# Generalized Rashba Electron-Phonon Coupling and Superconductivity in
Strontium Titanate

Maria N. Gastiasoro<EMAIL_ADDRESS>ISC-CNR and Department of
Physics, Sapienza University of Rome, Piazzale Aldo Moro 2, 00185, Rome, Italy
Maria Eleonora Temperini  ISC-CNR Institute for Complex Systems, via dei
Taurini 19, 00185 Rome, Italy  Paolo Barone  SPIN-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  Jose
Lorenzana  ISC-CNR and Department of Physics, Sapienza University of Rome,
Piazzale Aldo Moro 2, 00185, Rome, Italy

###### Abstract

SrTiO3 is 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 $T_{c}$ results 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.

††preprint: APS/123-QED

##  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 superconductivity
[3, 4] in samples with polar nanodomains [5, 6] and self-organized
dislocations with enhanced ferroelectric fluctuations [7]. Alternative methods
for tuning ferroelectricity such as Ca or 18O isotope substitution also
present enhanced superconducting critical temperatures $T_{c}$ [8, 9, 10]. In
doped samples with a global polar transition (and thus global broken inversion
symmetry) signatures of mixed-parity superconductivity have been reported
[11]. Theoretically, these doped polar samples have also been recently
proposed as a platform for the emergence of exotic phases such as Majorana-
Weyl superconductivity [12] and odd-frequency pair correlations [13].

Despite having experimentally established a qualitative connection between the
superconducting and ferroelectric 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 [14], there is still no
consensus about the pairing mechanism in this system [15]. One the of the
prominent theoretical challenges is its very low density of states and Fermi
energy due to low carrier densities, which places superconductivity in STO
outside of the standard BCS paradigm.

Proposed pairing theories include the dynamical screening of the Coulomb
interaction due to longitudinal modes [16, 17, 18, 19] recently challenged in
Ref. [20], bipolaron formation [21], and diverse approaches to linear coupling
[1, 22, 23, 24, 25, 26, 27, 28, 29, 30] or quadratic coupling to the FE mode
[31, 32, 33, 34, 35]. The last two proposals have the advantage that, coupling
electrons directly to the FE soft mode, provide a natural explanation to the
sensitivity to the FE instability.

In the more general context of polar or nearly polar metals, the coupling
between electrons and the soft FE modes has received attention only very
recently [24, 36, 25, 30, 37, 28, 35, 38]. The reason probably being that, as
already mentioned, the FE soft modes in these systems have a predominantly
_transverse_ polarization. Within the conventional electron-phonon interaction
scheme, this implies a decoupling of the soft modes from the electronic
density to linear order [39]. One promising alternative route involves going
to next order by coupling the electrons to pairs of TO modes [31, 32, 33, 34,
40]. 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) [41, 42, 43, 24, 44, 13, 25, 37].

In a recent work [28] we derived a vector coupling based on a Rashba-like
interaction within a minimal microscopic model and _ab initio_ frozen phonon
computations. The interaction originates from a combination of inter-orbital
coupling to the inversion breaking polarization of the mode and SOC [45]. 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 densities. Because of this, the problem of the dome
in STO could not be addressed in Ref. [28].

Here, we present a complete study of the spin-orbit assisted 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 eigenvector 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 electrons quite large. Furthermore, we find that a
naive, linear-in-$k$ Rashba coupling deviates strongly from the ab initio
computations when the electronic wave-vector exceeds a small fraction of the
inverse lattice constant. Incorporating these results into a generalized
Rashba coupling and using the soft-mode eigenvector deduced from hyper-Raman
scattering, we find a dome-like behavior of the superconducting $T_{c}$ with a
maximum value of the correct 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 parameters. Our work
shows that a generalized Rashba pairing mechanism can explain 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 [46, 47, 48]. It has recently been proposed that the extreme
sensitivity of superconductivity to the crystallographic orientation of KTaO3
(KTO) can be explained by invoking the linear coupling to TO modes [49]. 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 introduce the multiband
electronic structure of STO, which is successfully described by a tight-
binding model fit to ab initio band-structure computations within Density
Functional Theory (DFT). Because we are interested in coupling the electrons
to zone-center polar phonon modes, in Sec. III we present a complete basis
$\bar{S}_{i}$ to parametrize any polar mode belonging to tetragonal $E_{u}$
and $A_{2u}$ irreducible representations (irreps). In Sec. IV we show how a
linear-in-$k$ Rashba-like coupling between the electrons and zone-center polar
modes emerges from a microscopic model in the presence of SOC, and estimate
the coupling 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
dynamic 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

###  II.1 Electronic DFT bands

The low-energy electronic band structure of STO computed by DFT (see [50] for
details), shown in Fig. 1 (dashed lines), consists of three doubly degenerate
bands around the zone center. Below 105 K, an antiferrodistortive (AFD)
structural transition lowers the symmetry of STO from cubic [space group
$Pm\bar{3}m$, Fig. 1 (a)] to tetragonal [space group $I4/mcm$, Fig. 1 (b)],
which results in a split of the lower two bands at the zone center. At even
lower temperatures, upon electron doping STO, superconductivity develops at a
few hundred $m$K. The resulting superconducting state spans the filling of
those three bands 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.

Figure 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
pseudocubic directions respectively, and $a$ is the cubic lattice constant.

###  II.2 Minimal electronic model

A minimal tight-binding model with the $3d$ $t_{2g}$ orbitals of the Ti atom
$yz$, $zx$ and $xy$, denoted respectively $\mu=x,y$, and $z$ in this work,
successfully describes the low-energy electronic band dispersion (full lines
in Fig. 1) in both cubic and tetragonal phases [51, 52, 53, 15, 28]. The non-
interacting model Hamiltonian reads,

$\mathcal{H}=\mathcal{H}_{0}+\mathcal{H}_{\mathrm{SOC}}+\mathcal{H}_{\mathrm{AFD}}.$
(1)

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

$\mathcal{H}_{0}=\sum_{\bm{k}s\mu\nu}t_{\mu\nu}(k)c^{\dagger}_{\mu
s}(\bm{k})c_{\nu s}(\bm{k})$ (2)

between orbitals $\mu$ and $\nu$ with spin $s=\pm 1/2$ (shorthanded as $s=\pm$
in operator labels as in $c^{\dagger}_{\mu+}$). The atomic SOC of the $t_{2g}$
manifold reads,

$\mathcal{H}_{\mathrm{SOC}}=2\xi\bm{l}\cdot\bm{s}$ (3)

where we introduced an effective orbital moment operator with $l=1$ [54].
Finally, the tetragonal crystal field term is,

$\mathcal{H}_{\mathrm{AFD}}=\Delta\sum_{\mu s}\delta_{\mu,z}c^{\dagger}_{\mu
s}c_{\mu s}$ (4)

which effectively accounts for the AFD distortion by shifting the energy of
the $\mu=z$ orbital [28].

The $t_{2g}$ hopping term reproduces the low-energy quadratic dispersion given
by DFT along the high-symmetry cubic directions [Fig. 1(a)],

$\displaystyle t_{\mu\mu}(\bm{k})=$ $\displaystyle-2t_{1}\left(\cos
k_{\alpha}+\cos k_{\beta}\right)-2t_{2}\cos k_{\mu}$ $\displaystyle-4t_{3}\cos
k_{\alpha}\cos k_{\beta}+(4t_{1}+2t_{2}+4t_{3})$ (5) $\displaystyle
t_{\mu\nu}(\bm{k})=$ $\displaystyle-4t_{4}\sin k_{\mu}\sin k_{\nu},$ (6)

with hopping parameters $t_{1}=451$ meV, $t_{2}=40$ meV, $t_{3}=111$ meV and
$t_{4}=27$ meV. In Eq. (5) $\alpha\neq\beta$ and $\alpha,\beta\neq\mu$, while
in Eq. (6), $\mu\neq\nu$.

The SOC term breaks the six-fold degeneracy of the $t_{2g}$ manifold at the
zone center, opening a $3\xi=28$ 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 $t_{2g}$ orbital operators $c^{\dagger}_{\mu s}$ these new
eigenstates take the following form at the zone center [54]

$\displaystyle c^{\dagger}_{\frac{3}{2},\pm\frac{3}{2}}$
$\displaystyle=\mp\frac{1}{\sqrt{2}}\left(c^{\dagger}_{x,\pm}\pm
ic^{\dagger}_{y,\pm}\right)$ (7) $\displaystyle
c^{\dagger}_{\frac{3}{2},\pm\frac{1}{2}}$
$\displaystyle=\frac{1}{\sqrt{6}}\left(\mp
c^{\dagger}_{x,\mp}-ic^{\dagger}_{y,\mp}+2c^{\dagger}_{z,\pm}\right)$ (8)
$\displaystyle c^{\dagger}_{\frac{1}{2},\pm\frac{1}{2}}$
$\displaystyle=\frac{1}{\sqrt{3}}\left(-c^{\dagger}_{x,\mp}\mp
ic^{\dagger}_{y,\mp}\mp c^{\dagger}_{z,\pm}\right)$ (9)

The tetragonal crystal field term $\mathcal{H}_{\mathrm{AFD}}$ in Eq. (1),
does not affect $j_{z}=\pm 3/2$ states which remain therefore eigenstates of
the full Hamiltonian at the zone center. Instead, it mixes states with
$j_{z}=\pm 1/2$ (i.e. states with non-zero $\mu=z$ orbital 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-$T$ tetragonal phase [Fig. 1(b)]
gives $\Delta=17.7$ meV, and sets the following order of the three doubly-
degenerate bands at $\Gamma$:

$\displaystyle c^{\dagger}_{1,\pm}$
$\displaystyle=c^{\dagger}_{\frac{3}{2},\mp\frac{3}{2}}$ (10)
$\displaystyle\mathcal{E}_{1}$ $\displaystyle=0$ $\displaystyle
c^{\dagger}_{2,\pm}$ $\displaystyle=\cos\theta
c^{\dagger}_{\frac{3}{2},\pm\frac{1}{2}}\mp\sin\theta
c^{\dagger}_{\frac{1}{2},\pm\frac{1}{2}}$ (11) $\displaystyle\mathcal{E}_{2}$
$\displaystyle=\frac{1}{2}\left(3\xi+\Delta-\sqrt{9\xi^{2}-2\xi\Delta+\Delta^{2}}\right)$
$\displaystyle c^{\dagger}_{3,\pm}$ $\displaystyle=\pm\sin\theta
c^{\dagger}_{\frac{3}{2},\pm\frac{1}{2}}+\cos\theta
c^{\dagger}_{\frac{1}{2},\pm\frac{1}{2}}$ (12) $\displaystyle\mathcal{E}_{3}$
$\displaystyle=\frac{1}{2}\left(3\xi+\Delta+\sqrt{9\xi^{2}-2\xi\Delta+\Delta^{2}}\right)$

from lowest to highest energy
$\mathcal{E}_{1}<\mathcal{E}_{2}<\mathcal{E}_{3}$, and with $\tan
2\theta=\frac{-2\sqrt{2}\Delta}{-\Delta+9\xi}$. Note that the pseudospin index
$\pm$ 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 [55] ($\bm{l}(t_{2g})=-\bm{l}(p)$,
see also [50]).

Carrying the analysis for general momentum we can write the electronic
Hamiltonian Eq. (1) in the absence of a polar distortion as

$\mathcal{H}=\sum_{n\bm{k}}\psi_{n}^{\dagger}(\bm{k})\mathcal{E}_{n}(\bm{k})\sigma_{0}\psi_{n}(\bm{k})$
(13)

where we defined the spinor
$\psi^{\dagger}_{n}=(c^{\dagger}_{n+},c^{\dagger}_{n-})$ for band $n$, and
introduced the $2\times 2$ identity matrix $\sigma_{0}$ for pseudospin
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

Figure 2: Atomic distortions of the $E_{u}$ irrep for $Ima2$ ($C_{2v}$) with
polar axis $\bm{\hat{n}}_{p}\parallel[1\bar{1}0]$ for basis modes (a)
$\bar{S}_{1}$, (b) $\bar{S}_{2}$, (c) $\bar{S}_{3}$, (d) $\bar{S}_{4}$ and (e)
$\bar{S}_{5}$ given by Eqs. (15)-(19) and Table 1. 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 Oy atoms are given in [50]. The atomic displacements
for the basis of $A_{2u}$ mode with polar axis
$\bm{\hat{n}}_{p}\parallel[001]$ are equivalent to panels (a)-(c) with all
displacements pointing out of the plane, i.e. rotated by $\pi/2$ around the
$b_{t}$ tetragonal axis.

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 mode in terms of a complete set of basis modes defining symmetry
coordinates for the $T_{1u}$ irrep of $O_{h}$ in the high-$T$ cubic phase.
Indeed, Axe [56] 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 distortions of the various polar
modes that were compatible with reflectivity [56], neutron scattering [57] and
hyper-Raman experiments [58]. Since this coordinate set has been widely used
and referred to in the literature of polar 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

$\displaystyle\bm{\bar{U}}=(\bm{r}^{\mathrm{Sr}},\bm{r}^{\mathrm{Ti}},\bm{r}^{\mathrm{O}_{x}},\bm{r}^{\mathrm{O}_{y}},\bm{r}^{\mathrm{O}_{z}})=\sum_{i}\bm{\hat{n}}_{i}u_{i}\bar{S}_{i}.$
(14)

Here, $\bm{\hat{n}}_{i}$ is 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, $\bm{\hat{n}}_{p}$, do not need to be collinear;
this requires a different $\bm{\hat{n}}_{i}$ for each basis mode.
$\bar{S}_{i}$ defines the basis of eigenmodes expressed in terms of collinear
atomic displacements
$(s^{\mathrm{Sr}},s^{\mathrm{Ti}},s^{\mathrm{O}_{x}},s^{\mathrm{O}_{y}},s^{\mathrm{O}_{z}})$,

$\displaystyle\bar{S}_{1}$
$\displaystyle=\frac{1}{1+\kappa_{1}}(0,-\kappa_{1},1,1,1)$ (15)
$\displaystyle\bar{S}_{2}$
$\displaystyle=\frac{1}{1+\kappa_{2}}(-\kappa_{2},1,1,1,1)$ (16)
$\displaystyle\bar{S}_{3}$
$\displaystyle=\frac{2}{3}(0,0,-\frac{1}{2},-\frac{1}{2},1)$ (17)

and shown in Figs. 2(a)-(c). The coefficients
$\kappa_{1}=\frac{3m^{\mathrm{O}}}{m^{\mathrm{Ti}}}$ and
$\kappa_{2}=\frac{3m^{\mathrm{O}}+m^{\mathrm{Ti}}}{m^{\mathrm{Sr}}}$ ensure
the center of mass is not displaced for any of the $\bar{S}_{i}$ modes. That
is, $\sum_{j}m^{j}\bm{r}^{j}=0$ when summing over all the $j$ atoms with
atomic mass $m^{j}$ and displacement $r^{j}=u_{i}s^{j}$ in the unit cell for
each of the $\bar{S}_{i}$ modes. The coefficient $u_{i}$ sets the amplitude of
basis mode $i$ in the general displacement $\bm{\bar{U}}$. The $\bar{S}_{i}$
modes in Eqs. (15)-(17) have been normalized so that their amplitude $u_{i}$
is equal to the relative displacement of the two bodies in the mode. For
instance, $u_{1}$ is the relative atomic displacement between Ti and the O
cage in the $\bar{S}_{1}$ mode,
$r^{\mathrm{O}}-r^{\mathrm{Ti}}=u_{1}(s_{1}^{\mathrm{O}}-s_{1}^{\mathrm{Ti}})=u_{1}$.
Similarly, $u_{2}$ is the relative displacement between Sr and the Ti-O cage
in mode $\bar{S}_{2}$. 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 $\bar{S}_{i}$), whereas
the vector referring to the Cartesian coordinates of the atomic displacements
is specified by bold notation (as in $\bm{\hat{n}}_{i}$).

As it is well known, the long-range Coulomb interaction partially lifts the
three-fold degeneracy of polar modes into a high-energy longitudinal mode and
low-energy doubly degenerate transverse modes [59]. Thus, the soft mode of STO
is transverse and, in general, it is a linear combination of all three
$\bar{S}_{i}$ modes. According to several studies [59, 56, 57, 58], its atomic
displacements are close to the $\bar{S}_{1}$ mode [Eq. (15)], also known as
the Slater mode [60], 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
deviation from a pure Slater mode can have important consequences 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
($D_{4h}$ point group), the $T_{1u}$ polar mode of the cubic state splits
into: (a) a $A_{2u}$ irrep with a polar axis along $[001]$ ($C_{4v}$) and (b)
a $E_{u}$ irrep with a polar axis perpendicular to $[001]$. This split of the
soft mode has been tracked in $T$ by hyper-Raman spectroscopy [61]:
$\hbar\omega_{E_{u}}\sim 1$ meV and $\hbar\omega_{A_{2u}}\sim 2$ meV at 7K.

The analysis of polar modes for case (a) is simpler, as the basis of symmetry
modes Eqs. (15)-(17) for the $T_{1u}$ mode of $O_{h}$ in the high-$T$ cubic
phase is also a complete basis for the $A_{2u}$ mode in the low-$T$ tetragonal
phase. Therefore, in this case an enlargement of the basis is not needed. Of
course, the atomic displacements are restricted along the tetragonal $z$ axis
for this irrep, $\hat{n}_{i}^{A_{2u}}\parallel[001]$, leading to a polar
tetragonal structure with $I4cm$ lower symmetry. Although here we will focus
on the paraelectric phase, we note that the out-of-plane polar mode has also
been observed by electron microscopy and optical second harmonic generation in
strained STO films in the symmetry broken polar phase [62, 63, 64, 5]
highlighting the relevance of this symmetry.

For case (b), in general, we notice that a distortion belonging to the $E_{u}$
irrep can lead to various lower-symmetry structures ($Ima2$, $Fmm2$ or $Cm$),
all of which have a polar axis perpendicular to the tetragonal axis $[001]$.
Here we will focus on $Ima2$ ($C_{2v}$), a space group with a polar axis
parallel to the tetragonal in-plane axis, i.e. along the pseudocubic direction
$[1\bar{1}0]$ (see Fig. 2). This choice is justified by the fact that this
mode has been experimentally reported in the ferroelectric phase of isotope
O18 and Ca substitution systems [65, 66] and the optically excited metastable
polar phase of STO [67]. Because the in-plane O atoms are not equivalent in
$Ima2$, and Ti and in-plane O atoms are allowed to move orthogonally to the
polar axis, the dimension of the basis symmetry modes has to be expanded to
five. That is, besides the three $\bar{S}_{i}$ modes presented in Eqs.
(15)-(17), with displacements along the polar axis (i.e.
$\bm{\hat{n}}_{i}=\bm{\hat{n}}_{p}\parallel[1\bar{1}0]$ for $i=1,2,3$) one
needs to add to the subspace another two $\bar{S}_{i}$ modes with an amplitude
along the perpendicular direction (i.e.
$\bm{\hat{n}}_{i}\parallel[110]\perp\bm{\hat{n}}_{p}$ for $i=4,5$) to obtain a
complete basis,

$\displaystyle\bar{S}_{4}$ $\displaystyle=\frac{1}{2}(0,1,0,0,0),$ (18)
$\displaystyle\bar{S}_{5}$ $\displaystyle=\frac{1}{2}(0,0,1,-1,0).$ (19)

These two modes are shown in Figs. 2(d) and 2(e), respectively.

In general, normal modes will not be made of collinear displacements. Still,
they can be decomposed into the present basis, with each element representing
collinear displacements. Different elements of the expansion can have
displacements in different planes although they contribute to the same
polarization direction.

Table 1 summarizes the polar distortion directions $\bm{\hat{n}}_{i}$ of the
different $\bar{S}_{i}$ for both the $E_{u}$ ($Ima2$) and $A_{2u}$ ($I4cm$)
modes we will consider throughout this work.

In general for a mode $i$ with displacement amplitude $u_{i}$ and associated
with the polarization vector $\bm{\hat{n}}_{p}$, irrespectively of the
direction of atomic displacements, we define its associated polarization
vector as

$\displaystyle\bm{u}_{i}=u_{i}\bm{\hat{n}}_{p}.$ (20)

Notice that formally $\bm{u}_{i}$ should be multiplied by an effective charge
to be a real polarization. On the other hand, such charge does not play any
role in the present context and will be omitted.

Near the zone center but for finite $\bm{q}$ it is important to consider the
long-range Coulomb interaction which will split transverse and longitudinal
modes. In this case we will restrict to the symmetrized modes $i=1,2,3$ and we
will assume that for $\bm{q}$ along an arbitrary direction we can decompose
mode $i$ into the set of the same $i$ modes along high symmetry directions as
done in Ref. [28] for the Slater mode. For simplicity, deviations of the
polarization vector from pure longitudinal or transverse directions due to
non-spherical symmetry will be neglected. In this way we can generalize Eq.
(21) to

$\displaystyle\bm{u}_{i}(\bm{q})=u_{i}(\bm{q})\bm{\hat{n}}_{p}(\bm{q}),$ (21)

which will be used next to discuss the general interaction with polar modes.

Table 1: Symmetry coordinates of polar $E_{u}$ and $A_{2u}$ modes for a general distortion $\bm{\bar{U}}=\sum_{i}u_{i}\bm{\hat{n}}_{i}\bar{S}_{i}$ in the tetragonal phase of STO. The atomic displacement coordinates $\bar{S}_{i}$ are given by Eqs. (15)-(19). The polar axis of the modes shown here is along the in-plane direction, $\bm{\hat{n}}^{E_{u}}_{p}\parallel[1\bar{1}0],[110]$, for $E_{u}$ modes and out-of-plane $\bm{\hat{n}}^{A_{2u}}_{p}\parallel[001]$ for $A_{2u}$ modes. The symmetry modes analysis has been performed using the ISODISTORT tool [68, 69].  Irrep (space group, point group) | $\bm{\hat{n}}_{p}$ | $\bm{\hat{n}}_{i}$ | $\bar{S}_{i}$
---|---|---|---
$E_{u}$ ($Ima2$, $C_{2v}$) | $[1\bar{1}0]$ | $[1\bar{1}0]$ |  $\bar{S}_{1}$, $\bar{S}_{2}$, $\bar{S}_{3}$
| $[1\bar{1}0]$ | $[110]$ |  $\bar{S}_{4}$, $\bar{S}_{5}$
| $[110]$ | $[110]$ |  $\bar{S}_{1}$, $\bar{S}_{2}$, $\bar{S}_{3}$
| $[110]$ | $[1\bar{1}0]$ |  $\bar{S}_{4}$, $\bar{S}_{5}$
$A_{2u}$ ($I4cm$, $C_{4v}$) | $[001]$ | $[001]$ |  $\bar{S}_{1}$, $\bar{S}_{2}$, $\bar{S}_{3}$

##  IV Linear Rashba-like coupling

###  IV.1 _Ab initio_ computation of couplings

Having established the electronic structure of STO in Sec. II and the relevant
polar phonon modes around the zone center in Sec. III, in this section we
proceed to study and estimate their coupling to linear order. In particular,
we will show how a symmetry allowed linear coupling to transverse TO modes
emerges naturally from induced hopping channels, and estimate the
corresponding coupling constant for all electronic bands with the aid of _ab
initio_ frozen-phonon results in tetragonal STO.

The linear coupling Hamiltonian between a polar distortion and electronic
bands [Eqs. (10)-(13)] can be expressed as

$\mathcal{H}_{u}=\sum_{nn^{\prime}\bm{k}\bm{q},\bar{S}_{i}}\psi_{n}^{\dagger}(\bm{k}+\frac{\bm{q}}{2})\bm{\Lambda}^{\bar{S}_{i}}_{nn^{\prime}}(\bm{k},\bm{q})\psi_{n^{\prime}}(\bm{k}-\frac{\bm{q}}{2})$
(22)

with the coupling $2\times 2$ matrix
$\bm{\Lambda}^{\bar{S}_{i}}_{nn^{\prime}}(\bm{k},\bm{q})$ in pseudospin space
for a polar mode $\bar{S}_{i}$. The intra-band ($n^{\prime}=n$) coupling
matrix has the following form to linear order in $\bm{k}$ and $u_{\alpha}$ in
a $D_{4h}$ point group:

$\displaystyle\bm{\Lambda}^{\bar{S}_{i}}_{nn}(\bm{k},\bm{q})=$ $\displaystyle
ka\varepsilon_{\alpha\beta\gamma}u_{\alpha}\hat{k}_{\beta}\sigma_{\gamma}\left(\delta_{\alpha
z}\tau^{\bar{S}_{i}}_{n,C}+\delta_{\beta
z}\tau^{\bar{S}_{i}}_{n,B}+\delta_{\gamma z}\tau^{\bar{S}_{i}}_{n,A}\right)$
$\displaystyle=$ $\displaystyle
ka\Bigl{[}\tau^{\bar{S}_{i}}_{n,A}(\hat{k}_{y},-\hat{k}_{x},0)\sigma_{z}+\tau^{\bar{S}_{i}}_{n,B}\hat{k}_{z}(-\sigma_{y},\sigma_{x},0)$
(23)
$\displaystyle+\tau^{\bar{S}_{i}}_{n,C}(0,0,\hat{k}_{x}\sigma_{y}-\hat{k}_{y}\sigma_{x})\Bigr{]}\cdot\bm{u}_{i}(\bm{q}),$

where $a$ is the lattice constant, $\varepsilon_{\alpha\beta\gamma}$ the Levi-
Civita symbol, $\delta_{\alpha\beta}$ the Kronecker’s delta-function,
$\hat{k}_{i}=\bm{\hat{k}}\cdot\bm{\hat{\imath}}$ the Cartesian projection of
the unitary momentum vector, and $\sigma_{j}$ the Pauli matrices for the
pseudospin of the electronic bands. We also defined the couplings
$\tau^{\bar{S}_{i}}_{n,l}$ with symmetry allowed irrep labels $l=A,B,C$.

Equation (IV.1) describes a Rashba-like linear-in-$k$ coupling between a polar
distortion $\bm{u}_{i}(\bm{q})=u_{i}(\bm{q})\bm{\hat{n}}_{p}(\bm{q})$ of mode
$\bar{S}_{i}$ and the electronic band $n$ with pseudospin $\bm{\sigma}$. The
first form in Eq. (IV.1) makes evident that $z$ is a privileged axis in this
structure and clarifies the meaning of the $\tau^{\bar{S}_{i}}_{n,l}$
coefficients which are associated with one member of the triad
$(\bm{u}_{i},\bm{k},\bm{\sigma})$ having a projection on the $z$-direction.

In the following we drop the index $\bar{S}_{i}$ from the couplings
$\tau_{n,l}$ for simplicity, but emphasize that these couplings vary a lot
from mode to mode. The Rashba matrix Eq. (IV.1) has the most general form
allowed by the symmetry of our tetragonal system to linear order in $\bm{k}$.
It consists of couplings $\tau_{n,A}$ and $\tau_{n,B}$ ($\tau_{n,C}$) which
couple to the corresponding $E_{u}$ ($A_{2u}$) modes with polar axis
$\bm{\hat{n}}_{p}$ in the $xy$ plane (along the $z$ axis). These parameters
are not related by symmetry, and we will estimate them using _ab initio_
computations in the following. Note that in higher cubic symmetry Eq. (IV.1)
simplifies to
$ka[\tau^{\bar{S}_{i}}_{n}\bm{\hat{k}}\times\bm{\sigma}]\cdot\bm{u}(\bm{q})$
[42, 25, 26].

The interband coupling matrices ($n^{\prime}\neq n$) in Eq. (22) have a
similar $k$-linear Rashba form. However, because we are considering a long-
wavelength $\bm{q}\rightarrow 0$ phonon, the inter-band terms result in
$k$-cubic Rashba intra-band terms upon perturbation. We therefore focus solely
on intra-band terms with $n^{\prime}=n$ [Eq. (IV.1)] in this work, which
involve $k$-linear Rashba terms which can be directly extracted from _ab
initio_ frozen-phonon computations.

Finite Rashba couplings $\tau_{n,l}$ in Eq. (IV.1) cause the characteristic
linear-in-$k$ band splitting $E_{n+}(\bm{k})-E_{n-}(\bm{k})\equiv\delta
E_{n}(\bm{k})$. From the Rashba coupling matrix Eq. (IV.1) the splitting for
band $n$ is generally given by the following expression

$\displaystyle\delta E_{n}$
$\displaystyle(\bm{k},\bm{u}_{i}(\bm{q}))=2kau_{i}(\bm{q})\times$
$\displaystyle\times\sqrt{\tau_{n,A}^{2}\left(\hat{k}_{y}\hat{n}_{px}-\hat{k}_{x}\hat{n}_{py}\right)^{2}+\tau_{n,B}^{2}\hat{k}_{z}^{2}\left(\hat{n}_{px}^{2}+\hat{n}_{py}^{2}\right)+\tau_{n,C}^{2}\left(\hat{k}_{x}^{2}+\hat{k}_{y}^{2}\right)\hat{n}_{pz}^{2}-2\tau_{n,B}\tau_{n,C}\left(\hat{k}_{x}\hat{n}_{px}+\hat{k}_{y}\hat{n}_{py}\right)\hat{k}_{z}\hat{n}_{pz}},$
(24)

which peaks (vanishes) along momenta $\bm{k}$ perpendicular (parallel) to the
polar axis $\bm{\hat{n}}_{p}$ of the mode. As we show in the following, one
can use this band split to extract the Rashba couplings of each band
$\tau_{n,l}$ to each mode $\bar{S}_{i}$. For this purpose, we need to
particularize Eq. (IV.1) for the direction given by the polar axis of the
polar modes. We obtain the following pseudospin split for a mode $\bar{S}_{i}$

$\displaystyle\frac{\delta E_{n}(\bm{k},u_{i}[1\bar{1}0])}{2u_{i}}$
$\displaystyle=ka\sqrt{\frac{\tau_{n,A}^{2}}{2}(\hat{k}_{y}+\hat{k}_{x})^{2}+\tau_{n,B}^{2}\hat{k}_{z}^{2}}$
(25) $\displaystyle\frac{\delta E_{n}(\bm{k},u_{i}[001])}{2u_{i}}$
$\displaystyle=ka\tau_{n,C}\sqrt{\hat{k}_{x}^{2}+\hat{k}_{y}^{2}}$ (26)

with an amplitude $u_{i}$ and a polar axis
$\bm{\hat{n}}_{p}\parallel[1\bar{1}0]$ and $\bm{\hat{n}}_{p}\parallel[001]$,
respectively.

To obtain the couplings $\tau_{n,l}$, we computed by first principles the
reconstructed electronic band structure of tetragonal STO in the presence of a
frozen phonon $\bm{u}_{i}(\bm{q}=0)$ for all $\bar{S}_{i}$ modes listed in
Table 1 for both $E_{u}$ and $A_{2u}$ irreps. By fitting the linear-$k$ regime
of the DFT band-splitting in Fig. 3 along various momentum directions, we can
obtain for each band $n$ the $E_{u}$ Rashba couplings $\tau_{n,A}$ and
$\tau_{n,B}$, and the $A_{2u}$ coupling $\tau_{n,C}$. The sign of these
couplings is extracted from the averaged spin-polarization values of each band
obtained from the same _ab initio_ computation. Keeping track of these signs
is very important when computing the coupling to a mode that is combination of
the $\bar{S}_{i}$ modes, as will become clear below.

Figure 3: Electronic band split $\delta
E_{n}\left(\bm{k},u_{1}(q=0)\bm{\hat{n}}_{p}\right)$ in the presence of a
frozen Slater polar mode [Eq. (15)] normalized by twice the amplitude $2u_{1}$
along the specified $\bm{k}$ directions for the three bands of tetragonal STO
[Fig. 1(b)]. The polar axis of the mode is along (a)
$\bm{\hat{n}}_{p}^{E_{u}}\parallel[1\bar{1}0]$ and (b)
$\bm{\hat{n}}_{p}^{A_{2u}}\parallel[001]$ in pseudocubic coordinates. Dashed
lines are _ab initio_ results from frozen-phonon distortions. Full lines are
the $k$-linear Rashba split model Eq. (25) in (a) and Eq.(26) in (b), up to
$ka=0.15$. All shown $\bm{k}$ directions are perpendicular to the polar axis
$\bm{\hat{n}}_{p}$.

As an example we show the results of the frozen Slater mode $\bar{S}_{1}$ [Eq.
(15) and Fig. 2(a)]. The resulting band split of each $n$ band $\delta
E_{n}(\bm{k})$ found by _ab initio_ is shown by the dashed lines in Fig. 3(a)
and Fig. 3(b), for $\bm{u}_{1}$ with a polar axis along the in-plane
$\bm{\hat{n}}_{p}\parallel[1\bar{1}0]$ ($E_{u}$ mode) and out-of-plane
$\bm{\hat{n}}_{p}\parallel[001]$ ($A_{2u}$ mode) pseudocubic directions,
respectively. As seen in Fig. 3, for small enough momenta all bands show a
linear-$k$ split (full lines), but the momentum amplitude beyond which
deviations of $k$-linearity become significant depends on the band, the polar
axis and the direction of momentum. In fact, while for the $E_{u}$ mode the
split is robustly linear around the zone center with small deviations beyond
$ka\sim 0.2$, for the $A_{2u}$ mode strong non-linear features appear already
at small $ka\sim 0.05$ for the two lowest bands. This highlights the
limitation of a conventional Rashba linear-$k$ model Eq. (IV.1) to describe
the coupling between the bands and some of the polar modes in this system. We
will came back to this important point in Section VI.

Figure 4:  (a) $k$-linear Rashba couplings $\tau_{n,A}$, $\tau_{n,B}$ and
$\tau_{n,C}$ in Eq. (IV.1) in meV/$\AA$ for band $n$ and mode $\bar{S}_{i}$
[Eqs. (15)-(19)] in tetragonal STO. They were obtained from fits of the spin-
split Eqs (25)-(26) to frozen phonon _ab initio_ results. The numerical values
are listed in [50]. (b) Absolute value of the estimated electron-FE-mode
Rashba couplings [Eq. (35)] for band $n$ for three possible eigenvectors of
the FE mode: a pure $\bar{S}_{1}$ mode, eigenvector from neutron data [Eq.
(37)] and from Raman data [Eq. (38)]. We have used experimental optical gaps
$\hbar\omega^{E_{u}}_{\mathrm{TO}}=1$ meV and
$\hbar\omega^{A_{2u}}_{\mathrm{TO}}=2$ meV, and electronic momentum
$k_{F}a=0.3$.

The Rashba couplings to the rest of the $\bar{S}_{i}$ modes belonging to
$E_{u}$ and $A_{2u}$ irreps (see Table 1) have been also estimated by the same
fitting procedure to frozen phonon _ab initio_ computations; they are shown
together with those of the Slater mode in Fig. 4(a) (also listed in [50]). As
seen, for all $\bar{S}_{i}$ modes the $E_{u}$ coupling $\tau_{n,A}$ is larger
in the lowest two bands ($n=1,2$) than the highest band ($n=3$). This
hierarchy is reversed for the couplings $\tau_{n,B}$ and $\tau_{n,C}$ where
the highest two bands ($n=2,3$) show larger couplings than the lowest band
($n=1$). Remarkably, the Rashba coupling to the $\bar{S}_{3}$ mode, with
apical oxygen atoms Oz moving opposite to in-plane O atoms Ox,y distorting the
octahedra [see Eq. (17) and Fig. 2(c)] can be an order of magnitude larger
than the other couplings. As we will show in the following section, this
gigantic Rashba coupling has important consequences for the electron coupling
to the soft mode. Indeed, an enlarged electron-phonon coupling follows from a
modest contribution of $\bar{S}_{3}$ to any polar mode.

###  IV.2 Real-space origin of the coupling

We will now show how the symmetry allowed coupling Eq. (IV.1) emerges when
considering microscopic processes in real space. In the presence of a polar
distortion $\bm{\bar{U}}=\sum_{i}u_{i}\bm{\hat{n}}_{i}\bar{S}_{i}$ of the
lattice, new terms are allowed in the Hamiltonian Eq. (1) for $t_{2g}$
electrons around the zone center. These new terms include effects such as the
polarization of the orbitals and induced hopping channels which are symmetry
forbidden in the absence of the distortion [45, 70, 53, 71, 28, 37]. We thus
consider the following Hamiltonian with the spinor of the $t_{2g}$ orbitals
$\psi^{\dagger}_{\mu}=(c^{\dagger}_{\mu+},c^{\dagger}_{\mu-})$,

$\mathcal{H}_{u}=\sum_{\bm{k}\bm{q}\mu\nu
j}\psi^{\dagger}_{\mu}(\bm{k}+\frac{\bm{q}}{2})t_{\mu\nu
j}(\bm{k},\bm{q})\sigma_{j}\psi_{\nu}(\bm{k}-\frac{\bm{q}}{2})+\mathrm{h.c.}$
(27)

which describes the induced hopping between a $d$-orbital $\mu$ and a nearest
neighbor $d$-orbital $\nu$. The Pauli matrices $\sigma_{j}$ represent spin-
independent ($j=0$) as well as spin-dependent ($j=x,y,z$) hopping processes.
Fig. 5 shows some examples for $j=0,x$ and $y$. Around the zone center
($\bm{k}\rightarrow 0$, $\bm{q}\rightarrow 0$), the new allowed terms have the
following form

$t_{\mu\nu j}(\bm{k},\bm{q})\approx\sum_{ilm}\frac{\partial t_{\mu\nu
j}}{\partial u_{i,l}(\bm{q})}u_{i,l}(\bm{q})k_{m}a$ (28)

to linear order in the polar distortion
$\bm{u}_{i}(\bm{q})=u_{i}(\bm{q})\bm{\hat{n}}_{p}(\bm{q})$ for a mode
$\bar{S}_{i}$ with amplitude $u_{i,l}(\bm{q})$ in Cartesian coordinate $l$,
and polar axis $\bm{\hat{n}}_{p}$. In the case of tetragonal STO, as discussed
in the previous section, the polar axis we are considering are the in-plane
$\bm{\hat{n}}^{E_{u}}_{p}\parallel[1\bar{1}0]$ for $E_{u}$ modes and out-of-
plane $\bm{\hat{n}}^{A_{2u}}_{p}\parallel[001]$ for $A_{2u}$ modes in
pseudocubic coordinates. In general, the precise form of the terms allowed in
Eq. (28) is set by symmetry; and thus it depends on the pair of orbitals $\mu$
and $\nu$ involved, as well as the direction of the polar axis
$\bm{\hat{n}}_{p}$ associated with the distortion $\bm{\bar{U}}$.

Figure 5: Induced inter-orbital hopping elements in Eq. (29) between
$d$-orbital $\nu=y$ with spin $\downarrow$ (orange arrow) at the origin and
$d$-orbital $\mu=x$ with spin $s$ (orange arrow) at neighboring Ti atoms,
mediated by the $p$-orbitals with spin $s^{\prime}$ (gray arrow) in the
bridging oxygens. (a) Spin-conserving process $\partial_{y}t_{xy0}u_{1}$ with
$s=\downarrow$ mediated by $|p_{z}\downarrow>$. Spin-flip processes
$s=\uparrow$ when allowing for SOC of the oxygen $\xi_{O}$ for (b)
$\partial_{x}t_{xyx}u_{1}$ mediated by $|p_{z}\downarrow>$ and
$|p_{y}\uparrow>$ with matrix-element $i\xi_{O}$, and (c)
$\partial_{z}t_{xyx}u_{1}$ mediated by $|p_{x}\downarrow>$ and
$|p_{z}\uparrow>$ with matrix-element $\xi_{O}$. All three hopping elements
change sign along the horizontal bond (black curved arrows) with a finite
polar Slater displacement $\bm{u}_{1}$ along the vertical axis, specified in
each panel. The spins on the negative horizontal axis, not shown for clarity,
are mirror symmetric with respect to those shown on the positive side.

As a concrete example, we explicitly consider in the following the case of
$\mu=x$ and $\nu=y$ orbitals. The lowest electronic band in the tetragonal
state ($n=1$) is formed by only these two orbitals at the zone center [see Eq.
(10)], and thus induced hopping amplitudes involving these two orbitals are
the relevant terms for the coupling of the lowest band to polar modes. For a
general polarization vector defined by Eq. (21), for mode $\bar{S}_{i}$, the
following inter-orbital hopping elements are allowed in Eq. (27):

$\displaystyle\sum_{j}t_{xyj}(\bm{k},\bm{q})\sigma_{j}$ $\displaystyle\approx
2i\partial_{x}t_{xy0}u_{i}(\bm{q})ka\bigl{[}-\hat{k}_{y}\hat{n}_{px}+\hat{k}_{x}\hat{n}_{py}\bigr{]}\sigma_{0}$
$\displaystyle+2\partial_{x}t_{xyx}u_{i}(\bm{q})ka\bigl{[}-\hat{n}_{px}\sigma_{x}+\hat{n}_{py}\sigma_{y}\bigr{]}\hat{k}_{z}$
$\displaystyle+2\partial_{z}t_{xyx}u_{i}(\bm{q})ka\bigl{[}-\hat{k}_{x}\sigma_{x}+\hat{k}_{y}\sigma_{y}\bigr{]}\hat{n}_{pz}$
(29)

where we have used the shorthand notation $\frac{\partial t_{\mu\nu
j}}{\partial u_{i,l}(\bm{q})}\equiv\partial_{l}t_{\mu\nu j}$. The first term
in Eq. (29) corresponds to a spin-conserving ($\sigma_{0}$) hopping channel
with amplitude $\partial_{x}t_{xy0}$ which changes sign with hopping
direction, shown in Fig. 5(a). It couples only to the in-plane components of
the polar distortion axis $\hat{n}_{px}$ and $\hat{n}_{py}$. The second and
third terms describe spin-flip ($\sigma_{x,y}$) hopping processes instead, and
couple to the in-plane component of the polar axis, through the hopping
amplitude $\partial_{x}t_{xyx}$ [Fig. 5(b)], as well as to the out-of-plane
component $\hat{n}_{pz}$ through the hopping amplitude $\partial_{z}t_{xyx}$
[Fig. 5(c)]. Microscopically one obtains spin-flip hopping terms by extending
SOC to the bridging $p$-orbitals of the oxygen atom [as shown in Figs.
5(b)-(c)], or by considering virtual processes between the $t_{2g}$ and
$e_{g}$ manifolds [71]. Each $\bar{S}_{i}$ mode will generally induce
different hopping amplitudes, which then result in different couplings to the
electrons. Other $t_{\mu\nu j}(\bm{k},\bm{q})$ induced hopping elements
between different pairs of $t_{2g}$ orbitals in Eq. (27) can be similarly
obtained.

We can now connect the symmetry allowed couplings in Eq. (IV.1), to these
microscopic processes by projecting Eq. (27) to the band basis [Eqs.
(10)-(12)] of the non-interacting electrons. In general each coupling element
$\tau_{n,l}$ is a linear combination of the different induced hopping
derivatives $\partial_{l}t_{\mu\nu j}$ allowed in Eq. (28). For instance, the
induced hopping derivatives between the $x$ and $y$ orbitals considered in Eq.
(29), are connected to the Rashba couplings for the lowest band $n=1$ [Eq.
(10)] in the following way,

$\displaystyle\tau_{1,A}=-2\partial_{x}t_{xy0};\
\tau_{1,B}=2\partial_{x}t_{xyx};\ \tau_{1,C}=-2\partial_{z}t_{xyx}.$ (30)

Similar expressions for the other two electronic bands $n=2,3$ can also be
derived following the same procedure. These expressions are more involved than
the particularly simple expressions in Eq. (30) for $n=1$.

To close this section we comment on the importance of the different terms in
[Eq. (30)], relevant for the coupling to the lowest band $n=1$. We see from
Fig. 4(a) that for $n=1$ and for all modes $\bar{S}_{i}$ except Slater
($\bar{S}_{1}$), $\tau_{1,A}\gg\tau_{1,B},\tau_{1,C}$, implying that one can
safely neglect the spin-flip induced hopping terms $\partial_{x}t_{xyx}$ and
$\partial_{z}t_{xy}$ in Eq. (29). For a pure Slater mode $\bar{S}_{1}$ mode,
however, $\tau_{1,A}$ is of the same order of magnitude as $\tau_{1,B}$ and
$\tau_{1,C}$, hence the spin-flip processes (and the symmetry equivalent
virtual processes to the $e_{g}$ manifold) are in principle not negligible.
However, since the $\bar{S}_{3}$ spin-conserving coupling
$\tau_{1,A}^{(\bar{S}_{3})}$ is very large, a small admixture of this mode
allows to neglect spin-flip processes when considering the linear coupling
between the lowest band and the polar mode.

##  V Electron-polar-phonon Hamiltonian

In order to obtain an electron-phonon Hamiltonian, we quantize the general
atomic displacements of Eq. (14) by decomposing them into a set of normal
modes $\alpha$:

$\bm{r}^{j}(\bm{q})=\sum_{\alpha}\frac{\bm{e}^{j}_{\alpha}(\bm{q})}{\sqrt{m^{j}}}\sqrt{\frac{\hbar}{2\mathcal{N}\omega_{\bm{q}\alpha}}}\mathcal{\hat{A}}_{\bm{q}\alpha}$
(31)

Here $\mathcal{N}$ is the number of unit cells,
$\mathcal{\hat{A}}_{\bm{q}\alpha}=\hat{a}_{\bm{q}\alpha}+\hat{a}^{\dagger}_{-\bm{q}\alpha}$
is the phonon operator of mode $\alpha$ with frequency $\omega_{\bm{q}\alpha}$
and normalized eigenvector $\bm{e}^{j}_{\alpha}(\bm{q})$. To proceed, we need
an analogous expression for the polarization amplitude $u_{i}$ appearing in
Eqs. (21)-(IV.1). We write the ansatz,

$u_{i}=\sum_{\alpha}\frac{\varphi_{\alpha
i}}{\sqrt{m_{u}}}\sqrt{\frac{\hbar}{2\mathcal{N}\omega_{\bm{q}\alpha}}}\mathcal{\hat{A}}_{\bm{q}\alpha}$

with $m_{u}$ the atomic mass constant. Inserting it in Eq. (14) one obtains
that the coefficients $\varphi_{\alpha i}$ are determined by the decomposition
of the normalized displacement vector in the complete basis of the
$\bar{S}_{i}$ modes [Eqs. (15)-(19)]:

$\displaystyle\sqrt{m_{u}}\left(\frac{\bm{e}^{\mathrm{Sr}}_{\alpha}}{\sqrt{m^{\mathrm{Sr}}}},\frac{\bm{e}^{\mathrm{Ti}}_{\alpha}}{\sqrt{m^{\mathrm{Ti}}}},\frac{\bm{e}^{\mathrm{O_{x}}}_{\alpha}}{\sqrt{m^{\mathrm{O}}}},\frac{\bm{e}^{\mathrm{O_{y}}}_{\alpha}}{\sqrt{m^{\mathrm{O}}}},\frac{\bm{e}^{\mathrm{O_{z}}}_{\alpha}}{\sqrt{m^{\mathrm{O}}}}\right)$
$\displaystyle=\sum_{i}\varphi_{\alpha i}\hat{\bm{n}}_{i}\bar{S}_{i}$ (32)

with $\varphi_{\alpha i}$ set to normalize the polarization eigenvector of the
$\alpha$ mode, i.e. $\sum_{j}|\bm{e}^{j}_{\alpha}|^{2}=1$. As mentioned
before, for a given mode $\alpha$, displacements do not need to be collinear
even though they concur to the same polar axis $\bm{\hat{n}}_{p}^{\alpha}$.

With the above quantization, we obtain the following electron-polar-phonon
Hamiltonian for a mode $\alpha$

$\mathcal{H}_{u}=\frac{1}{\sqrt{\mathcal{N}}}\sum_{n\bm{k}\bm{q}\alpha}\psi_{n}^{\dagger}(\bm{k}+\frac{\bm{q}}{2})g^{\alpha}_{n}(\bm{k},\bm{q})\psi_{n}(\bm{k}-\frac{\bm{q}}{2})\mathcal{\hat{A}}_{\bm{q}\alpha}$
(33)

with coupling function

$\displaystyle g^{\alpha}_{n}(\bm{k},\bm{q})$
$\displaystyle=\Bigl{[}g^{\bm{q}\alpha}_{n,A}(\hat{k}_{y},-\hat{k}_{x},0)\sigma_{z}+g^{\bm{q}\alpha}_{n,B}\hat{k}_{z}(-\sigma_{y},\sigma_{x},0)$
$\displaystyle+g^{\bm{q}\alpha}_{n,C}(0,0,\hat{k}_{x}\sigma_{y}-\hat{k}_{y}\sigma_{x})\Bigr{]}\cdot\bm{\hat{n}}_{p}^{\alpha}(\bm{q}).$
(34)

For each $\alpha$ mode we have defined the electron-phonon matrix element:

$g^{\bm{q}\alpha}_{n,l}=ka\sqrt{\frac{\hbar}{2m_{u}\omega_{\bm{q}\alpha}}}\sum_{i}\varphi_{\alpha
i}\tau^{(\bar{S}_{i})}_{n,l}\equiv
ka\sqrt{\frac{\hbar}{2m_{u}\omega_{\bm{q}\alpha}}}\tau^{(\alpha)}_{n,l}$ (35)

where $l=A,B,C$. Equation (35) shows that the $k$-linear Rashba coupling
$\tau^{(\alpha)}_{n,l}$ of mode $\alpha$ is a weighted sum of the Rashba
couplings $\tau^{(\bar{S}_{i})}_{n,l}$ [shown in Fig. 4(a)] with the
coefficients $\varphi_{\alpha i}$ weighing the contribution of the
$\bar{S}_{i}$ modes to the normal mode $\alpha$.

Because of the strongly anharmonic nature of the problem [72, 73, 74, 75], the
eigenvector $\bm{e}^{j}_{\alpha}(\bm{q})$ [Eq. (V)] of the soft mode is
particularly difficult to determine accurately, both theoretically and
experimentally. Eqs. (31)-(35) allow to compute the coupling to polar modes
with arbitrary eigenvectors so they can be used to determine, for example, the
coupling to the soft polar mode from better refined eigenvectors in future
studies. In other words, we have separated the problem of determining the
coupling to the soft mode from the problem of determining its eigenvector.

We illustrate the evaluation of the electron-phonon matrix elements in Eq.
(35) by assuming first the $\alpha$ mode to be a pure Slater mode
$\bar{S}_{1}$ [Eq. (15)]. Then the only non-zero coefficient of the expansion
in Eq. (V) is $\varphi_{\alpha
1}=\sqrt{\frac{m_{u}}{\mu_{\bar{S}_{1}}}}=0.204$, where we have introduced the
reduced mass of the Slater mode
$\mu^{-1}_{\bar{S}_{1}}=\left(m^{\mathrm{Ti}}\right)^{-1}+\left(3m^{\mathrm{O}}\right)^{-1}$.
In this case the coupling Eq. (35) is then reduced to the following simple
expression [28]

$g^{\bm{q}\alpha}_{n,l}=ka\sqrt{\frac{\hbar}{2\mu_{\bar{S}_{1}}\omega_{\bm{q}\alpha}}}\tau^{(\bar{S}_{1})}_{n,l}.$
(36)

Substituting the estimated[50] Rashba couplings $\tau^{(\bar{S}_{1})}_{n,l}$
for the Slater mode $\bar{S}_{1}$ and the experimental zone center frequency
of the soft FE mode $\omega_{\bm{q}\alpha}=\omega_{\mathrm{TO}}^{E_{u}}$ and
$\omega_{\mathrm{TO}}^{A_{2u}}$ we obtain the electron-TO couplings for STO
listed in Table 2 (under $\bar{S}_{1}$ columns). The value for $l=A$ in the
first column coincides with the value reported in Ref. [28],
$g^{\mathrm{TO}}_{1,A}=65\mathrm{meV}\cdot k_{F}a$. The present results
generalize our previous computation for arbitrary polar modes and for all
symmetry allowed couplings.

The coupling constants for other $\bar{S}_{i}$ modes can be estimated in an
analogous way, by including the appropriate reduced mass for each mode, and
the optical gap $\omega_{\bm{q}\alpha}$ of the mode we are interested in. For
instance, for a pure $\bar{S}_{2}$ mode the only non-zero contribution in Eq.
(35) is $\varphi_{\alpha 2}=\sqrt{\frac{m_{u}}{\mu_{\bar{S}_{2}}}}=0.148$
where
$\mu^{-1}_{\bar{S}_{2}}=\left(m^{\mathrm{Sr}}\right)^{-1}+\left(m^{\mathrm{Ti}}+3m^{\mathrm{O}}\right)^{-1}$
is the reduced mass of the $\bar{S}_{2}$ mode.

Table 2: Electron-TO coupling of the soft FE mode $\alpha=\mathrm{TO}$ to electronic band $n$ of STO as obtained from Eqs. (V)-(35), assuming a Slater mode Eq. (36) ($\bar{S}_{1}$), a mode from neutron scattering [57] Eq. (37) ($\mathcal{N}$), and from Raman spectroscopy [58] Eq. (38) ($\mathcal{R}$). We have used the experimental phonon frequencies [61] at low-$T$ $\hbar\omega_{\mathrm{TO}}^{E_{u}}=1$ meV and $\hbar\omega_{\mathrm{TO}}^{A_{2u}}=2$ meV, and electronic momentum $k_{F}a=0.3$. |  |  $g^{\mathrm{TO}}_{n,A}$ [meV] |  |  |  $g^{\mathrm{TO}}_{n,B}$ [meV] |  |  |  $g^{\mathrm{TO}}_{n,C}$ [meV] |
---|---|---|---|---|---|---|---|---|---
$n$ | $\bar{S}_{1}$ | $\mathcal{N}$ | $\mathcal{R}$ | $\bar{S}_{1}$ | $\mathcal{N}$ | $\mathcal{R}$ | $\bar{S}_{1}$ | $\mathcal{N}$ | $\mathcal{R}$
1 | 19 | 21 | 51 | 11 | 10 | 8 | 5 | 5 | 5
2 | -21 | -22 | -47 | -13 | -16 | 7 | 7 | 9 | 1
3 | 12 | 12 | 23 | 15 | 18 | -5 | -10 | -12 | -4

Let us now turn instead to a more realistic eigenvector in Eq. (V), which
generally will have a contribution from different $\bar{S}_{i}$ modes. We can
use the normalized atomic displacements estimated for the soft ferroelectric
mode by neutron scattering [57] and hyper-Raman [58] experiments in the
high-$T$ cubic structure. According to these works the $\varphi_{\alpha i}$
coefficients for the expansion in Eq. (V) for the soft ferroelectric mode
$\alpha=\mathrm{TO}$ are respectively:

$\displaystyle(\varphi_{\alpha 1},\varphi_{\alpha 2},\varphi_{\alpha
3})_{\mathrm{neutron}}=(0.189,0.059,0.0014),$ (37)
$\displaystyle(\varphi_{\alpha 1},\varphi_{\alpha 2},\varphi_{\alpha
3})_{\mathrm{Raman}}=(0.198,-0.0154,-0.08).$ (38)

As already mentioned, both have a predominant $\bar{S}_{1}$ mode contribution,
but while the first case, Eq. (37), implies a small motion of the Sr atoms and
a mostly octahedral motion of the oxygens
($|\varphi_{1}|>|\varphi_{2}|\gg|\varphi_{3}|$), the second case, Eq. (38),
suggests a mode in which the Sr atoms are essentially at rest with a
significant distortion of the oxygen octahedra
($|\varphi_{1}|>|\varphi_{3}|\gg|\varphi_{2}|$) [see Figs. 2(b)-(c)].

In order to estimate the Rashba coupling constants arising from these two
cases we assume the soft mode in the low-$T$ tetragonal phase is weakly
changed and well described by the decomposition with coefficients
$\varphi_{\alpha i}$ given by Eqs. (37)-(38). Substituting these into the
weighted sum of Rashba-couplings in Eq. (35), we obtain a new set of electron-
TO couplings $g^{\mathrm{TO}}_{n,l}$ for the neutron ($\mathcal{N}$) and Raman
($\mathcal{R}$) eigenvectors. Figure 4(b) and Table 2 collect all the
estimated el-TO Rashba-like couplings in this work. As seen, while the values
from the eigenvector from neutron data [Eq. (37)] and the pure $\bar{S}_{1}$
Slater mode are very similar, the resulting couplings are quite different for
the eigenvector consistent with Raman data [Eq. (38)]. Indeed, the substantial
variation of the el-TO coupling constants in the latter case originates from
the intermediate contribution of the oxygen cage distortion of the
$\bar{S}_{3}$ mode (through the $\varphi_{\alpha 3}$ coefficient) which
couples to the gigantic Rashba coefficients $\tau^{(\bar{S}_{3})}_{n,l}$ [see
Fig. 4(a)]. In particular, as shown in Fig. 4(b), the absolute value of the
$E_{u}$ coupling $g_{n,A}$ of the Raman determined eigenvector ($\mathcal{R}$)
has more than doubled for all three bands and clearly dominates over the other
two couplings ($|g_{n,A}|\gg|g_{n,B}|,|g_{n,C}|$), which have been
significantly reduced in most cases. We remind the reader that the coupling
$g_{n,A}$ corresponds to the case in which the pseudospin is aligned in the
$z$ direction [c.f. Eq. (V)].

As a consistency check, we have recomputed the coupling to the Raman mode
directly from the band splittings, imposing its eigenvector in a frozen phonon
computation in DFT and obtained the same results as with the weighted sum of
Rashba-couplings [Eq. (35)].

Crucial for our results is the weight of the $\bar{S}_{3}$ component. One sees
that the oxygen cage appears very rigid in neutrons while it deforms
substantially in Raman. Theoretically, the determination of the soft-mode
eigenvector requires the solution of a highly non-harmonic dynamical phonon
problem which goes beyond our present scope. As a proxy for this eigenvector,
we can examine the fully relaxed broken symmetry ground state, which is polar,
since Born-Oppenheimer (adiabatic) DFT does not contain the quantum
fluctuations which make the system disorder[1]. Such DFT eigenvector has a
substantial $\bar{S}_{3}$ component, even larger than the one of the Raman
determined mode[50]. This suggests that the Raman determination is more
reliable than the one from neutrons, and hence we will consider its
eigenvector in the following computations.

##  VI  The superconducting dome

In this Section, starting from our findings on the Rashba electron-phonon
interaction in STO presented in Section V, we extend them to high momentum and
explore the consequences for superconductivity.

One important result from the previous section is that the electron-phonon
matrix-elements are very sensitive to the form of the eigenvector of the polar
mode. To address this sensitivity we estimate the BCS pairing coupling
constant for both the Slater and the Raman determined eigenvectors. For the
shake of comparison we restrict now to $E_{u}$ Rashba couplings of the lowest
band ($n=1$), and the $l=A$ irrep component. This is well justified for the
Raman eigenvector since $|g_{n,A}|\gg|g_{n,B}|,|g_{n,C}|$ and partially
justified also for the Slater mode since
$|g^{\mathrm{TO}}_{1,A}|>|g^{\mathrm{TO}}_{1,B}|,|g^{\mathrm{TO}}_{1,C}|$.
Since the magnitude of this matrix element is substantially larger for the
Raman determined eigenvector than for the pure $\bar{S}_{1}$ mode [see Fig.
4(b)] and the superconducting coupling constant is proportional to the square
of the Rashba electron-phonon matrix element [28],

$|g^{\mathrm{TO}(\mathcal{R})}_{1,A}|=2.6|g^{\mathrm{TO}(\bar{S}_{1})}_{1,A}|\longrightarrow\lambda_{\mathrm{BCS},A}^{(\mathcal{R})}\approx
7\lambda_{\mathrm{BCS},A}^{(\bar{S}_{1})},$ (39)

this translates into a factor of 7 larger SC coupling when taking the square.
Hence we see that the details of the eigenvector can strongly influence the
resulting bare couplings and in turn its pairing coupling strength. For a pure
Slater mode, the other $l=B,C$ couplings should also be taken into account and
the resulting gap structure and final pairing coupling constant will depend on
their ratio [26].

Given the DFT determined polar state[50], in the following we will assume the
soft-mode is best described by the hyper-Raman determined eigenvector.
Luckily, this simplifies computations as $|g_{n,B}|$ and $|g_{n,C}|$ can be
neglected, and estimates based on the sole contribution of the $l=A$ irrep
component are well justified.

###  VI.1 Generalized Rashba in DFT and dome behavior

So far we have explored the conventional Rashba-like $k$-linear model in Eq.
(IV.1), which describes the coupling between the soft FE phonon and the
electrons fairly well at low momenta, as we have shown by frozen phonon
computations [Fig. 3]. However, as already mentioned in Section IV.1, the band
split obtained by the _ab initio_ computations exhibits deviations from
linear-in-$k$ beyond a characteristic momenta $ka$ which generally depends on
the electronic band $n$, the momentum direction $\hat{\bm{k}}$ and the polar
mode $\bar{S}_{i}$.

Figure 6:  Electronic band split for the three bands of STO in the presence of
a frozen polar mode with eigenvector deduced from Raman [Eq. (38)] and with
polar axis $\bm{\hat{n}}_{p}^{E_{u}}\parallel[1\bar{1}0]$ and momentum
$\bm{k}\parallel[110]$. The right $y$-axis shows the corresponding generalized
Rashba electron-phonon matrix element [Eq. (40)]. Solid lines in (a) are _ab
initio_ results and dashed lines the conventional Rashba model [Eq. (25) and
Eq. (35)]. Panel (b) is the result for minimal $t_{2g}$ model in Eqs. (13),
(27) and (41) with the coupling matrix element corresponding to the Raman
determined eigenvector, $2\partial_{x}t_{xy0}=\tau^{\mathcal{R}}_{1,A}=115$
meV/Å (extracted from DFT results in (a)).

Figure 6(a) shows the _ab initio_ results (solid lines) of the pseudospin-
split of each $n$ band going beyond the small $ka$ values presented in Fig. 3.
We chose a frozen-phonon $E_{u}$ Raman mode [Eq. (38)] with polar axis
$\hat{\bm{n}}_{p}\parallel[1\bar{1}0]$, and show the band split along the
perpendicular momentum direction $\hat{\bm{k}}\parallel[110]$. As seen, for
all bands the band-split $\delta E_{n}$ is initially conventional Rashba-like
(growing linearly with momenta), but deviates from linearity and peaks at
intermediate values of momenta after which steadily decreases in a form close
to $1/k$.

To linear-in-$k$ order the splitings are given by the Rashba couplings through
Eq. (25) and Eq. (35). We can generalize Eq. (35) to an arbitrary odd function
of $k$ by introducing for each electronic band $n$,

$g^{\bm{q}\alpha}_{n,l}=ka\mathcal{F}_{n,l}(ka)\sqrt{\frac{\hbar}{2m_{u}\omega_{\bm{q}\alpha}}}\tau^{(\alpha)}_{n,l},$
(40)

were we defined $\mathcal{F}_{n,l}(ka)$ such that it is even in $k$ and
$\mathcal{F}_{n,l}(ka)\rightarrow 1$ for $k\rightarrow 0$. By definition, the
electron-phonon matrix element is proportional to the band split so
$\mathcal{F}_{n,l}(ka)$ was extracted directly from the _ab initio_ results.
The corresponding $g_{n,A}^{\mathrm{TO}}$ is given by the right $y$-axis in
Fig. 6(a). We anticipate that this dome in $k$ results in a dome in electronic
density for both $\lambda_{\mathrm{BCS}}$ and $T_{c}$ so it is important to
discuss its origin, which we do next.

###  VI.2 Minimal model and dome behavior

The dome-like behavior of the band split can be traced back to a $k$-dependent
quenching of angular momentum. The essential physics is captured by the
minimal model in Eq. (1) supplemented by a one-parameter simplification of the
polar interaction in Eq. (27). Namely we keep only the spin-conserving $j=0$
term and restrict to mixing of $x$ and $y$ orbitals,

$t_{xy0}(\bm{k},\bm{q})=2i\partial_{x}t_{xy0}u(\bm{q})\left[-\sin\left(k_{y}a\right)\hat{n}_{px}+\sin\left(k_{x}a\right)\hat{n}_{py}\right]$
(41)

Computing the band splitting along $\hat{\bm{k}}\parallel[110]$ for
$\hat{\bm{n}}_{p}\parallel[1\bar{1}0]$ one obtains panel (b) of Fig. 6. As
seen, this simple approximation captures very well the coupling of the first
two bands including the dome behavior. It underestimates the coupling of the
third band which therefore calls for additional parameters beyond the scope of
this subsection.

Figure 7: (a) Band split for two-orbital toy-model Eq. (42), and the
perturbative limits Eqs. (43a)-(43b). (b) $k$-dependence of eigenstates, in
the perturbative limits shown in (a). The color legend represents the argument
of the wave function.

Focusing on the lowest band, the wave function near $\Gamma$ [Eq. (7)],
suggests to simplify even more the non-interacting $t_{2g}$ model Eq. (1) by
restricting it to two orbitals $\mu=x,y$, which is formally equivalent to
taking the AFD parameter to infinity, $\Delta\rightarrow\infty$ in Eq. (4).
The band split of this toy-model for a finite $u(\bm{q})$ and the same
$\hat{\bm{k}}$ and $\hat{\bm{n}}_{p}$ orientations as above can be
analytically obtained, and reads

$\frac{\delta
E(k[110],u[1\bar{1}0])}{2u}=\frac{\tau_{1,A}\sqrt{2}\sin\left(\frac{ka}{\sqrt{2}}\right)}{\sqrt{1+\left[\frac{4t_{4}\sin^{2}\left(\frac{ka}{\sqrt{2}}\right)}{\xi}\right]^{2}}},$
(42)

where we have used the relation $2\partial_{x}t_{xy0}=\tau_{1,A}$ [ Eq. (30)].
Expanding to linear order in $k$ one recovers the conventional Rashba form of
Ref. [28] which here is generalized to arbitrary momentum.

Equation (42) is plotted in Fig. 7(a) for the electronic parameters in STO
$t_{4}$ and $\xi$ listed in Sec. II.2, and $\tau^{\mathcal{R}}_{1,A}=115$
meV/Å corresponding to the Raman deduced soft-mode eigenvector. As seen, the
two-orbital toy model excellently captures the band split of the lowest band,
$n=1$, computed by DFT [c.f. Fig. 6(a) and Fig. 7(a)]. Furthermore, the
analytical result for $n=2$ is identical to the $n=1$ case while in DFT both
are very similar. Thus, surprisingly, the two-orbital toy-model provides a
good approximation to the second band despite its non-negligible weight of the
$z$-orbital near $\Gamma$ [Eq. (8)]. This is attributed to the rapid decrease
of the $z$-orbital character as momentum increases along $[110]$. Indeed,
including the $z$ orbital results in little change on the splittings for
$n=1,2$ [Fig. 6(b)].

The $k$-dependence in Eq. (42) is determined by the competition between the
SOC energy $\xi$ in $\mathcal{H}_{\mathrm{SOC}}$ (which dominates at $k=0$),
and the hopping term $4t_{4}\sin^{2}\left(\frac{ka}{\sqrt{2}}\right)$ [Eq.
(6)] which induces a mass mismatch of the bands in $\mathcal{H}_{0}$, and
increases with $k$. The former term promotes a state with $l_{z}=\pm 1$
angular momentum $c^{\dagger}_{x,\pm}\pm ic^{\dagger}_{y,\pm}$ [Eq. (7)]
leading to $j_{z}=\pm 3/2$ states, whereas the latter term constrains the
system towards $c^{\dagger}_{x,\pm}\pm c^{\dagger}_{y,\pm}$ states which have
$\langle{\bm{l}}\rangle=0$. This $k$-dependent quenching of angular momentum
is illustrated in Fig. 7(b) where the complex, $l_{z}=\pm 1$ (real,
$\langle\bm{l}\rangle=0$) orbitals for small (large) $k$ are shown.

Doing perturbation in Eq.(42) in the two opposite limits, where the SOC term
dominates over the hopping term and vice versa, one obtains the following
expressions for the band split in the continuum limit ($ka\ll 1$),

$\displaystyle\frac{\delta E}{2u}\approx$
$\displaystyle\tau_{1,A}ka\left[1+\mathcal{O}\left(\frac{t_{4}^{2}(ka)^{4}}{\xi^{2}}\right)\right],\
\ \ \ t_{4}(ka)^{2}\ll\xi$ (43a) $\displaystyle\frac{\delta E}{2u}\approx$
$\displaystyle\frac{\tau_{1,A}\xi}{2t_{4}ka}\left[1+\mathcal{O}\left(\frac{\xi^{2}}{t_{4}^{2}(ka)^{4}}\right)\right],\
\ \ \ \xi\ll t_{4}(ka)^{2}$ (43b)

We recover the Rashba linear-in-$k$ term Eq. (25) when the SOC energy term
dominates over the hopping term [Eq. (43a)], and the $1/k$ dependence in the
opposite limit, when the kinetic term takes over [Eq. (43b)]. These two
perturbative expressions are shown in Fig. 7(a) together with the full
expression Eq. (42), by dashed orange and purple lines, respectively.
Deviations of the expansion at large momentum are due to lattice effects which
were neglected as they do not change the qualitative picture. Clearly, when
the angular momentum becomes quenched the spin-orbit assisted electron-phonon
interaction becomes ineffective and dies out. In the continuum limit the
maximum of the coupling is given by

$k^{max}a=\sqrt{\frac{\xi}{2t_{4}}}\approx 0.42,$ (44)

which again illustrates the competition between spin orbit and band mass
mismatch energies. The right hand side corresponds to the present parameters,
relevant for STO.

Because the pairing interaction arising from the polar coupling is in turn
proportional to the square of the electron-phonon matrix element [28] (shown
in the right $y$-axis of Fig. 6),
$V_{\mathrm{TO}}\propto|g^{\mathrm{TO}}|^{2}$, it also acquires a pronounced
peak as a function of $k=k_{F}$, the Fermi momentum. The initial quadratic
increase with $k_{F}$ peaks and decreases as $V_{\mathrm{TO}}\propto
1/k_{F}^{2}$ for all three bands. Consequently the pairing coupling
constant[28] $\lambda_{\mathrm{BCS},A}=\frac{2}{3}N_{F}V_{\mathrm{TO}}\propto
k_{F}V_{\mathrm{TO}}$ of each band (assuming parabolic bands) shows a dome-
like form with increasing $k_{F}$ [inset of Fig. 8]. We took a constant factor
of 2 effective mass enhancement in $N_{F}$, chosen to match the values from
specific heat [76] at low carrier densities.

Figure 8: $T_{c}$ dome vs carrier density $n=\frac{k_{F}^{3}}{3\pi^{2}}$ for
the lowest band $n=1$, normalized to its maximum value. Full (dashed) line
neglects (includes) the hardening of the TO mode. Inset: band resolved
$\lambda_{\mathrm{BCS},A}=\frac{2}{3}N_{F}\frac{|g_{n,A}^{\mathrm{TO}}|^{2}}{\omega_{\mathrm{TO}}}$
using the _ab initio_ results $g_{n,A}^{\mathrm{TO}}$ of Fig. 6(c). Open
symbols are experimental data from Ref. [77] (circles), Ref. [78](triangles)
and Ref. [79] (diamonds) with $T_{c,max}=0.35$K.

The obtained values of $\lambda_{\mathrm{BCS},A}$ are in the weak coupling
limit so we can use a simple BCS formalism. This is in agreement with the
$2\Delta(T=0)/T_{c}$ ratio obtained with tunneling and microwave spectroscopy
studies [79, 80] which suggest that a weak coupling picture applies.
Neglecting inter-band (finite-$q$) couplings in Eq. (22), we obtain three
uncoupled SC gap equations, one for each electronic band. In this
approximation the higher $T_{c}$ corresponds to the bulk critical temperature
of the material.

This simplified picture already predicts a dome of
$T_{c}\propto\exp[-1/\lambda_{BCS,A}]$ within the generalized Rashba coupling
pairing mechanism, shown in Fig. 8 for the first band $n=1$. Notice that the
largest $T_{c}$ corresponds to $n=2$, but since both curves are very similar
there is little difference on which one is chosen. Note also that we are
assuming a rigid band picture, which seems to be a good approximation for bulk
SC obtained by Nb and La doping [81].

Within the present mechanism we believe that the dominant correction to the
above computation of the dome is given by the hardening of the TO mode with
carrier density [15, 29],
$\tilde{\omega}_{\mathrm{TO}}(k_{F}a)=\sqrt{\omega_{\mathrm{TO}}^{2}+D(k_{F}a)^{3}}$
with $D=1.64\mathrm{meV}^{2}$. This parametrization was chosen to match the
factor of three hardening of the soft mode at low temperatures when reaching
$n=1.5\times 10^{20}$ as measured by infra-red spectroscopy[82]. As seen by
the dashed lines in Fig. 8, since $\lambda_{\mathrm{BCS}}\propto
1/\omega^{2}_{\mathrm{TO}}$, the hardening reduces the pairing constant even
faster at high Fermi momenta where the hardening effect is largest.
Consequently the $T_{c}$ dome is narrowed and shifted to lower densities.

###  VI.3 Comparison with experiments

A key prediction of the present mechanism is the density where $T_{c}$ peaks,
which including (neglecting) the hardening of the soft mode is found to be
$n_{opt}\approx 7\times 10^{19}$ cm-3 ($n_{opt}\approx 1\times 10^{20}$ cm-3
), in excellent agreement with diamagnetic experiments[83, 77, 79], without
free parameters. To the best of our knowledge the fine resolution in carrier
density achieved in early Refs. [83, 77] has not since been attained in bulk
superconductivity measurements. To determine the experimental optimum density,
we have neglected two outlier points in Ref. [77] that were neglected also in
their fit and a similar outlier point in the data of Ref. [78], which
unfortunately does not cover the maxima of Refs. [83, 77, 79]. More
experimental and theoretical work is needed to understand if those outliers
are a systematic effect. Eventually, they may be attributed to multiband
effects neglected here. Furthermore, we are not taking into consideration
resistivity data, as it is not a bulk superconductivity probe. Indeed, $T_{c}$
in bulk probes (specific heat, thermal conductivity and diamagnetism) is
consistently lower than the zero-resistance $T$, which points towards
filamentary superconductivity at higher $T$s and extremely dilute samples [2,
84, 81].

When plotted in linear scale it becomes clear that the dome is very asymmetric
with a rapid rise and a much slower decrease [Fig. 8]. Also this asymmetry is
well reproduced by the theory presented here. The rise of $T_{c}$ appears at a
density slightly higher than in experiments while the decrease is very
sensitive to what is assumed for the hardening of the soft mode. Within the
present uncertainties on the experimental data [Fig. 8], also the width and
the asymmetry of the dome are in very good agreement with experiment.

An important remaining question is if the present theory can explain the
observed maximum value of $T_{c}$. In Ref. [28] we used the naive $k$-linear
Rashba form, neglected the phonon hardening, and obtained a good estimate of
$T_{c}$ near optimum density. The present computations show that both
approximations can lead to an overestimation of $\lambda_{\mathrm{BCS}}$. On
the other hand, a pure Slater mode $\bar{S}_{1}$ was assumed; Eq. (39) shows
that the details of the soft mode eigenvector strongly affects the coupling
strength. For the soft mode eigenvector found by hyper-Raman [Eq. (38)], the
overestimation error of Ref. [28] tends to cancel with the amplification of
Eq. (39) so the computed $\lambda_{\mathrm{BCS}}$ is again close to the one
required by experiments. Indeed, taking the BCS form
$k_{B}T_{c}=1.13\hbar\omega_{TO}\exp[-1/\lambda_{\mathrm{BCS}}]$ and assuming
the soft mode frequency to be $\omega_{TO}=3$ meV near the optimum density
requires a $\lambda_{\mathrm{BCS}}=0.21$ to obtain a maximum transition
temperature near the experimental one[83, 77, 79, 2] ($T_{c,max}\approx 0.35$
K). In our computation the maximum $\lambda_{\mathrm{BCS}}$, including phonon
hardening, is $\lambda_{max}=0.13$ [see inset of Fig. 8] which is fairly close
to this BCS estimate without any free parameter. Thus, it is clear that the
present mechanism can explain the observed $T_{c}$ as $\lambda_{max}$ is in
the correct range.

As explained above, our estimate of $\lambda_{\mathrm{BCS}}$ is a lower bound.
A more accurate computation should take into account that around optimum
carrier density three bands are filled; our estimate takes into account only
one band. Also the contributions of $g_{n,B}$ and $g_{n,C}$ matrix elements to
the $A_{1g}$ pairing channel have been neglected. Both of these aspects are
expected to increase $T_{c}$ and hence improve the agreement with the
experiment. While one can incorporate the above corrections in the
computations, in practice, due to the exponential dependence of $T_{c}$ with
$\lambda_{\mathrm{BCS}}$ and the approximate treatment of the Coulomb
interaction [85] (neglected in this work), a high level of accuracy in $T_{c}$
should not to be expected. Therefore, for the time being, we consider the
present results robust enough to claim that the magnitude of $T_{c}$ can be
explained within the present mechanism.

##  VII Summary and Conclusions

In this work, we have derived the most general Rashba-like linear coupling
between the electronic bands and the polar modes at the zone center of
tetragonal STO [Eq. (IV.1)]. Fitting the electronic band split of the Rashba
model [Eqs. (25)-(26)] to _ab initio_ $q=0$ frozen-phonon calculations [Fig.
3] we have estimated the corresponding Rashba couplings,
$\tau^{(\bar{S}_{i})}_{n,l}$, to zone center polar modes $\bar{S}_{i}$ [Fig.
4(a)]. These $\bar{S}_{i}$ modes form a complete basis of in-plane, $E_{u}$,
and out-of-plane, $A_{2u}$, modes at the zone center in STO [Eqs. (15)-(19)]
and hence we have mapped out the entire Rashba-like linear coupling subspace
of these zone-center polar modes.

The origin of the Rashba couplings can be understood as arising from induced
hopping channels between neighboring $t_{2g}$ orbitals in the presence of
polar distortions [Eqs. (27)]. We have explicitly shown how to connect the
symmetry allowed Rashba-coupling constants $\tau_{n,l}$ in Eq. (IV.1) to the
microscopic hopping processes $\partial_{l}t_{\mu\nu}$ [Eqs. (29) and (30)].
Indeed, a minimal, three-orbital model with only one induced hopping parameter
reproduces qualitatively and to some extent quantitatively many results of the
ab initio computations [Fig. (6)].

We have shown how to estimate the electron-polar-phonon coupling function of a
general polar mode by decomposing it into the $\bar{S}_{i}$ basis [Eqs.
(V)-(V)] using the obtained Rashba-couplings $\tau^{(\bar{S}_{i})}_{n,l}$
[Fig. 4(a)]. Since an accurate determination of the soft-mode eigenvector is
difficult, this allows us to separate the problem of determining the coupling
to the soft mode from the problem of determining its eigenvector.

Starting from eigenvector cases of a Slater mode $\bar{S}_{1}$ and those
consistent with neutron data [Eq. (37)] and hyper-Raman data [Eq. (38)] in
STO, we have obtained and compared three sets of electron-TO-phonon couplings
$g_{n,l}^{\mathrm{TO}}$ [Fig. 4(b) and Table 2]. We find substantial coupling
values for the three electronic bands, with the $E_{u}$ coupling
$g_{n,A}^{\mathrm{TO}}$ being larger than the other two couplings
$g_{n,B}^{\mathrm{TO}}$ and $g_{n,C}^{\mathrm{TO}}$. Physically the $A$
component of the $E_{u}$ irrep corresponds to the case in which the pseudospin
lies along the $z$-direction, as considered in Ref. [28] and in the toy-model
of Sec. VI.2.

We found that the details of the eigenvector can substantially alter the
magnitude of the couplings. In particular, the intermediate contribution of
the $\bar{S}_{3}$ mode (which distorts the oxygen octahedra) to the Raman
determined eigenvector [Eq. (38)] significantly increases the $E_{u}$ coupling
$g_{n,A}^{\mathrm{TO}}$ for all three bands [see Fig. 4(b)].

Certainly, the sensitivity of the couplings to the eigenvector has important
implications for the Rashba pairing mechanism [28], since the BCS coupling
constant $\lambda_{BCS}$ is proportional to the square of the electron-TO-
phonon coupling in its simplest form. Comparison with the DFT broken symmetry
state suggests that the oxygen cage should deform substantially in the soft
mode as found by the hyper-Raman eigenvector. More experimental and
theoretical work will be highly desirable to refine the eigenvector of the
soft mode and improve the estimate of $\lambda_{\mathrm{BCS}}$.

While the linear-in-$k$ conventional Rashba model works well at low momenta,
our _ab initio_ frozen phonon results generally indicate a deviation from the
linear-$k$ Rashba split of the three electronic bands beyond a critical wave
vector for all polar modes. As a result, we find a dome-like behavior of the
electron-TO-phonon coupling $g_{n,l}^{\mathrm{TO}}$ [Eq. (40)]: beyond the
critical wave vector the linear-$k$ growth slows and peaks, subsequently
evolving into a $1/k$ decrease [Fig. 6]. This behavior is due to a
$k$-dependent quenching of the angular momentum, as explicitly shown by
reducing the three-orbital model to a two-orbital toy model [Eq. (42) and Fig.
7].

Assuming a rigid band shift for electronic doping, and without introducing
electronic screening effects, this deviation from Rashba already entails a
dome for $T_{c}$ as a function of electronic density. We remark that this
result does not depend crucially on the particular form of the polar mode
eigenvector. A popular explanation of the dome invokes the proximity to a
ferroelectric quantum critical point [1]. In the simplest picture, the dome is
attributed to the change of the frequency of the soft-mode with density,
possibly with the structural transition laying below the dome [86]. This last
simplified picture, however, requires that the system breaks inversion
symmetry to the right or to the left of the optimum density which, to the best
of our knowledge, is not the case.

In the mechanism we presented the proximity to the ferroelectric quantum
critical point is important to have a soft-mode in the first place, which
increases $\lambda_{\mathrm{BCS}}\propto 1/\omega_{\mathrm{TO}}^{2}$, but it
is not responsible for the non-monotonous behavior of $T_{c}$. Including the
hardening with doping increases the agreement with the experimental data of
Refs. [77, 83] [Fig. 8].

The obtained maximum value of $T_{c}$, the density at which it peaks, and the
characteristic asymmetry of the dome are in surprisingly good agreement with
experiments without free parameters [Fig. 8]. Small deviations remain, which
we attribute to several simplifications we have made to avoid introducing more
parameters in the theory. For example, additional Rashba coupling
contributions to the $A_{1g}$ pairing channel can be considered, which will
increase $T_{c}$ and possibly decrease the smaller density at which bulk
superconductivity becomes robust.

Our results place the present pairing mechanism as a leading candidate to
explain the observed bulk superconductivity in doped STO, as
$\lambda_{\mathrm{BCS}}$ is both of the right order of magnitude and in the
correct range of electronic carrier density.

The approach presented here can be generalized to study the linear coupling
characteristics between electrons and FE soft TO modes, as well as the
corresponding pairing mechanism in other incipient ferroelectrics. A
noteworthy example are KTO interfaces, where the linear coupling to the TO
mode has recently been invoked for pairing [49].

## Acknowledgements

We thank P. Volkov for useful discussions. We acknowledge financial support
from the Italian MIUR through Projects No. PRIN 2017Z8TS5B, and No.
20207ZXT4Z. M.N.G. is supported by the Marie Skłodowska-Curie individual
fellowship Grant Agreement SILVERPATH No. 893943. We acknowledge the CINECA
award under the ISCRA initiative Grants No. HP10C72OM1 and No. HP10BV0TBS, for
the availability of high-performance computing resources and support.

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