of the mode and SOC [47]. In the minimal model

we assumed a conventional Rashba coupling linear in the

electronic momentum k. In the present work we show

that this approximation is valid only at very low densi-

ties. Because of this, the problem of the dome in STO

could not be addressed in Ref. [30].

Here, we present a complete study of the spin-orbit as-

sisted coupling between the low-energy electronic bands

and the FE soft TO modes in tetragonal doped STO.

We ﬁnd that the magnitude of the Rashba coupling is

strongly sensitive to the particular form of the eigen-

vector of the soft mode. Indeed, we discover a gigantic

coupling to the polar mode deforming the oxygen cage,

so that even a small admixture of this distortion in the

eigenvector of the soft mode makes the coupling to elec-

trons quite large. Furthermore, we ﬁnd that a naive,

linear-in-kRashba coupling deviates strongly from the ab

initio computations when the electronic wave-vector ex-

ceeds a small fraction of the inverse lattice constant. In-

corporating these results into a generalized Rashba cou-

pling and using the soft-mode eigenvector deduced from

hyper-Raman scattering, we ﬁnd a dome-like behavior of

the superconducting Tcwith a maximum value of the cor-

rect order of magnitude. The origin of the dome can be

explained with a minimal model of generalized Rashba

coupling. Also the position of the dome maximum and

its characteristic asymmetry as a function of doping are

in good agreement with experiment without free param-

eters. Our work shows that a generalized Rashba pairing

mechanism explains bulk SC in doped STO. We refer

here to the standard deﬁnition of bulk superconductivity

as the one which shows the Meissner eﬀect.

This mechanism may also be relevant in two-

dimensional electron gases at oxide interfaces [48–50]. It

has recently been proposed that the extreme sensitivity

of superconductivity to the crystallographic orientation

of KTaO3(KTO) can be explained by invoking the lin-

ear coupling to TO modes [51]. KTO is also an incipient

ferroelectric, and hence the coupling to the soft FE mode

may be important for pairing as well.

The paper is organized as follows. In Sec. II we intro-

duce the multiband electronic structure of STO, which is

successfully described by a tight-binding model ﬁt to ab

initio band-structure computations within Density Func-

tional Theory (DFT). Because we are interested in cou-

pling the electrons to zone-center polar phonon modes,

in Sec. III we present a complete basis ¯

Sito parametrize

any polar mode belonging to tetragonal Euand A2uirre-

ducible representations (irreps). In Sec. IV we show how

a linear-in-kRashba-like coupling between the electrons

and zone-center polar modes emerges from a microscopic

model in the presence of SOC, and estimate the cou-

pling constants with the aid of ab initio frozen-phonon

computations in STO. The corresponding electron-polar-

phonon coupling Hamiltonian is then derived in Sec. V;

we ﬁnd all three electronic bands have a substantial dy-

namic Rashba coupling to the soft TO mode in STO. In

Sec. VI we use the ab initio results and a minimal model

to explore the superconducting properties derived from

the generalized Rashba mechanism. We ﬁnally present

our conclusions in Sec. VII.

II. ELECTRONIC STRUCTURE

A. Electronic DFT bands

We ﬁrst discuss the electronic band structure of

STO as computed by DFT. We adopted the projec-

tor augmented-wave (PAW) method as implemented

in VASP [52,53] and the Perdew-Burke-Ernzerhof

generalized gradient approximation revised for solids

(PBEsol) [54]. An antiferrodistortive (AFD) struc-

tural transition is known to occur below 105 K, there-

fore we considered both the high-temperature cubic

(space group P m¯

3m) and the low-temperature tetrag-

onal (space group I4/mcm) unit cell. We ﬁrst relaxed

both structures until forces were smaller than 1 meV/˚

using a plane-wave cutoﬀ of 520 eV and a Monkhorst-

Pack grid of 8×8×8 and 6×6×6 k-points for cubic and

tetragonal phases, respectively. Optimized lattice con-

stants are a= 3.907 ˚

A for cubic STO and at= 5.508 ˚

ct= 7.845 ˚

A for tetragonal STO. Electronic structure

calculations have then been performed with the inclusion

of SOC, as implemented in VASP[55].

The low-energy electronic band structure is shown in

Fig. 1(dashed lines) and consists of three doubly de-

generate bands around the zone center. AFD distortions

result in a split of the lower two bands at the zone center,

as displayed in Fig. 1(b).

Superconductivity develops upon electron doping the

tetragonal STO at a few hundred mK. The resulting su-

perconducting state spans the ﬁlling of the three bands

shown in Fig. 1(b) before vanishing [2], starting from a

zero-resistance state in the very dilute single-band regime

with a Fermi energy of a few meV, and evolving into bulk

multi-band SC with a Fermi energy of a few tens of meV.

B. Minimal electronic model

A minimal tight-binding model with the 3d t2gorbitals

of the Ti atom yz,zx and xy, denoted respectively µ=

x, y, and zin this work, successfully describes the low-

energy electronic band dispersion (full lines in Fig. 1) in

both cubic and tetragonal phases [17,30,56–58]. The

non-interacting model Hamiltonian reads,

H=H0+HSOC +HAFD.(1)

Here, we have included a hopping term up to next-nearest

neighbors,

H0=X

ksµν

tµν (k)c†

µs(k)cνs(k) (2)

(

)

(

)

0.3 0.15 0. 0.15 0.3

[

100

]

k

[

110

]

Energy [meV]

j=3/2

j=1/2

0.3 0.15 0. 0.15 0.3

[

100

]

k

[

110

]

Energy [meV]

n=1

n=2

n=3

FIG. 1. Low energy band structure of STO in the (a) cubic

state and (b) tetragonal state. Dashed lines are computed

from DFT and full lines the tight-binding model in Eq. (1).

The momenta in (a) and (b) are along the cubic and pseudocu-

bic directions respectively, and ais the cubic lattice constant.

between orbitals µand νwith spin s=±1/2 (short-

handed as s=±in operator labels as in c†

µ+). The

atomic SOC of the t2gmanifold reads,

HSOC =−2ξl·s(3)

where we introduced an eﬀective orbital moment oper-

ator with l= 1 [59–61]. The physical orbital angular

momentum l(t2g) is directed in the opposite direction

with respect to l.

Finally, the tetragonal crystal ﬁeld term is,

HAFD = ∆ X

µs

δµ,zc†

µscµs (4)

which eﬀectively accounts for the AFD distortion by

shifting the energy of the µ=zorbital [30].

The t2ghopping term reproduces the low-energy

quadratic dispersion given by DFT along the high-

symmetry cubic directions [Fig. 1(a)],

tµµ(k) = −2t1(cos kα+ cos kβ)−2t2cos kµ

−4t3cos kαcos kβ+ (4t1+ 2t2+ 4t3) (5)

tµν (k) = −4t4sin kµsin kν,(6)

with hopping parameters t1= 451 meV, t2= 40 meV,

t3= 111 meV and t4= 27 meV. In Eq. (5)α6=βand

α, β 6=µ, while in Eq. (6), µ6=ν.

The eigenstates of Eq. (3) can be classiﬁed with an

eﬀective total angular momentum j=l+swith the as-

sociated quantum number j. The SOC term breaks the

six-fold degeneracy of the t2gmanifold at the zone center,

opening a 3ξ= 28 meV gap between the lower multiplet

j= 3/2 and the higher doublet j= 1/2 in the high-T

cubic state [Fig. 1(a)]. In terms of the t2gorbital opera-

tors c†

µs these new eigenstates and associated operators,

c†

j,jz, take the following form at the zone center [60]:

c†

2,±3

=∓1

√2c†

x,±±ic†

y,±(7)

c†

2,±1

√6∓c†

x,∓−ic†

y,∓+ 2c†

z,±(8)

c†

2,±1

√3−c†

x,∓∓ic†

y,∓∓c†

z,±.(9)

The tetragonal crystal ﬁeld term HAFD in Eq. (1), does

not aﬀect jz=±3/2 states which remain therefore eigen-

states of the full Hamiltonian at the zone center. Instead,

it mixes states with jz=±1/2 (i.e. states with non-zero

µ=zorbital character) and thus splits the degeneracy

of the lowest multiplet j= 3/2 at the zone center. A

ﬁtting to the DFT band structure in the low-Ttetrago-

nal phase [Fig. 1(b)] gives ∆ = 17.7 meV, and sets the

following order of the three doubly-degenerate bands at

Γ:

c†

1,±=c†

2,∓3

(10)

E1= 0

c†

2,±= cos θc†

2,±1

2∓sin θc†

2,±1

(11)

E2=1

23ξ+ ∆ −p9ξ2−2ξ∆+∆2

c†

3,±=±sin θc†

2,±1

+ cos θc†

2,±1

(12)

E3=1

23ξ+∆+p9ξ2−2ξ∆+∆2

from lowest to highest energy E1<E2<E3, and with

tan 2θ=−2√2∆

−∆+9ξ. Note that the pseudospin index ±of

the bands in Eqs. (10)-(12) is chosen to coincide with the

projection of the electronic spin along the real orbital

moment instead of the eﬀective orbital moment within

the T-P equivalence [61] (l(t2g) = −l, see also [62]).

Carrying the analysis for general momentum we can

write the electronic Hamiltonian Eq. (1) in the absence

of a polar distortion as

H=X

ψ†

n(k)En(k)σ0ψn(k) (13)

where we deﬁned the spinor ψ†

n= (c†

n+, c†

n−) for band

n, and introduced the 2 ×2 identity matrix σ0for pseu-

dospin degeneracy. Figure 1(b) shows that this model

with the parameters quoted above gives an excellent ﬁt

of the bands obtained by DFT in the presence of both

AFD and SOC.

III. POLAR SOFT MODE IN STO

Although we are focusing on the temperature region

where an AFD is present, it is customary to discuss the

atomic displacements of the near zone-center polar soft

FIG. 2. Atomic distortions of the Euirrep for Ima2 (C2v) with polar axis ˆnpk[1¯

10] for basis modes (a) ¯

S1, (b) ¯

S2, (c) ¯

S3, (d)

S4and (e) ¯

S5given by Eqs. (15)-(19) and Table I. In all cases distortions are on the plane of the drawing and the arrowheads

identify the atoms moving. The left inset shows the pseudocubic and tetragonal in-plane axes. The Wickoﬀ positions of Ti, Ox

and Oyatoms are given in [62]. The atomic displacements for the basis of A2umode with polar axis ˆnpk[001] are equivalent

to panels (a)-(c) with all displacements pointing out of the plane, i.e. rotated by π/2 around the bttetragonal axis.

mode in terms of a complete set of basis modes deﬁn-

ing symmetry coordinates for the T1uirrep of Ohin the

high-Tcubic phase. Indeed, Axe [63] introduced one such

possible set of coordinates to describe the eigenvectors

of polar normal modes in cubic perovskite structures,

which has been used to restrict the possible atomic dis-

tortions of the various polar modes that were compatible

with reﬂectivity [63], neutron scattering [64] and hyper-

Raman experiments [65]. Since this coordinate set has

been widely used and referred to in the literature of po-

lar modes in STO, we shall use it in our work as well.

Within this framework, a general polar distortion can be

decomposed into symmetry coordinates in the following

way

U= (rSr,rTi,rOx,rOy,rOz) = X

ˆniui¯

Si.(14)

Here, ˆniis a unit vector setting the direction of atomic

displacements for basis mode i. We shall see that, in

general, displacements with the same polar axis, ˆnp,

do not need to be collinear; this requires a diﬀerent

ˆnifor each basis mode. ¯

Sideﬁnes the basis of eigen-

modes expressed in terms of collinear atomic displace-

ments (sSr, sTi, sOx, sOy, sOz),

S1=1

1 + κ1

(0,−κ1,1,1,1) (15)

S2=1

1 + κ2

(−κ2,1,1,1,1) (16)

S3=2

3(0,0,−1

2,−1

2,1) (17)

and shown in Figs. 2(a)-(c). The coeﬃcients κ1=3mO

mTi

and κ2=3mO+mTi

mSr ensure the center of mass is not dis-

placed for any of the ¯

Simodes. That is, Pjmjrj= 0

when summing over all the jatoms with atomic mass

mjand displacement rj=uisjin the unit cell for each

of the ¯

Simodes. The coeﬃcient uisets the amplitude

of basis mode iin the general displacement ¯

U. The ¯

modes in Eqs. (15)-(17) have been normalized so that

their amplitude uiis equal to the relative displacement

of the two bodies in the mode. For instance, u1is the

relative atomic displacement between Ti and the O cage

in the ¯

S1mode, rO−rTi =u1(sO

1−sTi

1) = u1. Similarly,

u2is the relative displacement between Sr and the Ti-O

cage in mode ¯

S2. This normalization reduces the two-

body problem into a one-body problem with a reduced

mass when deriving the electron-phonon Hamiltonian, as

will be shown in Section V. Note the bar symbol indicates

a vector spanned by the atoms of the unit cell (as in ¯

Si),

whereas the vector referring to the Cartesian coordinates

of the atomic displacements is speciﬁed by bold notation

(as in ˆni).

As it is well known, the long-range Coulomb inter-

action partially lifts the three-fold degeneracy of polar

modes into a high-energy longitudinal mode and low-

energy doubly degenerate transverse modes [66]. Thus,

the soft mode of STO is transverse and, in general, it

is a linear combination of all three ¯

Simodes. Accord-

ing to several studies [63–66], its atomic displacements

are close to the ¯

S1mode [Eq. (15)], also known as the

Slater mode [67], where the Ti atom vibrates opposite to

the O octahedron [see Fig. 2(a)]. Because of the strong

sensitivity of the electron-phonon coupling to the soft-

mode eigenvector, we anticipate that even a small devia-

tion from a pure Slater mode can have important conse-

quences for superconductivity.

Rigorously speaking, the above analysis in terms of

three basis modes is only valid in the cubic phase. The

presence of the AFD distortion requires an enlargement

of the basis. Indeed, below 105 K, as the symmetry of

STO is lowered to a tetragonal structure belonging to

the I4/mcm space group (D4hpoint group), the T1upo-

lar mode of the cubic state splits into: (a) a A2uirrep

with a polar axis along [001] (C4v) and (b) a Euirrep

with a polar axis perpendicular to [001]. This split of

the soft mode has been tracked in Tby hyper-Raman

spectroscopy [68]: ~ωEu∼1 meV and ~ωA2u∼2 meV

at 7K.

The analysis of polar modes for case (a) is simpler, as

the basis of symmetry modes Eqs. (15)-(17) for the T1u

mode of Ohin the high-Tcubic phase is also a com-

plete basis for the A2umode in the low-Ttetragonal

phase. Therefore, in this case an enlargement of the ba-

sis is not needed. Of course, the atomic displacements

are restricted along the tetragonal zaxis for this irrep,

ˆnA2u

ik[001], leading to a polar tetragonal structure with

I4cm lower symmetry. Although here we will focus on

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

GeneralizedRashbaelectron-phononcouplingandsuperconductivityinstrontiumtitanateMariaN.Gastiasoro,1,2,MariaEleonoraTemperini,3PaoloBarone,4andJoseLorenzana1,y1ISC-CNRandDepartmentofPhysics,SapienzaUniversityofRome,PiazzaleAldoMoro2,00185,Rome,Italy2DonostiaInternationalPhysicsCenter,20018Donostia-S...

展开>> 收起<<

Generalized Rashba electron-phonon coupling and superconductivity in strontium titanate Maria N. Gastiasoro1 2Maria Eleonora Temperini3Paolo Barone4and Jos e Lorenzana1y.pdf

共18页,预览4页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Generalized Rashba electron-phonon coupling and superconductivity in strontium titanate Maria N. Gastiasoro1 2Maria Eleonora Temperini3Paolo Barone4and Jos e Lorenzana1y

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: