# Layer and spontaneous polarizations in perovskite oxides and their interplay
in multiferroic bismuth ferrite

Nicola A. Spaldin  Department of Materials, ETH Zurich, CH-8093 Zürich,
Switzerland  Ipek Efe  Department of Materials, ETH Zurich, CH-8093 Zürich,
Switzerland  Marta D. Rossell  Electron Microscopy Center, Swiss Federal
Laboratories for Materials Science and Technology, Empa, 8600, Dübendorf,
Switzerland  Chiara Gattinoni  Department of Materials, ETH Zurich, CH-8093
Zürich, Switzerland

###### Abstract

We review the concept of surface charge, first in the context of the
polarization in ferroelectric materials, and second in the context of layers
of charged ions in ionic insulators. While the former is traditionally
discussed in the ferroelectrics community, and the latter in the surface
science community, we remind the reader that the two descriptions are
conveniently unified within the modern theory of polarization. In both cases,
the surface charge leads to electrostatic instability — the so-called “polar
catastrophe” — if it is not compensated, and we review the range of phenomena
that arise as a result of different compensation mechanisms. We illustrate
these concepts using the example of the prototypical multiferroic bismuth
ferrite, BiFeO3, which is unusual in that its spontaneous ferroelectric
polarization and its layer charges can be of the same magnitude. As a result,
for certain combinations of polarization orientation and surface termination
its surface charge is self-compensating. We use density functional
calculations of BiFeO3 slabs and superlattices, analysis of high-resolution
transmission electron micrographs as well as examples from the literature to
explore the consequences of this peculiarity.

##  I Introduction

Bound charge at the surface of an insulator, or at an interface between two
insulating materials, must be screened in order to avoid a so-called polar
catastrophe caused by a divergence of the electrostatic energy Stengel (2011).
Such bound surface charge, $\sigma_{\text{surf}}$, exists whenever there is an
uncompensated component of the bulk polarization, $\vec{P}_{\text{bulk}}$,
perpendicular to a surface, and is given by

$\sigma_{\text{surf}}=\vec{P}_{\text{bulk}}\cdot\vec{n}$ (1)

where $\vec{n}$ is the unit vector along the surface normal. A bulk
polarization can occur of course in ferroelectric materials due to their
spontaneous polarization. It can also occur in centrosymmetric crystals in
which the ions form charged layers; we refer to this latter contribution as
the layer polarization.

Many mechanisms are known for screening bound surface charge, ranging from
metal electrodes Fong  _et al._ (2006); Sai  _et al._ (2005), to formation of
charged defects Wang  _et al._ (2009); Highland  _et al._ (2011); Gao  _et
al._ (2017); Chisholm  _et al._ (2010); Levchenko and Rappe (2008) or
adsorption of charged species Setvin  _et al._ (2018); Strkalj  _et al._
(2019); Gattinoni  _et al._ (2020); Garrity  _et al._ (2013), and even loss or
reorientation of ferroelectric polarization Junquera and Ghosez (2003);
Ishikawa  _et al._ (1996); Mundy  _et al._ (2020). Note that it is not
possible to screen the bound surface charge by surface relaxation alone,
without the addition or removal of charged species.

In this work we explore the special case of multiferroic perovskite-structure
bismuth ferrite, BiFeO3, which has both a spontaneous polarization from its
ferroelectric distortion and a layer polarization from its ionic charges.
BiFeO3 is particularly unusual because the size of its spontaneous
ferroelectric polarization in the common [001] and [111] growth directions is
close to the size of the layer polarization from the charged ionic layers in
flat (100) and (111) planes. As a result, for certain choices of polarization
orientation and surface termination, the spontaneous and layer polarizations
self-compensate, leading to uncharged surfaces that are stable without
external screening mechanisms.

We begin by reviewing in the next section (Sec. II) two key results of the
modern theory of polarization — the multivaluedness of the polarization
lattice and the concept of the polarization quantum — that are key to this
work. We then briefly review compensating mechanisms at the surfaces of
ferroelectric materials with charge-neutral layers (Sec. III). Next, we
discuss centrosymmetric materials with charged layers (Sec. IV), and show how
the layer polarization associated with charged ionic layers is conveniently
described within the modern theory of polarization. We then combine the
concepts of layer polarization and spontaneous polarization in the example of
bismuth ferrite (Sec. V), discussing in turn its interface with metallic
electrodes, and with insulators with different layer polarizations.

##  II Reminder of key results from the modern theory of polarization

We begin with a reminder of a fundamental result of the modern theory of
polarization King-Smith and Vanderbilt (1994), that the polarization,
$\vec{P}$, of a bulk periodic solid, is not a single number but rather a
lattice of values, separated by the polarization quantum,
$\vec{P}_{q}=\frac{e\vec{R}}{V}$. The polarization quantum corresponds to the
change in polarization on moving an electronic charge $e$ by a lattice vector
$\vec{R}$ ($V$ is the unit cell volume), which changes the polarization by an
amount $\vec{P}_{q}$, but does not change the physical system. (For a more
extensive introductory discussion, see Ref. Spaldin, 2012.)

A consequence of this property is that the polarization lattice of a
centrosymmetric crystal, which must also be centrosymmetric by symmetry, can
take one of two sets of values,

$\displaystyle\vec{P}$ $\displaystyle=$ $\displaystyle
0+n\vec{P}_{q}\quad,\quad\text{or}$ (2) $\displaystyle\vec{P}$
$\displaystyle=$ $\displaystyle\frac{\vec{P}_{q}}{2}+n\vec{P}_{q}\quad,$ (3)

where $n$ is any integer. All insulating, centrosymmetric, periodic solids can
be classified as belonging to one of these classes, which we will refer to as
having “zero-containing” (Eq. 2) or “half-quantum-containing” (Eq. 3)
polarization lattices. (In principle, it is possible for a centrosymmetric
material to contain zero for one component of its polarization and a half
quantum along another component, although we do not know of an example).
Examples of half-quantum-containing centrosymmetric crystals are the high-
symmetry paraelectric phase of BiFeO3, and the wide-band-gap insulating
perovskite, LaAlO3. Note also that these materials have charged ionic layers
perpendicular to their usual [001] growth direction; in section IV we will
show that these properties are formally connected. SrTiO3 is an example of the
zero-containing polarization lattice type; correspondingly charge-neutral
layers, such as SrO and TiO2 (001) planes, can be readily identified.

While the fact that a centrosymmetric crystal can have a polarization lattice
that does not contain zero is somewhat unintuitive, it is reconciled by a
second important result of the modern theory of polarization: Differences in
polarization, defined as the change in polarization along a given branch of
the polarization lattice as the system is modified along an insulating
pathway, are single-valued. As a result, the spontaneous polarization, which
is the difference in polarization between the ferroelectric structure and its
high-symmetry centrosymmetric counterpart, is single-valued. Since only
polarization differences (for example when a ferroelectric is switched between
domains, or heated above its Curie temperature) are experimentally accessible,
the theory is consistent with experimental reality. Indeed, when discussing
the properties of infinite, bulk periodic crystals, the question of whether
the polarization lattice contains zero or a half quantum is not generally
relevant.

At surfaces and interfaces, however, the question of the origin of the
polarization lattice should not be disregarded. The bulk polarization,
$\vec{P}_{\text{bulk}}$, which gives rise to the bound charge at the surface
of a crystal (Eqn. 1) contains the contributions from both the half-
polarization quantum (layer charges) and the spontaneous polarization
Vanderbilt and King-Smith (1993). Considering only the spontaneous
polarization in evaluating the bound charge on the surface of a material will
yield an incorrect result for centrosymmetric crystals with half-quantum-
containing polarization lattices, as well as for ferroelectrics whose
paraelectric reference structures’ polarization lattices contain the half
quantum.

To illustrate these concepts in this work, we use ABO3 perovskite-structure
oxides in the usual pseudo-cubic [001] growth orientation. (For the
generalization to other surface planes see Ref. Stengel, 2011). In this
orientation, the layers have alternating AO / BO2 chemistry, and the planar
surfaces are formed from either entirely AO or entirely BO2 layers. Since
oxygen has formal charge -2, the (001) layers in so-called II-IV perovksite
oxides (in which the A-site cation is divalent and the B-site cation has
formal charge +4) are charge neutral. In III-III perovskite oxides (in which
both A- and B-site cations have formal charge +3), the AO layer has charge +1
and the BO2 layer has charge -1.

As example II-IV ferroelectric materials with charge-neutral layers we choose
PbTiO3 and BaTiO3; the polar discontinuity at their surfaces (that is their
interface with the vacuum), and hence their bound surface charge, derives
entirely from their spontaneous polarization. To illustrate the two possible
behaviors of centrosymmetric ionic insulators, we choose SrTiO3, with its
neutral layers and zero-containing polarization lattice, and LaAlO3, whose
layers are charged, and whose polarization lattice contains the half quantum.
The polar discontinuity between these two materials at their (001) interface
has been of particular recent interest Ohtomo and Hwang (2004); Nakagawa  _et
al._ (2006); Reyren  _et al._ (2007). The main part of this paper combines the
concepts developed for both of these pairs of example materials to treat the
case of BiFeO3, with its combined spontaneous and layer polarizations.

##  III Surface effects in ferroelectric materials with zero layer
polarization

Figure 1: Crystal structure and (001) layer charges of PbTiO3 (top) and BaTiO3
(bottom). a) Arrangement of ions in the high-symmetry centrosymmetric
reference structures. On the left of the crystal structure is the total charge
for each layer, on the right the composition of each layer. b) Arrangement of
ions in the ferroelectric structure. The arrows indicate the direction of the
ferroelectric polarization. Pb is in black, Ti in blue, Ba in green and O in
red.

We begin with a brief review of relevant results for the surface properties of
the prototypical II-IV perovskite ferroelectrics, PbTiO3 and BaTiO3. In both
cases, when the ions have the centrosymmetric arrangement of the high-symmetry
reference phase, then the polarization lattice contains zero, and is given by
Eqn. 2. Correspondingly, the (001) layers have zero formal charges (Fig. 1).
As a result, the only contribution to the $\vec{P}_{\text{bulk}}$ of Eqn. 1,
and hence to the surface charge, is from the spontaneous polarization. In
PbTiO3 the spontaneous polarization in the ferroelectric phase is oriented
along a Cartesian axis ([001] in Fig. 1) and has the value $\sim$80 $\mu$C
cm-2; in BaTiO3 there are a series of phase transitions with the polarization
(of magnitude $\sim$25 $\mu$C cm-2) reorienting from the cubic [111] to [011]
to [001] directions as the temperature is lowered; we show the low-temperature
[001] case in Fig. 1b. The bound surface charge on a (001) surface (Fig. 2) is
then given trivially by the component of the spontaneous polarization
perpendicular to the surface (80 or 25 $\mu$C cm-2 for PbTiO3 and BaTiO3
respectively). It is negative (positive) on the upper surface for downward
(upward) pointing polarization, and does not depend on the choice of layer (AO
or BO2) termination. In all cases, the uncompensated (001) surface is
electrostatically unstable when the ferroelectric polarization is along the
[001] axis.

Figure 2: In ferroelectric materials with uncharged layers, such as BaTiO3 and
PbTiO3, the sign and magnitude of the surface charge are determined by the
spontaneous polarization, $\vec{P}_{\text{spont}}$.

The electrostatic instability associated with the spontaneous polarization in
ferroelectrics is often discussed in terms of a depolarizing field, which acts
in the opposite direction to the polarization to suppress the ferroelectricity
in thin films. For a film with no compensation from external species, four
main responses are known, as sketched in Fig. 3. Polarization reorientation
into the plane of the film (Fig. 3a) completely eliminates the bound surface
charge, and so is energetically favorable if it is not prohibited by, for
example, strain effects Sarott  _et al._ (2020). If in-plane rotation of the
polarization is unfavorable, the formation of domains can occur. The example
of small domains of opposite orientation Fong  _et al._ (2004); Lubk  _et al._
(2012); Yang  _et al._ (2010) shown in Fig. 3b reduces the overall charge on
each surface; other more exotic textures such as polar skyrmions have also
been reported Stachiotti and Sepliarsky (2011); Yadav  _et al._ (2016);
Damodaran  _et al._ (2017); Das  _et al._ (2019). Screening surface charges
can in principle be generated by electron-hole excitation across the band gap
Stengel (2011) (Fig. 3c), although, since band gaps are typically of the order
of an eV in ferroelectrics this is energetically expensive. Finally, complete
suppression of the polarization (Fig. 3d) can occur, usually manifesting as a
critical thickness of the paraelectric reference structure before the
ferroelectric phase emerges Fong  _et al._ (2004); Stengel  _et al._ (2012);
De Luca  _et al._ (2017).

Screening can also occur from extrinsic factors. When metallic electrodes are
present, then carriers from the metal can provide compensating external
surface charge to screen the polarization discontinuity. Unless the screening
is completely effective, however, the magnitude of the polarization and the
ferroelectric Curie temperature, $T_{c}$, tend to be reduced from their bulk
values Junquera and Ghosez (2003); Dawber  _et al._ (2003); Kim  _et al._
(2005); Sai  _et al._ (2005); Tenne  _et al._ (2009) and there still tends to
be a critical thickness below which the paraelectric phase is stable Junquera
and Ghosez (2003); Puggioni  _et al._ (2018). In the absence of electrodes (or
in combination with a bottom electrode), compensating charge can be provided
by ions from the environment Levchenko and Rappe (2008); this is the physics
behind the well-known pyroelectric effect, in which the reduction in
polarization on heating releases charged species from the surface Kakekhani
and Ismail-Beigi (2016). Recently, the reciprocal effect has been
demonstrated, in which adsorption of adsorbates carrying a specific charge was
shown to switch the ferroelectric polarization to achieve an electrostatically
stable surface configuration Wang  _et al._ (2009); Gattinoni  _et al._
(2020); Levchenko and Rappe (2008). Finally, we mention that the presence of
charged ions in the growth chamber atmosphere has been exploited to enable
growth of single-domain ultra-thin ferroelectric films of PbTiO3 on SrRuO3
through metalorganic chemical vapor deposition of PbTiO3 on SrRuO3 Fong  _et
al._ (2006). A charged atmosphere has even been shown to be more effective in
screening the polarization than a top electrode for BaTiO3 films grown on
SrRuO3 using pulsed laser deposition Strkalj  _et al._ (2019).

Figure 3: Effects of the depolarizing field on a ferroelectric thin film. a)
In-plane polarization; b) Formation of domains; c) Accumulation of interfacial
charge, for example by electron-hole excitation across the gap; d) Phase
transition to a non-polar phase.

##  IV Surface effects arising from the layer-charge polarization in non-
ferroelectric materials

A pair of well-known centrosymmetric materials that illustrate the two cases
of “zero-containing” or “half-quantum-containing” polarization lattices are
the perovskite-structure oxides strontium titanate, SrTiO3, and lanthanum
aluminate, LaAlO3, shown in Fig. 4. (Note that rotations of the oxygen
octahedra, which lower the symmetry from the ideal cubic perovskite structure,
occur in both materials; since these rotations preserve the center of
inversion they do not change the polarization behavior and we do not consider
them here.) We discuss next how the different polarization lattice types
correspond to their different layer charges and result in different surface
charges, focussing on the (001) surface for conciseness. For a more
comprehensive discussion we direct the reader to Ref. Stengel, 2011.

Figure 4: Crystal structure and (100) layer charges of a) SrTiO3 and b)
LaAlO3. On the left of the crystal structure is the total charge for each
layer, on the right the composition of each layer.  Figure 5: a) Unit cell of
LaAlO3 with AlO2 (left) and LaO (right) termination. The dotted-line box
indicates the basis used to calculate $\vec{P}_{\mathrm{bulk}}$, and the
surface termination is the topmost layer in the box. $\vec{P}_{bulk}$ in the
[001] direction is of opposite sign for the two systems. b-c) Electric field E
and potential V along the [001] direction for LaAlO3 with b) AlO2 and c) LaO
termination. The system on the left in each panel has no compensation and so V
diverges with thickness. The system on the right has a surface compensation of
$1/2$ of the layer charge and therefore is stable.

Since SrTiO3 is a II-IV perovskite, and LaAlO3 is a III-III perovskite, it is
trivial to show, by calculating the polarization as
$\vec{P}=\frac{1}{V}\Sigma_{i}Z_{i}\vec{r}_{i}$, that SrTiO3 has the zero-
containing polarization lattice of Eqn. 2, and LaAlO3 the half-quantum-
containing form of Eqn. 3. Here, $Z_{i}$ are the formal ionic charges and
$\vec{r}_{i}$ their positions within any choice of unit cell. (Note that
rigorous calculation using the Berry phase formalism gives an identical
result, and the use of the formal charges when calculating the lattice
polarization of the paraelectric structure is formally correct Stengel
(2011).) As a result, the (001) surface of SrTiO3 has a charge of zero or
$n\vec{P}_{q}$, whereas that of LaAlO3 has a charge of
$\frac{n\vec{P}_{q}}{2}$.

A convenient recipe was provided in Ref. Stengel, 2011 to determine which
branch of the polarization lattice (that is, which value of $n$) is the
relevant one for a particular choice of surface plane and chemistry: Take the
unit cell that tiles the semi-infinite slab containing the surface of
interest, and calculate the dipole per unit volume for that unit cell. The
answer is $\vec{P}_{\text{bulk}}$ with the appropriate choice of polarization
lattice branch. For the (001) surface of SrTiO3, both smooth surfaces
(containing SrO or TiO2) yield $\vec{P}_{\text{bulk}}=0$ with this recipe.
Therefore they have no bound charge and do not require any external charge
compensation. For the (001) surface of LaAlO3 (Fig. 5a), the LaO surface has
[001] polarization value
$\vec{P}_{\text{bulk}}=+\frac{1}{2}\frac{|e|}{a_{0}^{2}}=+\frac{\vec{P}_{q}}{2}$,
requiring compensation by a negative of charge of this size and the AlO2
surface has
$\vec{P}_{\text{bulk}}=-\frac{1}{2}\frac{|e|}{a_{0}^{2}}=-\frac{\vec{P}_{q}}{2}$,
requiring compensation by the corresponding positive charge. ($a_{0}$ is the
length of the pseudo-cubic unit cell, and the polarization quantum is
$\vec{P}_{q}=\frac{|e|}{a_{0}^{2}}$.) The required compensation of half an
electronic charge per simple cubic unit cell corresponds to the convenient
value of 50 $\mu$C cm-2 in the conventional units used in the ferroelectrics
literature, taking a cubic lattice constant of 4 Å which is slightly larger
than the values for SrTiO3 ($\sim$ 3.9 Å ) and LaAlO3 ($\sim$ 3.8 Å ).

An alternative picture that is intuitively appealing, although not as
rigorously well-founded, is to decompose the materials into planes of ions and
consider the net charges of these planes, as shown in Fig. 4. For the case of
II-IV perovskites such as SrTiO3, the (001) planes are alternately SrO and
TiO2, both of which are charge neutral. Therefore any planar (001) surface in
a II-IV perovskite carries no net surface charge and so is stable. In III-III
perovskites such as LaAlO3, the (001) planes are alternately LaO and AlO2 with
charges $+1$ $e$ and $-1$ $e$ per unit cell respectively. And so, depending on
the choice of termination, the surface has the corresponding positive or
negative charge per surface unit cell, that is $\pm 100$ $\mu$C cm-2. As
illustrated in Fig. 5b, the positive layer charge of the LaO surface requires
a compensating negative charge of half the layer charge, that is $-0.5$ $e$
per unit cell or $-50$ $\mu$C cm-2, to prevent a divergence of the
electrostatic potential and stabilize the surface. (A compensating charge
equal to the surface charge just displaces the problem to a new terminating
layer Hwang (2006).) Likewise, the formally negatively charged AlO2 surface
(Fig. 5c) requires a compensating positive charge of the same amount. Thus we
reach the same conclusion as that derived from consideration of the bulk
polarization.

The implication of the different bulk polarization lattices of LaAlO3 and
SrTiO3 for the interface between the two materials is profound: The
polarization discontinuity between the two materials means that it is not
possible to make a stoichiometric interface that is electrostatically stable
Stengel and Vanderbilt (2009). Specifically, an SrO / AlO2 interface requires
a compensating positive charge of magnitude half an electronic charge per unit
cell, and the LaO / TiO2 interface requires half an electronic charge per unit
cell of negative charge. In the latter case, the extra electrons occupy the
broad Ti $3d$-derived energy bands at the bottom of the valence band. The
compensating electrons are therefore mobile and form an interfacial two-
dimensional electron gas Ohtomo and Hwang (2004), a remarkable behavior for
the interface of two robust band insulators. The electron gas has even been
shown to be superconducting at low temperature Reyren  _et al._ (2007).

Note that these considerations are not limited to III-III perovskites, but are
relevant for the surfaces and interfaces of all centrosymmetric insulators
that have a “half-quantum containing” polarization lattice. Another example is
provided by the I-V perovskites, such as KTaO3, where surface reconstructions
Setvin  _et al._ (2018) and surface and interface 2D electron gases Santander-
Syro  _et al._ (2012); King  _et al._ (2012) have been observed.

##  V Surfaces of ferroelectric materials with charged layers – the interplay
of layer charge and spontaneous polarization in bismuth ferrite

Next we turn to the case of ferroelectric materials whose polarization lattice
in their high-symmetry centrosymmetric prototype structure contains the half-
polarization quantum. We choose the example of the III-III ferroelectric
perovskite BiFeO3, which, as mentioned in the introduction, combines a half-
quantum-containing polarization lattice in its centrosymmetric reference
structure, with a spontaneous polarization of almost exactly 50 $\mu$C cm-2 in
the [001] direction. In particular, we will explore the consequences of the
accidental layer- and spontaneous polarization-surface charge compensation on
the stability of thin films and heterostructures of BiFeO3.

The ground state of bulk BiFeO3 has the $R3c$ structure, which is reached from
the prototypical cubic perovskite structure by alternating rotations of the
oxygen octahedra around the [111] axis, combined with opposite displacements
of anions and cations along the [111] direction. The latter results in a large
spontaneous polarization of magnitude $\sim$ 90 $\mu$C cm-2 oriented along
[111]. In Fig. 6 we show the evolution of the polarization (calculated using
the Berry phase approach in Ref. Neaton  _et al._ , 2005) as a function of the
amplitude of the ferroelectric distortion from the high-symmetry reference
structure (0% distortion) to the ground-state ferroelectric structure (100%
distortion), for several branches of the polarization lattice. The spontaneous
ferroelectric polarization along [111] is highlighted in red. Interestingly,
and completely coincidentally, the value of the spontaneous polarization is
very close to half of the polarization quantum of $\sim$ 180 $\mu$C cm-2 along
the [111] direction (highlighted in blue in Fig. 6) for BiFeO3. Since the
polarization lattice for the centrosymmetric reference structure is of the
half-quantum type, we see that there are two combinations of the
centrosymmetric layer polarization and the spontaneous polarization
($+\frac{\vec{P}_{q}}{2}$ with $P_{\text{spont}}=-90$ $\mu$C cm-2, and
$-\frac{\vec{P}_{q}}{2}$ with $P_{\text{spont}}=+90$ $\mu$C cm-2) that combine
to give a bulk polarization value, $\vec{P}_{\text{bulk}}$, in the
ferroelectric structure that is very close to zero (in fact $\pm 2.3$ $\mu$C
cm-2 in Fig. 6).

This in turn leads to a cancellation of the bound surface charge,
$\sigma_{\text{surf}}\approx 0$. A consequence of this cancellation,
therefore, is that free-standing thin films of BiFeO3 are electrostatically
stable for one choice of polarization for each surface; this has been referred
to as the “happy” configuration in the literature Efe  _et al._ (2021).

Figure 6: Polarization in the [111] direction for BiFeO3, calculated using the
LSDA$+U$ and Berry phase methods within density functional theory in Ref.
Neaton  _et al._ , 2005. $\pm 100$% distortion corresponds to the ground-state
$R3c$ structures of opposite polarization; 0% distortion corresponds to the
ideal cubic perovskite structure. Three full branches of the polarization
lattice are shown. The central branch illustrates that, starting from a
centrosymmetric polarization of 92.8 $\mu$C
cm${}^{-2}(=\frac{\vec{P}_{q}}{2})$ and introducing a negative spontaneous
polarization yields a bulk polarization close to zero (in fact -2.3 $\mu$C
cm-2) and a correspondingly small surface charge. Introducing a positive
spontaneous polarization in this branch results in a very large bulk
polarization (187.8 $\mu$C cm-2) and an unfavorably large surface charge. The
lower branch illustrates the opposite scenario. Reproduced from Ref. Neaton
_et al._ , 2005. Copyright 2005 by the American Physical Society.

In Fig. 7 we illustrate this with a cartoon of a free-standing BiFeO3 slab in
the commonly grown [001] orientation. The projection of the [111]-oriented
ferroelectric polarization into the [001] direction results in a spontaneous
[001] polarization of $\sim\pm 50$ $\mu$C cm-2. As we saw in the case of
LaAlO3 in Section IV, the unit cell corresponding to the BO2 (FeO2 in this
case) surface selects for the branch on the centrosymmetric polarization
lattice with value $-\frac{1}{2}\frac{e}{a_{0}^{2}}=-50$ $\mu$C cm-2.
Therefore an FeO2 surface with a positive (i.e. pointing towards it, or
upwards in Fig. 7a) value of spontaneous polarization has zero surface charge
and is stable; conversely the AO (BiO in this case) surface selects for the
$+\frac{1}{2}\frac{e}{a_{0}^{2}}=+50$ $\mu$C cm-2 half quantum, and requires a
negative (i.e. pointing away from it) polarization to ensure stability. Note
that the opposite combinations are twice as unfavorable (“unhappy”) as they
would be in a II-IV perovskite with the same magnitude of spontaneous
polarization but uncharged layers (Fig. 7b), since they would have a surface
charge of $\pm 100$ $\mu$C cm-2. In the alternative charged-layers picture,
the centrosymmetric BiFeO3 is composed of alternating (001) layers of
positively charged BiO (+1 $e$ per unit cell or $\sim+100\mu$C cm-2) and
negatively charged FeO2 (-1 $e$ per unit cell or $\sim-100\mu$C cm-2). The
appropriately oriented spontaneous polarization of magnitude $50$ $\mu$C cm-2
then provides the required compensating surface charge of half that amount.

Figure 7: Combinations of ferroelectric polarization direction and surface
termination leading to a) stable (happy) and b) unstable (unhappy) BiFeO3
(001) surfaces. In a), the spontaneous polarization, $P_{\text{spont}}=50$
$\mu$C cm-2 is compensated by the layer polarization. In b) the layer
polarization adds to the spontaneous polarization $P_{\text{spont}}=-50$
$\mu$C cm-2 to give a surface charge of $\pm 100$ $\mu$C cm-2.

Next, we discuss the consequences of this layer and spontaneous polarization
cancellation, examining examples from the literature as well as presenting new
results of behaviors that are caused by the happiness or unhappiness of BiFeO3
surfaces and interfaces. We consider three scenarios: first, BiFeO3 on a
metallic substrate, followed by interfaces with centrosymmetric II-IV then
III-III insulators.

###  V.1 Interaction of BiFeO3 thin films with a metal substrate

We begin with the case of BiFeO3 films grown on substrates that are metallic,
and therefore provide good screening of any interfacial charge at the bottom
interface. We expect, therefore, that the orientation of the spontaneous
polarization will be determined by the nature of the top surface with the
vacuum.

We take the examples of BiFeO3 on two metallic oxides, La0.7Sr0.3MnO3 (LSMO)
and SrRuO3. Heterostructures of these combinations were grown and
characterized in Ref. Yu  _et al._ , 2012, and we begin by analyzing the
results of that work in the context of the surface electrostatics introduced
above.

In Ref. Yu  _et al._ , 2012, the SrRuO3 substrate was terminated with an SrO
layer, and so the bismuth ferrite film, which grows in complete BiFeO3 unit
cells, began with an FeO2 layer and ended with a BiO surface. As expected, the
polarization spontaneously adopted the down orientation, corresponding to zero
surface charge. An FeO2 surface was achieved for BiFeO3 on SrRuO3 by inserting
a monolayer of TiO2 at the interface so that the BiFeO3 film began with a BiO
layer. This caused a spontaneous upwards polarization in the BiFeO3, again
corresponding to the zero surface-charge configuration as expected.

Growth of BiFeO3 on LSMO shows a similar behavior. In Fig. 8 we show two high-
angle annular dark-field scanning transmission electron microscopy (HAADF-
STEM) images of the BiFeO3 on LSMO heterostructures grown in Ref. Yu  _et al._
, 2012. (For additional details about the thin film growth see section VII and
Ref. Yu  _et al._ , 2012.) In panel a the LSMO is terminated with MnO2, so the
BiFeO3 layer has an FeO2 surface, while in panel b the LSMO is (La,Sr)O
terminated, so the bismuth ferrite starts with a FeO2 layer and has a BiO
surface. We have overlaid arrows, which are vector maps indicating the local
polarization extracted from the measured atomic positions, in the BiFeO3
layers. Again, as expected from the surface electrostatics, we see that case
(a) develops a spontaneous up-pointing polarization ($\vec{P}$ pointing
towards the FeO2 surface) and case (b) a spontaneous down-pointing ($\vec{P}$
pointing away from the BiO surface). In all four scenarios switching of the
polarization was achieved using a tip in a piezoforce geometry, but with
considerable exchange bias favoring the spontaneous orientation.

In our discussion so far, we have assumed that the LSMO and SrRuO3 substrates
behave like ideal metals, and have disregarded the fact that their constituent
ions have different formal layer charges ($\pm\sim 0.7$ _e_ for LSMO at 0.3 Sr
concentration, and neutral for SrRuO3). The different formal layer charge
discontinuities could play a role if the metallic screening is incomplete.
Indeed, it is known that ionic relaxation is an important contributor to
surface-charge screening in oxide electrodes Stengel and Spaldin (2006), and
the use of polar metals as electrodes has been proposed as a route to
overcoming the critical thickness in ferroelectric capacitors Puggioni  _et
al._ (2018). To investigate the role of the formal layer charges in metallic
oxide electrodes, we next perform density functional calculations of
[001]-oriented BiFeO3/SrRuO3 superlattices in both the happy (Fig. 7a) and sad
(Fig. 7b) interfacial orientations (for details see the methods in Section
VII).

Figure 8: Cross-sectional HAADF-STEM images of two distinct
BiFeO3/La0.7Sr0.3MnO3 heterointerfaces with overlaid vector maps showing the
polarization in the BiFeO3 layers. a) The sample with the MnO2-terminated
(La0.7Sr0.3O-MnO2-BiO-FeO2) interface develops a spontaneous up-pointing
polarization ($\vec{P}$ pointing towards the FeO2 surface). b) The sample with
the La0.7Sr0.3O-terminated (MnO2-La0.7Sr0.3O-FeO2-BiO) interface develops a
spontaneous down-pointing polarization ($\vec{P}$ pointing away the BiO
surface). The scale bar is 1 nm.

We constructed two superlattices each containing six layers of BiFeO3 and four
layers of SrRuO3, with one SrO/FeO2 and one BiO/RuO2 interface, see Fig. 9.
The two supercells had opposite orientation of the BiFeO3 ferroelectric
polarization, such that one system had self-compensating (Fig. 9a) and the
other charged interfaces (Fig. 9b). In both cases, the entire heterostructure
adopts the a-a-c- tilt pattern of BiFeO3 (Fig. 9, left panels), and both
materials maintain their bulk magnetic orderings (G-type antiferromagnetic for
BiFeO3 and ferromagnetic for SrRuO3). As expected, both from our electrostatic
arguments and from the experimentally observed exchange bias Yu  _et al._
(2012), the happy system is energetically the most stable, $\sim$ 2 eV per
supercell lower in energy than the unhappy system.

The calculated structures, layer-by-layer polarizations and layer-resolved
densities of states are shown in Fig. 9. Note that, in both the “happy” and
“unhappy” systems, the SrRuO3 layers are metallic (in green in the density of
states graph on the right-hand-side on Fig. 9), with a finite density of
states at the Fermi energy in all layers, and the BiFeO3 (in purple) is
insulating, with the Fermi energy (shown as a vertical red line) lying in the
gap. The layer-resolved densities of states for the happy system (right-most
panel of Fig. 9a) indicate that there is no band bending and hence no internal
electric field in the happy system, consistent with the absence of surface
charge in the happy BiFeO3 slabs. The polarization (middle panel) has its full
bulk value throughout the slab, and drops abruptly to zero in the first layer
of the SrRuO3.

For the six-unit-cell BiFeO3 heterostructures that we present here, we find
that the unhappy system is metastable in our DFT calculations. (For thinner
films the polarization orientation reverses and the structure relaxes to the
happy system.) We find, however, a suppressed layer polarization compared with
the happy system, as visible in the middle graph of Fig. 9b, as well as a
pronounced shift in the BiFeO3 band edges from layer to layer indicating a
strong internal electric field resulting from the large uncompensated surface
charges in the BiFeO3 slab. In addition, the interfacial SrRuO3 layers undergo
a polar ionic distortion to further reduce the polar discontinuity, similar to
that observed in Ref. 35 for the SrRuO3/BaTiO3 interface.

In summary, our calculations indicate that, even with metallic screening, the
direction of polarization preferred by the interplay between the lattice and
spontaneous polarization of the ferroelectric layer is strongly preferred.
While screening by the metal is able to stabilize the unhappy polarization
orientation it is still energetically unfavorable. This behavior explains the
strong electric-field exchange bias effects Maksymovych  _et al._ (2009), as
well as the highly asymmetric resistive switching Shuai  _et al._ (2011) found
in BiFeO3 capacitors.

Figure 9: Calculated structures (left), layer by layer polarizations (middle)
and layer density of states (right) for the two BiFeO3/SrRuO3 heterostructures
studied in this work. Panel a) shows the happy BiFeO3 slab configuration;
panel b) the unhappy. Green symbols and shading indicate SrRuO3; BiFeO3 is
shown in purple.

As a final example from the literature, we choose the case of heterostructures
between BiFeO3 and the metallic lightly-doped II-IV magnetic insulator,
CaMnO3. Heterostructures of Ca1–xCexMnO3/BiFeO3 were grown on YAlO3, whose
small lattice constant causes a strongly compressive biaxial in-plane strain
and correspondingly large out-of-plane lattice constant and polarization (see
detailed discussion in Subsection C) Marinova  _et al._ (2015). The
ferroelectric polarization was determined to have an out-of-plane value of
$\sim 100$ $\mu$C/cm-2, and to point towards the BiFeO3/CaMnO3 interface,
which was of the FeO${}_{2}^{-}$ – CaO type. While this is the least unhappy
arrangement, in this case, because of the unusually large out-of-plane
polarization, the FeO${}_{2}^{-}$ layer provides only partial compensation of
the bound surface charge. An additional electronic charge accumulation of
$\sim 0.65$ electrons per unit cell area was found using electron energy loss
spectroscopy (EELS) to accumulate in the near-interfacial Ca1–xCexMnO3 layers.

###  V.2 Interfaces of bismuth ferrite with centrosymmetric II/IV insulating
perovksites

Figure 10: Polarization along the [001] direction of a (SrTiO3)4/(BiFeO3)6
heterostructure with one BiO/TiO2 interface and one FeO2/SrO interface. The
polarization direction leading to a happy system is in panel a, and to an
unhappy system in panel b. The systems are shown on the left, in the middle
their corresponding unit cell-by-unit cell polarization and on the right the
unit cell-by-unit cell local density of states. Orange data points correspond
to SrTiO3 and purple ones to BiFeO3. In the atomic structure Sr is green, Ti
blue, Bi purple, Fe brown and O red.

Since the (100) surface of a centrosymmetric II/IV insulating perovskite such
as SrTiO3 has zero bound charge, we expect it to behave electrostatically
similarly to the vacuum in its interface with BiFeO3. That is, we expect that
BiFeO3 surfaces that are happy in free-standing slabs to form stable
interfaces with SrTiO3, with bulk-like ferroelectricity in the BiFeO3 layer
down to small thicknesses, whereas electrostatically unstable surfaces will
have similarly unhappy interfaces with SrTiO3 and their ferroelectric
polarization will tend to reverse Efe  _et al._ (2021). All the examples that
we have been able to find of experimental BiFeO3/SrTiO3 superlattices and
heterostructures have their as-grown polarization orientation in the direction
which compensates the layer polarization, consistent with this assumption
Bruyer  _et al._ (2015); Ranjith  _et al._ (2010); Yang  _et al._ (2013); Gao
_et al._ (2019). Likewise, literature calculations of PbTiO3/BiFeO3
superlattices, with the polarization of the PbTiO3 entirely in the plane of
the superlattice Yang  _et al._ (2012a), found a stable solution with BiFeO3
in its happy configuration.

To explore the details of the behavior, we perform DFT calculations for two
[001] superlattices, each containing four layers of SrTiO3 and six layers of
BiFeO3 with one SrO / FeO2 and one BiO / TiO2 interface, but with the BiFeO3
polarization initialized to opposite orientations. As expected, we find that
the happy case is stable, with the layer polarization close to the bulk value
throughout the film (Fig. 10a, middle panel). As in the case of the interface
with SrRuO3, the unit cell-by-unit cell density of states indicates zero
internal electric field in ferroelectric BiFeO3 since the charge compensation
between the spontaneous and layer polarization leads to zero surface charge.
In contrast, we are unable to stabilize the unhappy state in our DFT
calculations unless we constrain the polarization orientation in the middle
layers of the BiFeO3 slab (Fig. 10b). The layer-by-layer density of states
indicates a strong band bending due to the large internal electric field, and
the formation of metallic layers by electron-hole excitation across the band
gap at the interfaces. (Similar metallicity at the unhappy BiO / TiO2
interface was seen in an earlier DFT calculation for a non-stoichiometric
BiFeO3 slab with both of its interfaces set to BiO sandwiched between two
SrTiO3 layers Zhang  _et al._ (2011).) Additionally, the SrTiO3, which is an
incipient ferroelectric and therefore readily polarizable, develops a
polarization parallel to the spontaneous polarization of BiFeO3 to reduce the
polar discontinuity.

###  V.3 Interfaces of bismuth ferrite with insulating centrosymmetric III-III
perovskites

Finally, we consider the case of the interface between BiFeO3 and a
centrosymmetric insulating III-III perovskite, which contains
$\frac{\vec{P}_{q}}{2}$ in its polarization lattice. Here the centrosymmetric
contributions to the polarization lattice are similar in both materials (they
will differ slightly if the lattice vectors and unit cell volumes are
different) and so the interfacial polar discontinuity is given by the
difference in spontaneous polarizations. If the second material is
centrosymmetric, then the polar discontinuity is equal to the spontaneous
polarization of BiFeO3, similar to the case of the interfaces between the II-
IV ferroelectrics PbTiO3 or BaTiO3 and vacuum discussed above.

A number of different routes to avoiding the polarization discontinuity at the
interface, of the types we summarized in Fig. 3, have been observed. An in-
plane polarization (Fig. 3a) associated with an orthorhombic phase has been
reported for BiFeO3 on a NdScO3 substrate, which also imparts a small biaxial
tensile strain Yang  _et al._ (2012b). Since many low-energy metastable non-
polar, anti-polar and even antiferroelectric phases of BiFeO3 are known
Stengel and Íñiguez (2015); Dieguez  _et al._ (2011), the BiFeO3 film can also
lose its polarization entirely (Fig. 3d). For example, in superlattices and
heterostructures of BiFeO3 with centrosymmetric $Pnma$ LaFeO3, BiFeO3 has been
reported to adopt the antiferroelectric PbZrO3 structure Carcan  _et al._
(2017, 2018), or even observed in an entirely new antiferroelectric structure,
which has not been reported in the bulk Mundy  _et al._ (2020). Density
functional calculations indicated that, for the strain conditions of the
sample, this antiferroelectric phase is only slightly higher in energy than
the ground-state polar phase, and it is favored because of its lower
electrostatic energy cost Mundy  _et al._ (2020). (Note that a DFT calculation
for a BiFeO3/LaFeO3 slab in vacuum suggested the formation of a metallic layer
at the interface, although that study did not explore the formation of non-
polar BiFeO3 phases and it is unclear how the polar discontinuties at the
surface were treated in the calculation Sun  _et al._ (2019).)

Another route to the compensation of the polar discontinuity is the creation
of extended defects; we discuss the example of LaAlO3/BiFeO3, where this
behavior has been observed, next. Under strong biaxial compressive strain,
imposed by a small-lattice-constant substrate such as LaAlO3, BiFeO3 is known
to undergo a phase transition to a tetragonal or tetragonal-like phase
(T-BiFeO3) with a large $c/a$ ratio of $\sim 1.3$ and a giant, almost entirely
out-of-plane spontaneous polarization of $\sim 150$ $\mu$C cm-2 Zeches  _et
al._ (2009); Hatt  _et al._ (2010); Rossell  _et al._ (2012). This spontaneous
polarization is roughly three times the [001] spontaneous polarization of the
usual rhombohedral phase of BiFeO3, and correspondingly roughly three times
the half-polarization quantum. (Note that the half-polarization quantum for
T-BiFeO3 is slightly larger, at 59 $\mu$C/cm2, than that of the usual
rhombohedral phase, because of its different lattice parameters. We obtain
values of $a=b=3.67$ Å and $c=4.64$ Å in our calculations for the lowest
energy tetragonal structure.) This giant spontaneous polarization has two
implications: First, the spontaneous polarization can be at best only
partially compensated by the layer polarization at a flat and stoichiometric
BiO or FeO2 (001) surface. Second, the giant polarization means that the
electrostatic potential diverges strongly at an interface, and only a few
layers can form before a compensation mechanism is required. In Fig. 11 we
show a HAADF-STEM image of a 100nm-thick film of T-BiFeO3 on LaAlO3, in which,
we observe such a compensation mechanism in the formation of an extended
planar defect just a few unit cells above the T-BiFeO3/LaAlO3 interface. We
indicate the local ferroelectric polarization (plotted opposite to the atomic
displacements of the Fe cations) by the yellow arrows in Fig. 11; this vector
map reveals that the first five to seven T-BiFeO3 unit cells above the
T-BiFeO3/LaAlO3 interface develop a spontaneous up-pointing polarization.
Then, perhaps unexpectedly, above the extended planar defect the polarization
in the T-BiFeO3 lattice is reversed and a down-pointing polarization forms.
This results in a head-to-head polarization configuration, with a giant
discontinuity of the spontaneous polarization of $\sim 300$ $\mu$C cm-2. For
both the top and bottom layers, however, the stoichiometric BiFeO3 terminates
with an FeO2 layer, so the absolute polarization of each layer is reduced from
the spontaneous polarization by half a quantum, to $\sim 100$ $\mu$C cm-2.
Note that this is the “happiest” configuration possible for T-BiFeO3, which,
we emphasize again, does not have the accidental cancellation between its
spontaneous polarization and the half-polarization quantum seen in the
rhombohedral ground state. Correspondingly, the absolute polar discontinuity
between the two T-BiFeO3 layers is reduced to the (still very large!) value of
$\sim 200$ $\mu$C cm-2.

The planar defect (highlighted with red rectangles in Fig. 11) consists of a
characteristic Bi2O${}_{2}^{2+}$ layer, which shifts the perovskite layers
above the defect half a perovskite block along the orthogonal horizontal
directions in the manner of an Aurivillius phase, surrounded by two O2- layers
(see cartoon on the right side of Fig. 11; the horizontal red lines indicate
the boundary of the defect). The total stoichiometry of the defect consists of
one formally Bi2O${}_{2}^{2+}$ block plus two O2- ions per surface BiFeO3 unit
cell, leading to a net defect charge of two electrons, or $\sim-100\mu$C cm-2,
per primitive unit cell cross-sectional area. This is exactly half of the
polar discontinuity, and so, as sketched in the lower panel of Fig. 5, is
exactly the layer charge needed for compensation. Note that oxygen non-
stoichiometry in the Aurivillius-like layer, yielding Bi2O2±δ rather than
precisely Bi2O2, is likely, and will change the exact amount of compensating
charge that it provides. Indeed, a similar planar defect, with a double
Bi2O${}_{2}^{2+}$ layer, was observed in a BiFeO3 film grown on an LaAlO3
substrate in spite of an intermediate metallic electrode between the film and
the substrate Li  _et al._ (2017). In this case the BiFeO3 layer adjacent to
the electrode was non-polar, and that above the defect developed a downward-
pointing polarization as in our example.

Interestingly, a similar extended defect has been reported as a surface “skin”
in BiFeO3, in all cases when the polarization points in the upwards (towards
the surface) direction Kim  _et al._ (2013); Jin  _et al._ (2017); Xie  _et
al._ (2017). In Ref. Jin  _et al._ , 2017 a film of rhombohedral BiFeO3 was
grown in [001] orientation on an insulating DyScO3 substrate, and two opposite
domains, separated by a 180∘ domain wall were imaged using HAADF-STEM. The
down-polarization domain had a pristine BiO surface and so was in the ‘happy’
configuration. The up-polarization domain, which would have been in the
‘unhappy’ configuration with an excess positive charge in its pristine form,
had a capping layer of the negatively-charged Bi2O2 Aurivillius-type extended
defect to compensate. This finding has clearly unfavourable implications for
the switching of BiFeO3 domains. In Ref. Xie  _et al._ , 2017, the BiFeO3 film
on TbScO3 had domains of strongly suppressed polarization alternating with
domains of enhanced upward-pointing polarization; the latter had the surface
skin overlayer. Finally, we mention that similar Aurivillius structures have
also previously been observed as intergrowths in rhombohedral BiFeO3 thin
films Deniz  _et al._ (2014). It would be interesting to analyze the
interfaces between the intergrowths and the surrounding BiFeO3 regions to
determine the nature of the interface chemistry and the polarization
orientation in the context of the compensation principles discussed here.

Figure 11: Cross-sectional HAADF-STEM image showing the presence of planar
defects just above the BiFeO3/LaAlO3 interface. The stepped Bi2O2 structural
units are highlighted by the red rectangles and the overlaid vector map of the
ferroelectric polarization plotted opposite to the displacement of the Fe
cations (yellow arrows) reveals that the polarization in the pseudo-tetragonal
BiFeO3 lattice changes from up to down direction across the defect. An atomic
model of the defect is shown on the right. The scale bar is 1 nm.

##  VI Summary and Outlook

In summary, we have reviewed how the spontaneous polarization associated with
ferroelectricity combines with the layer polarization associated with the
ionic charges of the lattice to determine the electrostatic stability of the
surfaces and interfaces of insulators. We reminded the reader that the two
contributions are conveniently treated on the same footing by the modern
theory of polarization, allowing straightforward determination of the amount
of bound charge at a general surface or interface. The bound charge is
important for the design of thin-film heterostructures and for the stability
of surfaces, since any non-zero surface or interfacial charge must be
compensated to avoid divergence of the electrostatic potential.

After briefly discussing examples of materials with a spontaneous polarization
but no layer polarization ([001]-oriented PbTiO3 and BaTiO3) and a material
with no spontaneous polarization but a non-zero layer polarization
([001]-oriented LaAlO3) we focused on the case of multiferroic BiFeO3, which
combines both spontaneous and layer contributions. BiFeO3 is of particular
interest, because the spontaneous and layer contributions to the polarization
in the ground-state $R3c$ structure have the same size along the usual [001]
growth direction, leading to combinations of polarization and surface
termination that are uncharged and therefore electrostatically stable. The
opposite combinations have double the surface charge of a ferroelectric with
the same spontaneous polarization but with uncharged layers. These have been
referred to as the happy and unhappy combinations in earlier work Efe  _et
al._ (2021).

We considered three scenarios: BiFeO3 on a metallic substrate, BiFeO3 in a
superlattice or heterostructure with a zero-layer-charge insulator, and
finally a BiFeO3/insulator superlattice or heterostructure in which the
insulator has the same layer charges as the BiFeO3. In each case we
illustrated the different possible behaviors with examples from the
literature, as well as with density functional calculations and HAADF-STEM
analyses performed for this work.

We can summarize the differences in behavior in the three cases as follows: i)
Both the happy and the unhappy polarization orientations can be stabilized by
a metallic electrode, although the unhappy case is higher in energy and has a
lower spontaneous polarization. This is consistent with the known large
electric-field exchange bias of BiFeO3 films, and implies that symmetric
switching of a BiFeO3 capacitor will be difficult to achieve. ii) The
interface of BiFeO3 with a zero-charge-layer insulator behaves like a free
BiFeO3 surface, with the happy combination of polarization and surface
termination strongly favored. The unhappy combination can be stabilized by
strong band bending to generate metallic layers at the interface and/or
polarization of the adjacent insulator. iii) The interface of BiFeO3 with
another III-III perovskite has a polar discontinuity equal to the spontaneous
polarization, and so is equally energetically unfavorable for both
orientations of the polarization. As a result of the rich Bi-Fe-O low energy
phase space, many responses are possible, including stabilization of phases
with zero out-of-plane polarization, and the formation of extended planar
defects.

While the electrostatic concepts discussed in this paper are not new, we hope
that their collection in this article will be helpful in guiding the design of
BiFeO3 and related thin films or heterostructures with targeted electrical
properties, as well as in interpreting experimental observations.

##  VII Methods

Density functional Theory. Density functional calculations were performed
within the periodic supercell approach using the VASP code Kresse and Hafner
(1993, 1994); Kresse and Furthmueller (1996a, b). We chose the PBEsol
functional Perdew  _et al._ (2008) for all calculations because i) it gives a
good band alignment between metallic SrRuO3 and BiFeO3 and no pathological
situation arises Stengel  _et al._ (2011), and ii) it yields a paraelectric
ground state for SrTiO3. In order to obtain a band gap for BiFeO3 close to the
experimental value the Hubbard $U$, in the Dudarev Dudarev  _et al._ (1998)
approach, was used with $U-J=4$ eV on the Fe $3d$ states, and $U-J=2.0$ eV on
the Ti and Ru $d$ states. Core electrons were replaced by projector augmented
wave (PAW) potentials Kresse and Joubert (1999), while the valence states (5e-
for Bi, 8e- for Fe, 6e- for O, 10e- for Sr, 4e- for Ti, 8e- for Ru) were
expanded in plane waves with a cut-off energy of 500 eV. In all calculations
the in-plane lattice parameters were set to that of SrTiO3,
$a_{\mathrm{PBE}_{\mathrm{sol}}}=3.90$ Å, as it is the substrate commonly used
in epitaxial growth of BiFeO3 thin films. The in-plane surface area is
$\sqrt{2}a\times\sqrt{2}a$. A Monkhorst-Pack $k$-point grid of ($5\times
5\times 1$) was used for all ionic relaxations, which had an optimization
threshold on the forces of 0.01 eV/Å. For the density of states calculations,
Monkhorst-Pack $k$-point grid of 11 $\times$ 11 $\times$ 1 was used. An
antiferromagnetic G-type ordering was imposed in BiFeO3, which gave a magnetic
moment of 4.15 $\mu_{\mathrm{B}}$ per Fe ion in the bulk. SrRuO3 instead is
ferromagnetic and the magnetic moment is 1.4 $\mu_{\mathrm{B}}$ per Ru ion in
the bulk.

The unit cell-by-unit cell polarization along the [001] direction shown in
Fig. 9 and 10 was calculated by computing the displacement of each ion from
the high symmetry position and multiplying it by the Born effective charges
from Ref. Neaton  _et al._ (2005).

Thin Film Growth. The BiFeO3/La0.7Sr0.3MnO3 (BFO/LSMO) heterostructures shown
in Fig. 8 were grown by pulsed laser deposition on SrTiO3 (001) (STO) single
crystal substrates. Before the growth, a buffered HF acid-etch and thermal
treatment process was used to obtain fully TiO2-terminated surfaces. The
sample with the MnO2-terminated (La0.7Sr0.3O-MnO2-BiO-FeO2) interface was
designed by growing whole LSMO unit cells directly on the STO substrate,
followed by the growth of the BFO layer. For the sample with the
La0.7Sr0.3O-terminated (MnO2-La0.7Sr0.3O-FeO2-BiO) interface, 1.5 unit cells
of SrRuO3 (SRO) were deposited on STO to switch the termination of the STO
from TiO2 to SrO. The SRO layer was grown at 650 ∘C in 100 mTorr of oxygen
pressure. Both the LSMO and BFO layers were grown at 690 ∘C in 150 mTorr of
oxygen pressure. A postannealing process was carried out at 400 ∘C under an
oxygen ambient for 1 h to ensure the samples were fully oxidized. For
additional information, see Ref. Yu  _et al._ (2012). The BFO thin film shown
in Fig. 11 was grown by molecular beam epitaxy (MBE) on single-crystal
substrates of (001) LaAlO3 (LAO). The studied 100-nm-thick film showed the
coexistence of two interspersed BFO phases: a rhombohedral-like (R) phase and
a tetragonal-like (T) phase. Further growth and characterization details can
be found in Ref. Zeches  _et al._ (2009).

Transmission Electron Microscopy. Cross-sectional specimens for transmission
electron microscopy analysis were prepared by mechanical polishing using a
tripod polisher followed by argon ion milling until electron transparency.
High-angle annular dark-field scanning transmission electron microscopy
(HAADF-STEM) was carried out using the TEAM 0.5 microscope located at the
National Center for Electron Microscopy (NCEM). The TEAM 0.5 is a FEI Titan
80-300 microscope equipped with a high-brightness Schottky-field emission
X-FEG electron source, a source monochromator, a CEOS DCOR spherical-
aberration probe corrector, and a CEOS CETCOR spherical-aberration image
corrector. The microscope was operated at 300 kV, the probe semi-convergence
angle set to 16.5 mrad (which yields a calculated probe size of 0.63 Å), and
the annular semi-detection range of the HAADF detector calibrated at 45–290
mrad. This setting was chosen to allow for a sufficiently large depth of field
in order to enhance the contrast of the atomic columns. The positions of the
atomic columns were first fitted by means of a center of mass peak-finding
algorithm, and subsequently refined by solving a least-squares minimization
problem (using the Levenberg–Marquardt algorithm). This iterative refinement
was carried out using a custom-developed script that makes use of 7-parameter
two-dimensional Gaussians and allows estimation of the atomic column peak
positions with picometer precision Yankovich  _et al._ (2014); Campanini  _et
al._ (2018). Then, polarization maps were calculated from the relative
displacements of the two cation sublattices present in the ferroelectric
perovskite-type structures with general formula ABO3. Thus, the local
ferroelectric polarization was calculated by measuring the polar displacement
in the image plane of the B position from the center of mass of its four
nearest A neighbors. Here, in the polarization maps derived from HAADF-STEM
images, the polarization vectors are plotted opposite to the displacement of
the B cations.

###### Acknowledgements.

N. A. S. acknowledges funding from the European Research Council (ERC) under
the European Union’s Horizon 2020 research and innovation programme grant
agreement No 810451. I. E. and C. G. acknowledge the use of the Euler cluster
managed by the HPC team at ETH Zurich. M. D. R. acknowledges support by the
Swiss National Science Foundation under Project No 200021-175926, and is
thankful to R. J. Zeches, P. Yu and R. Ramesh for the samples used in this
study.

## Data Availability

The data that support the findings of this study are available within the
article.

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