Supporting Material
Experimental condition of Lorentz microscopy and electron holography
A conventional transmission electron microscope (TEM) (JEOL, Ltd., 3000F) with 300 kV acceleration
voltage was used to identify the structural arrangement of nanoparticles. The arrays were imaged at room
temperature (24 °C) after heating experiment in order to verify that there were no changes to the particle
ordering. Fresnel Lorentz microscopy (FLM) and electron holography (EH) were performed using a 200 kV
field emission TEM (Hitachi, HF2000) equipped with an electron biprism. A magnetic-shield objective lens
(Lorentz lens) was installed in the holography TEM and turned off during the experiment, so the sample was
in a field-free environment. An intermediate lens current was adjusted to focus the image for EH and to
defocus the image for FLM. Before loading the sample into the TEM holder, the film was scratched by a
sharp tungsten needle for the EH experiments, to obtain a wide area for the reference wave passing through
vacuum adjacent to the arrays. The sample was loaded in a double-tilt heating holder (Hitachi) to see the
domain structures at high temperature.
Principle of electron holography
Electron holographyR1 is a TEM technique to quantitatively observe the electric and the magnetic fields in
micro/nano meter scale. R2, R3 Figure S1(a) shows a schematic illustration of the experimental set-up in a
TEM. Coherent electron wave produced from a field emission gun penetrates a sample, and then the wave is
1
split into two parts by an electron biprism, to create an "object wave" modulated by the sample and a
"reference wave" passed through the vacuum. The electron biprism consists of two grounded electrode plates
and a fine filament electrode with a positive voltage. The object wave and the reference wave are overlapped
beneath the filament and interfere with each other. The information of the magnetic flux in the sample is
recorded in the modulated interference fringe pattern that is called "hologram". To reconstruct the object
wave (Figure S1(b)), the hologram is processed by 2-dimensional Fourier transformation. One of the
side-bands, corresponding to the Fourier transformation of the object wave, is selected by a spatial frequency
filter and shifted to the center, and then the side-band is processed by inverse Fourier transformation, finally
the object wave is reconstructed as complex number matrix. The phase image is obtained from the
reconstructed object wave. The contour map of the phase image corresponds to the magnetic flux distribution,
as shown in Figure S1(b). The magnetic poles are determined from the phase slope with Lorentz law.
Measurement of the magnetic order parameter and calculation of the average deviation angle
The object wave is modulated by not only the in-plane magnetic flux but also the electrostatic inner
potential of the sample. To remove the phase shift due to the inner potential, the reconstructed phase
measured at 632 °C (higher than Tc of bulk Fe3O4, 585 °C) was subtracted from each reconstructed phase
image.
The
phase
shift
 M
due
to
Lorentz
force
is
expressed
as
M  2 e h Ads  2 e h BdS  2 e h M ,enclosed , where A , e , h , and M ,enclosed are the vector
c
S
2

potential, the electron charge, Planck's constant and the enclosed magnetic flux between the sample and
reference beams, respectively. R4 For a spherical particle as shown in Figure S2, the enclosed magnetic flux
M ,enclosed is given by  M ,enclosed  B rpt2 , where rpt is the radius. Considering the radius (5.95 - 7.45 nm)
of our Fe3O4 nanoparticles, the edge-to-edge separation (1.6 nm) and the saturation magnetization (B = 0.60

T), we calculated the average phase slope of 0.010 rad/nm for full alignment of magnetic dipoles in array
films. The experimental values of the phase slope in each domain were measured from the reconstructed
phase images (Figure 3), as shown in Figure S3. The local magnetic order parameter shown in Figure 4 was
obtained as the ratio of the measured phase slope to the calculated value (0.010 rad/nm). When the dipoles
fluctuate, the parameter is less than 1.0 (Figure S4(a)). The average deviation angle of the dipoles in plane is
estimated by arc-cosine of the order parameter, for example, when the order parameter is 0.26, the angle is
75° (Figure S4(b)).
3
(a)
electron wave
sample
electron biprism
reference
wave
object
wave
interference
fringe pattern
hologram
(b)
hologram
filtering + sideband shift
Object wave
Amplitude
Phase
magnetic flux
distribution
Figure S1
Principle of electron holography
(a) Object wave and reference wave split by an electron biprism interfere with each other, resulting in a
hologram. (b) Reconstruction procedure to obtain a phase image. Contour map of the phase image shows the
magnetic flux distribution. The sample in the hologram is barium ferrite particle, 1 μm in diameter. (T.
Hirayama, J. Chen, Q. Ru, K. Ishizuka, T. Tanji, A. Tonomura, J. Electron Microsc. 43, 190-197 (1994))
4
Particle diameter(2r): 13.4 ± 1.5 nm
(radius(r): 5.95 - 7.45 nm)
r
Edge-to-edge separation between particles: 1.6 nm
Saturation magnetization (B): 0.60 T
B
0.60T
ΔφM = 2π
5.95 - 7.45 nm
e
Bπr 2 = 0.101 - 0.159 rad = 0.130 rad
h
on average
13.4 + 1.6 nm
30°
cross section
Phase slope = 0.010 [rad/nm]
for full alignment
0.130 rad
(13.4 + 1.6) x cos(30°)
= 13.0 nm
Figure S2
Phase slope when the dipoles in array are fully aligned
One magnetic dipole gives the electron wave to shift 0.130 rad on average. When the magnetic dipoles are
fully aligned in plane, the electron holography detects the phase slope of 0.010 rad/nm in a domain.
5
(a)
vacuum
24℃
Fe3O4 array
②
①
2 μm
phase [rad]
(b)
1
0
-1
A
-2
0
C
B
1
2
①
D
3
4
② [μm]
Figure S3 Measurement of the phase slope in each domain
(a) Typical phase image shown in Figure 3(a). (b) Phase line profile along ① - ② drawn in Figure S3(a).
The magnetic order parameter can be obtained as the ratio of the measured phase slope to the calculated
value (0.010 rad/nm, from Figure S2)
6
(a)
cross section
Phase slope measured by EH
? rad
(13.4 + 1.6) x cos(30°)
= 13.0 nm
(b)
average deviation angle "θ"
θ
B
ordering parameter
obtained by EH
B cos (θ)
= 0.26 ～ 0.44
B
θ = arccos (0.26) = 75 °
θ = arccos (0.44) = 64 °
Figure S4
Calculation of the average deviation angle
(a) When the dipoles fluctuate, the phase slope detected by EH decreases, resulting in a low ordering
parameter. (b) The average deviation angle is calculated by arc-cosine of the measured ordering parameter.
7
References
R1
D. Gabor, Nature 161, 777 (1948).
R2
A. Tonomura, Electron Holography, Springer Series in Optical Sciences, Vol. 70 (Springer, Berlin, 1999).
R3
E. Völkl, L. F. Allard, D. C. Joy (Eds), Introduction to Electron Holography, (Kluwer Academic/Plenum
Publishers, New York, 1999).
R4
Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959).
8