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aligned to the zone axis, and local conditions, such as strain, composition, or a small change in crystal
orientation across a boundary, can strongly affect the transmission of the electron probe. As a result,
domain boundaries appear slightly darker. Their orientation varies depending on the direction of view. In
the (110) section [Figure 1B], straight bands are parallel to the (001) surface, while in the (010) section
[Figure 1C], they are inclined at an angle of approximately 25°, running along [ 01]. Sawtooth walls present
different V-shapes when observed from different directions, and their structure is considered in detail
[19]
[16]
elsewhere . The domain appearance and PFM measurements agree with the observations of Berger et al. .
However, they interpreted the flat bands of contrast seen in Figure 1B and C as domains with a width of
approximately 10 nm. The varying thickness of the lamellae in Figure 1 shows this to be incorrect. In
Figure 1C, the lamella decreases to a thickness of just a few nm at the edge of the specimen on the top-right
of the image. The width of the bands decreases as the specimen thickness decreases, indicating that they
correspond to planes in the material that are inclined to the direction of view. The same effect can also be
seen in Figure 1B, where the specimen thickness increases towards the bottom of the image. Therefore, the
darker bands are not slabs of material that form domains, different to the adjacent lighter material. They are
the projection of a plane that is inclined to the point of view.

The proposal that the flat bands of contrast in Figure 1B and C are planes rather than slabs of finite
thickness is confirmed by the ( 10) section shown in Figure 2A. In this projection, their width does not
depend on specimen thickness. Here, they are seen edge-on and appear as sharp, straight lines at an angle of
35° to (001) surface, i.e., lying on (112) planes. The sawtooth domain pattern can be recognised in the
thinnest part of the specimen but becomes indistinct where the lamella is thicker due to overlapping
projections. Higher magnification images [Supplementary Figure 2B] show that the average habit plane is
slightly away from (112); the flat domain walls have a terraced structure, with (112) facets seen edge-on,
separated by steps 1-2 nm in height.

High-resolution ( 10) STEM images of one of these domain walls, seen edge-on in the thinnest part of the
specimen, are shown in Figure 2C and D. In any {110} section of BiFeO , the [111] polar axis can either lie
3
within the plane of the section or at 35° to it, (i.e. with an out-of-plane component). The a a a antiphase
- - -
tilting of oxygen octahedra in the R3c structure gives different projections for the two cases. When the polar
axis is in the image plane, the antiphase tilting of octahedra is not visible, and oxygen columns appear as a
[21]
dumbbell at the same location in every unit cell . Conversely, when it has an out-of-plane component,
O-Fe-O-Fe-O c h a i n s f o r m a n a r c w i t h a c u r v a t u r e i n d i c a t i n g t h e d i r e c t i o n o f P s [17]

[Supplementary Figure 3]. In Figure 2C, the polar axis is in the image plane for both domains, consistent
with a [ 10] BiFeO projection [Figure 2B]. On a unit-cell level, polarisation P is often taken to be
3
s
proportional to the negative displacement of Fe relative to the centre of the rectangle defined by its four Bi
neighbours, namely -δ FB [22,23] . Since the polar axis lies in the image plane, we observe a full projection of the
displacement (in comparison, one can only observe approximately 57% or 81% of the displacement in [110]
[23]
or [100] projections, respectively ). The quiver map of -δ vectors overlaid on the ADF-STEM image of
FB
Figure 2D shows that polarisation changes by 180°, pointing towards the wall on both sides. (N.B. at the
domain wall itself, δ is not calculated because it is unclear how to define the unit cell.) Away from the
FB
domain wall, the magnitude of δ is generally around 40 pm, in good agreement with the theoretical value
FB
of 41 pm for P of approximately 100 μC cm -2[7,23,24] . As a final observation in Figure 2D, Bi atom columns in
s
the domain wall are not as bright as equivalent atom columns in the surrounding material, indicating a
lower occupancy of Bi at the domain wall.