Strong Shape Dependence of the Morin Transition

Nano Research
Nano Res
DOI
10.1007/s12274-014-0699-1
Strong Shape Dependence of the Morin Transition in
α-Fe2O3 Single-Crystalline Nanostructures
Jun Wang1,a(), Victor Aguilar2, Le Li1, Fa-gen Li1, Wen-zhong Wang3, Guo-meng Zhao1,2,b()
Nano Res., Just Accepted Manuscript • DOI: 10.1007/s12274-014-0699-1
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The Moring transition in -Fe2O3 nanostructures depends strongly on
the morphology: There is a sharp Morin transition at about 210 K for
nanorings while it disappears for nanotubes.
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Nano Res
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Research Article
Strong Shape Dependence of the Morin Transition in α-Fe2O3
Single-Crystalline Nanostructures
Jun Wang1,a(), Victor Aguilar2, Le Li1, Fa-gen Li1, Wen-zhong Wang3, Guo-meng Zhao1,2,b()
1Department
of Physics, Faculty of Science, Ningbo University, Ningbo, P. R. China
of Physics and Astronomy, California State University, Los Angeles, CA 90032, USA
3School of Science, Minzu University of China, Beijing 100081, P. R. China
2Department
Received: day month year / Revised: day month year / Accepted: day month year (automatically inserted by the publisher)
© Tsinghua University Press and Springer-Verlag Berlin Heidelberg 2011
ABSTRACT
Single-crystalline α-Fe2O3 nanorings (short nanotubes) and nanotubes were synthesized by a hydrothermal
method. High-resolution transmission electron microscope and selected-area electron diffraction confirm that
the axial directions of both nanorings and nanotubes are parallel to the crystalline c-axis. What is intriguing is
that the Morin transition occurs at about 210 K in the short nanotubes with a mean tube length of about 115 nm
and a mean outer diameter of about 169 nm while it disappears in the nanotubes with a mean tube length of
about 317 nm and a mean outer diameter of about 148 nm. Detailed analyses of magnetization data, x-ray
diffraction spectra, and room-temperature Mössbauer spectra demonstrate that this very strong shape
dependence of the Morin transition is intrinsic to hematite. We can quantitatively explain this intriguing shape
dependence in terms of opposite signs of the surface magnetic anisotropy constants in the surface planes
parallel and perpendicular to the c-axis.
KEYWORDS
Hematite nanostructures, Morin transition, shape dependence, surface magnetic anisotropy
1. Introduction
Hematite (α-Fe2O3) has a corundum crystal
structure and orders antiferromagnetically below
its Néel temperature of about 950 K. Bulk hematite
exhibits a Morin transition [1] at about 260 K,
below which it is in an antiferromagnetic (AF)
phase, where the two antiparallel sublattice spins
are aligned along the rhombohedral [111] axis.
Above the Morin transition temperature TM,
α-Fe2O3 is in the weak ferromagnetic (WF) phase,
where the antiparallel spins are slightly canted and
lie in the basal (111) plane rather than along [111]
axis. The Morin transition is companied by the
————————————
Address correspondence to Guo-meng Zhao, [email protected]; Jun Wang, [email protected]
2
change of the total magnetic anisotropic constant
from a negative value at T > TM to a positive value
at T < TM. Interestingly, this AF-WF transition was
found to depend on magnetic field. An applied
magnetic field parallel to the rhombohedral [111]
axis below TM was shown [2-4] to induce the
spin-flip transition in the entire temperature range
below TM. The AF-WF transition can also be
induced
by
an
applied
magnetic
field
perpendicular to the [111] direction [5]. The
magnetic structure, the Morin transition, and the
field dependence of TM were explained [5, 6] in
terms of phenomenological thermodynamic
potential of Dzyaloshinsky.
Figure 1 Transmission electron microscopic (TEM) images of
the -Fe2O3 nanostructures prepared with different phosphate
concentrations. (a) TEM image for the sample prepared with
the phosphate concentration of 0.05 mM/L. A ring-like (short
nanotube) morphology is seen in the sample. (b) TEM image
for the sample prepared with the phosphate concentration of
0.50 mM/L. A tube-like morphology is observed in the sample.
In recent years, magnetic nanostructures have
attracted much attention, not only because of their
interesting physical properties but also because of
their broad technological applications. Of
particular interest is a finite-size effect on
ferromagnetic
or
ferromagnetic
transition
temperature. Finite-size effects have been studied
in quasi-two-dimensional ultra-thin ferromagnetic
films [7-10] and in quasi-zero-dimensional
ultra-fine
ferromagnetic
or
ferromagnetic
nanoparticles [11-15]. The studies on thin films
[7-10] and recent studies on nanoparticles [13-15]
have consistently confirmed the finite-size scaling
relationship predicted earlier [16]. Similarly, a
finite-size effect on the Morin transition
temperature was observed in nanosized α-Fe2O3
spherical particles [17-21]. The data show that TM
decreases with decreasing particle size [17, 19],
similar to the case of ferromagnetic or
ferrimagnetic nanoparticles [13-15]. The reduction
in the Morin transition temperature was
interpreted as due to inherent lattice strain (lattice
expansion) of nanocrystals [17, 20]. It was also
shown [19] that the TM suppression is caused by
both strain and the finite-size effect, commonly
observed in ferromagnetic/ferromagnetic materials.
More recently, Mitra et al. [22] have found that TM
shifts from 251 K for ellipsoidal to 221 K for
rhombohedra nanostructure, which suggests
observable shape dependence of the Morin
transition. Here we show that the Morin transition
temperature depends very strongly on the shape of
nanocrystals: TM shifts from 210 K for the nanorings
(short nanotubes with a mean tube length of about
115 nm) to <10 K for the nanotubes with a mean
tube length of about 317 nm. The very strong shape
dependence of the Morin transition is quite
intriguing considering the fact that the lattice
strains of both nanoring and nanotube crystals are
negligibly small, and that the sizes of nanotubes
are too large to explain their complete suppression
of TM by a finite-size effect. Instead, we can
quantitatively explain this intriguing shape
dependence in terms of opposite signs of the
surface magnetic anisotropy constants in the
surface planes parallel and perpendicular to the
c-axis
2
2. Experimental
α-Fe2O3
nanorings
treatment at 220 ℃ for 48 h. After the autoclave
were
a
was allowed to cool down to room temperature,
hydrothermal method, which is similar to that
the precipitate was separated by centrifugation,
reported in [23]. In the typical process, FeCl 3,
washed with deionized water and ethanol several
NH4H2PO4
times, and finally dried in air at 80℃. While
(phosphate),
prepared
and
Na2SO4
by
were
dissolved in deionized water with concentrations
keeping
of 0.002, 0.05 and 0.55 mM/L, respectively. After
unchanged,
vigorous stirring for 15 min, 80 mL of solution
concentration from 0.05 to 0.50 mM/L to produce
was transferred into a 100 mL Teflon-lined
nanotubes.
stainless
steel
autoclave
for
all
other
experimental
increasing
the
parameters
phosphate
hydrothermal
Figure 2 Length (L) and wall-thickness (t) histograms of the -Fe2O3 nanostructures. (a), (b) Length and wall-thickness histograms of
the -Fe2O3 nanorings. (c), (d) Length and wall-thickness histograms of the -Fe2O3 nanotubes. The solid lines are the best fitted curves
by log-normal distribution functions.
performed on a field-emission transmission
The morphology of the samples was
electron microscope (TEM, JEOL, JEM 2000,
checked by transmission electron microscopy
accelerating voltage 200 KV). X-ray diffraction
(JEOL-2010, operated at 200 kV). High-resolution
(XRD) spectra were taken by Rigaku Rint
transmission electron microscopy (HRTEM) was
D/Max-2400
X-ray
diffractometer.
Magnetic
3
moment was measured using a Quantum Design
phosphate
vibrating sample magnetometer (VSM) with a
morphology is seen in the sample prepared with
resolution better than 1×10-6 emu. The Mössbauer
the phosphate concentration of 0.05 mM/L (Fig.
spectra (MS) were taken by a WISSEL MB-500
1(a)) and a tube-like morphology is observed in
(Germany) using a standard absorption method
the
with a source of
concentration of 0.50 mM/L (Fig. 1(b)). In fact,
Co under room temperature
57
and fitted using the Gauss-Newton method.
sample
concentrations.
prepared
with
A
the
ring-like
phosphate
these nanorings can be described as short
nanotubes with tube lengths shorter than outer
diameters.
3. Results
From
TEM
images
we
obtain
histograms of their lengths and wall thicknesses,
which are displayed in Fig. 2.
Both length (Figs.
2(a),
(Fig.
2(c))
and
thickness
2(b),
2(c))
distributions are well described by a log-normal
distribution function [24]:
P(x) 
 ln 2 x x 
0
exp
 2 2 

x 2


1
(1)
where σ is the standard deviation and lnx0 is the

mean value of lnx. This distribution function was
used to fit the diameter histogram of -Fe2O3
nanoparticles well [24]. The best fit of Eq. 1 to the
data yields L0 =113.8±0.9 nm and t0 = 47.8±1.3 nm
for the nanorings; L0 = 309.5±3.7 nm and t0 =
25.6±0.9 nm for the nanotubes.
Figure 3 TEM images (left panels) and SAED patterns (right
panels) for a single nanoring. The results consistently
demonstrate the single-crystalline nature of the nanoring with
its axis parallel to the [001] direction.
Figure 4 TEM image (left panel) and SAED pattern (right
panel) for a single nanotube. The result demonstrates the
single-crystalline nature of the nanotube with its axis parallel
Figure
1
shows
transmission
electron
to the [001] direction.
microscopic (TEM) images of the two α-Fe2O3
nanostructures
prepared
with
different
4
Since
or
direction or the crystalline c-axis. From the SAED
magnetic moment of a particle is proportional to
pattern, we can evaluate the c-axis lattice constant.
its volume, the mean value of length or thickness
The obtained c = 13.77(2) Å is close to that
should be length- or thickness-weighted [21], that
determined from the XRD data (see below).
is,
x a 

x-ray
diffraction
intensity
For the nanotube sample, it is very

0

2
x P(x)dx

0
(2)
xP(x)
Based on Eq. 2 and the fitting parameters

for the histograms, the mean length and thickness
are calculated to be 115 and 50 nm for the
nanorings, respectively, and 317 and 30 nm for
the nanotubes, respectively. The mean outer
diameter of the short nanotubes is 169 nm, much
larger than the mean tube length (115 nm), which
is consistent with the ring morphology. The mean
outer diameter of the nanotubes is 148 nm, much
smaller than the mean tube length (317 nm),
which is consistent with the tube morphology. It
is remarkable that the mean thicknesses (50 and
unlikely to get a top-view TEM image since the
axes of the tubes tend to be parallel to the surface
of the sample substrate. So we can only take TEM
images of a single nanotube from the side-wall
view. The left panel of Fig. 4 displays a
side-wall-view TEM image of a nanotube. The
image indicates that the length of the tube is
about 200 nm. The single-crystalline nature of the
nanotube is clearly confirmed by the SAED
pattern (see the right panel of Fig. 4). The red
arrow marks the [001] direction, which is
determined by the SAED pattern. It is striking
that the tube axis is also parallel to the [001]
direction.
30 nm) of the nanorings and nanotubes inferred
from the TEM images are very close to those (58
and 32 nm) deduced from x-ray diffraction
spectra (see below).
In the left panel of Figure 3(a), we show the
TEM image of a single nanoring (the top view).
This ring has a wall thickness of about 50 nm,
which is very close to the mean value deduced
from the histogram above and slightly smaller
than the mean value of 58 nm deduced from the
XRD peak widths (see below). The selected-area
electron diffraction (SAED) pattern (right panel
of Fig. 3(a)) with a clear hexagonal symmetry
indicates that the nanoring is a single crystal with
a ring axis parallel to the crystalline c-axis. In
order to further prove the single-crystalline
nature of the nanoring, we show the side-wall
view of the ring (left panel of Fig. 3(b)) and the
corresponding SAED pattern (right panel of Fig.
3(b)). The red arrow indicates the [001] direction,
which is determined by the SAED pattern. It is
apparent that the ring axis is parallel to the [001]
5
Figure 5 X-ray diffraction (XRD) spectra of the two -Fe2O3
nanostructures. (a), B) XRD spectra for the nanorings and
Figure 5 shows x-ray diffraction (XRD)
nanotubes, respectively. Rietveld refinement of the XRD data
spectra of two α-Fe2O3 nanostructures prepared
(solid blue lines) with a space group of R3c (trigonal hematite
with the NH4H2PO4 concentrations of 0.05 and
lattice) was carried out to obtain the lattice parameters and
0.50 mM/L, respectively. These samples are phase
fractional coordinates of the atoms. The red lines represent the
pure, as the spectra do not show any traces
differences between the data and the refined curves.
Figure 6 Expanded views of some X-ray diffraction peaks of the -Fe2O3 nanostructures. (a), (b), (c) The (110), (300), (220) peaks of
the nanorings, respectively. (d), (e), (f) The (110), (300), (220) peaks of the nanotubes, respectively. The peaks are best fitted by two
Lorentzians (solid lines), which are contributed from the Cu K1 and K2 radiations. The intrinsic peak width  (after correcting for the
instrumental broadening) is indicated in each figure.
of other phases. Rietveld refinement of the XRD
uncertainties of these fitting parameters are even
data (see solid blue lines) with a space group of
much larger than themselves. A large reliability
R3c (trigonal hematite lattice) was carried out to
factor (Rwp~11%) of the refinement makes it
obtain
fractional
impossible to yield reliable fitting parameters for
coordinates of the atoms. The atomic occupancy
lattice strains (which are negligibly small) and
was assumed to be 1.0 and not included in the
particle sizes (which are quite large). In contrast,
refinement. We tried to include the lattice strains
the
and particle sizes in the refinement but the
refinements are quite accurate: a = b = 5.0311(12)
the
cell
parameters
and
lattice
parameters
obtained
from
the
6
Å , c = 13.7760(33) Å for the nanotube sample; a = b
by Eq. 3. The fitting parameters are indicated in each figure.
= 5.0340(6) Å , c = 13.7635(17) Å for the nanoring
The error bars in (a) are inside the symbols and not visible.
sample. These parameters are slightly different
from those for a bulk hematite [25]: a = b = 5.0351
Å , c = 13.7581 Å .
It is known that the x-ray diffraction peaks
are broadened by strain, lattice defects, and small
Since the axes of both nanorings and
particle size. When the density of lattice defects is
nanotubes are parallel to the crystalline c-axis, the
negligibly small, the broadening is contributed
mean wall thickness of the nanorings and
from both strain ε and particle size taυ. In this case,
nanotubes can be quantitatively determined by
there is a simple expression [26]:
 cos 0.89 2 sin 
(3)



t a

where the first term is the same as Scherrer's
the peak widths of the x-ray diffraction peaks
that are associated with the diffraction from the
planes perpendicular to the c-axis. Figure 6
shows x-ray diffraction spectra of the (110), (300), 
and (220) peaks for the nanoring and nanotube
samples. The peaks are best fitted by two
Lorentzians (solid lines), which are contributed
from the Cu Kα1 and Kα2 radiations. The fit has a
constraint that the ratio of the Kα1 and Kα2
intensities is always equal to 2.0, the same as that
used in Rietveld refinement.
equation that is related to the particle size taυ, the
second term is due to strain broadening, and ξ
was found to be close to 2ε [27]. In Fig. 7, we plot
βcosθ/λ versus 2ξsinθ/λ for the nanorings (Fig.
7(a)) and nanotubes (Fig. 7(b)). According to Eq. 3,
a linear fit to the data gives information about the
mean wall thickness taυ and strain εa along a and b
axes. The strain is small and negative for both
samples (see the numbers indicated in the
figures). It is interesting that the magnitudes of
the strain inferred from the XRD peak widths are
very close to those calculated directly from the
measured lattice parameters. For example, the
strain is calculated to be 0.023(13)% from the
lattice parameters for the nanorings and for the
bulk, in excellent agreement with that (0.017(2)%)
inferred from the XRD peak widths. Moreover,
the mean wall thicknesses inferred from the XRD
peak widths are very close to those determined
from TEM images. This further justifies the
validity of our Williamson-Hall analysis of the
XRD peak widths.
Figure 8 shows temperature and field
dependences of the normalized magnetizations
M(T)/M(350K) for the α-Fe2O3 singe-crystalline
nanorings and nanotubes. The samples were
initially cooled to 10 K in zero field and a field of
100 Oe was set at 10 K, and then the moment was
Figure 7 Dependence of βcosθ/λ on 2sinθ/λ for the
nanotings and nanotubes. The linear lines are the fitted curves
taken upon warming up to 350 K and cooling
down from 350 K to 10 K. At 10 K, other higher
7
fields (1 kOe, 10 kOe, and 50 kOe) were set and
state) and it enhances significantly above TM (WF
the moment was taken upon warming up to 350
state). It is interesting that TM for heating
K and cooling down from 350 K to 10 K. It is
measurements is significantly higher than that for
remarkable that the magnetic behaviors of the
cooling measurements (the arrows in the figure
two nanostructures are very different. For the
indicate the directions of the measurements). This
nanorings,
rapid
difference is far larger than a difference (about 6
increase around 200 K upon warming, which is
K) due to extrinsic thermal lag. This thermal
associated with the Morin transition (see Fig.
hysteresis was also observed in spherical α-Fe2O3
8(a)). It is worth noting that there seem to be two
nanoparticles [19]. The observed intrinsic thermal
transitions
Morin
hysteresis shows that the nature of the Morin
transition temperatures. The reason for this is
transition is of first-order. The result in Fig. 8(a)
unclear.
also
the
magnetization
with
slightly
shows
different
suggests
that
the
Morin
transition
temperature decreases with the increase of the
applied magnetic field. The zero-field TM in the
nanorings is about 210 K. What is striking is that
the
Morin
transition
is
almost
completely
suppressed in the nanotubes (see Fig. 8(b)).
Figure 8 Temperature and field dependences of the normalized
magnetizations M(T)/M(350K) for the nanorings (a) and
nanotubes (b). For the nanorings, the mean Morin transition
temperatures for heating and cooling measurements are 210.5,
208.6, 199.5, and 175.2 K in the fields of 0.1, 1.0, 10, and 50
kOe, respectively. For the nanotubes, the Morin transition
disappears down to 10 K.
Figure 9 Magnetic hysteresis loops at 300 K (a) and 10 K (b)
Furthermore, the magnetization below the
for the nanorings and nanotubes. The inset in (a) is the
Morin transition temperature TM is small (AF
expanded view of the loop. The room-temperature saturation
8
magnetization Ms, as inferred from a linear fit to the
magnetization data above 15 kOe, is the same (0.303±0.001
4. Discussion
emu/g) for both samples. The saturation magnetizations for
both samples are nearly the same as that (0.29±0.02 emu/g) [21]
The completely different magnetic behaviors
for a polycrystalline sample with a mean grain size of about 3
observed in the nanoring and nanotube samples
m.
are intriguing considering the fact that the two
samples have the same saturation magnetization
In Figure 9(a), we compare magnetic
at 300 K and nearly the same lattice parameters. It
hysteresis loops at 300 K for the nanorings and
is known that the lattice strain can suppress TM
nanotubes. There is a subtle difference in the
according to an empirical relation deduced for
magnetic hysteresis loops of the two samples.
spherical nanoparticles [19]: ∆TM = −600ε K,
The remanent magnetization Mr for the nanotube
where ε is isotropic lattice strain in %. For a
sample is about 30% higher than that for the
uniaxial strain, the formula may be modified as
nanoring sample, which is related to a higher
∆TM = −200εi K, where εi is the strain along
coercive field in the former sample (see the inset
certain crystalline axis. For the nanoring sample,
of
saturation
a = 5.0340(6) Å , which is slightly smaller than
magnetization Ms, as inferred from a linear fit to
(5.0351 Å ) for a bulk hematite [25]. This implies
the magnetization data above 15 kOe, is the same
that εa = −0.023(13)% for the nanoring sample, in
(0.303±0.001 emu/g) for both samples. The
excellent
saturation magnetizations for both samples are
inferred from the XRD peak widths. For the
also the same as that (0.29±0.02 emu/g) [21] for a
nanotube sample, a = 5.0311(12) Å , so εa =
polycrystalline sample with a mean grain size of
−0.080(24)%, in good agreement
about 3 μm. Since the saturation magnetization is
(-0.048(6)%) inferred from the XRD peak widths.
very sensitive to the occupancy of the Fe
site,
The negative strain would imply an increase in
the fact that the saturation magnetizations of both
TM according to the argument presented in Ref.17.
nanaoring and nanotube samples are nearly the
Therefore the suppression of TM cannot arise
same as the bulk value suggests that the
from the lattice strains along the a and b
occupancies of the Fe site in the nanostructural
directions. On the other hand, the lattice strain
samples are very close to 1.0, which justifies our
along the c direction is positive. Comparing the
XRD Rietveld refinement. The fact that the
measured c-axis lattice parameters of the two
saturation magnetizations of the two nanostructures
are close to the bulk value argues against the
possibility of α-Fe2O3 to Fe3O4 phase
transformation. Figure 9(b) shows magnetic
samples with that for a bulk hematite [24], we can
hysteresis loops at 10 K for the two samples. It is
by 8(2) K and 26(5) K for the nanoring and
clear that the nanotube sample remains weak
nanotube
ferromagnetic at 10 K (the absence of the Morin
negative strains along a and b directions are
transition down to 10 K) while the nanoring
compensated by the positive strain along c
sample is antiferromagnetic with zero saturation
direction (also see Table 2 below) so that the
magnetization.
volume of unit cell remains nearly unchanged.
Fig.
9(a)).
However,
the
3+
3+
agreement
with
that
(−0.017(2)%)
with
that
readily calculate that εc = 0.040(12)% for the
nanoring sample and 0.130(24)% for the nanotube
sample. This would lead to the suppression of TM
samples,
respectively.
The
small
This implies that the TM suppression due to
lattice strains should be negligibly small.
9
As mentioned above, there is also an
independent finite-size effect on TM unrelated to
Another possibility is that the nanotubes
the strain. For spherical nanoparticles, TM is
may contain more lattice defects than the
suppressed according to ∆TM = −1300/d K (Ref. 19),
nanorings. If this were true, the line width of the
where d is the mean diameter of spherical
Mössbauer spectrum for the nanotube sample
particles in nm. For the nanoring and nanotube
would be broader than that for the nanoring
samples, the smallest dimension is the wall
sample because the line width is sensitive to
thickness, which should play a similar role as the
disorder, inhomogeneity, and lattice defects. In
diameter of spherical particles [14]. With taυ = 58
contrast, the observed line width for
nm and 32 nm for the nanoring and nanotube
nanotube sample is smaller than that for the
samples, respectively, the suppression of TM is
nanoring sample by 33% and very close to that
calculated to be 22 K and 41 K, respectively.
for the bulk sample (see Fig. 10 and Table 1). In
Therefore, due to the finite-size effect, TM would
fact, all the Mössbauer parameters for the
be reduced from the bulk value of 258 K (Ref. 17)
nanotube sample are the same as those of the
to 236 K and 217 K for the nanoring and
bulk
nanotube samples, respectively. For the nanoring
uncertainties (see Table 1). If there would exist
sample, the zero-field TM is about 211 K, which is
substantial lattice defects, they would mostly be
25 K lower than the expected value from the
present
finite-size
effect
only.
This
additional
sample
in
within
surface
the
layers.
the
experimental
The
narrower
TM
Mössbauer line width observed in the nanotube
suppression of 25 K should be caused by other
sample is consistent with the fact that the
mechanism discussed below. For the nanotube
nanotubes have a smaller fraction of surface
sample, the Morin transition is almost completely
layers.
suppressed, which cannot be explained by the
Mössbauer spectra of both samples show only
strain and/or finite-size effect.
one
Moreover,
set
of
superparamagnetic
the
sextet,
room-temperature
suggesting
relaxation
at
no
room
temperature. This is consistent with the observed
magnetic hysteresis loops (see Fig. 9(a)).
Table 1 The fitting parameters for the Mössbauer spectra of
the nanorings and nanotubes. The parameters for the bulk
samples are taken from Fig. 3 of Ref. [28].
Width
(mm/s)
Hyperfine
field
(kOe)
Isomer
shifts
(mm/s)
Nanoring
0.48±0.01
513.3±0.2
0.44±0.01
shifts
(mm/s)
-0.22±0.01
Nanotube
0.36±0.01
511.9±0.2
0.44±0.01
-0.22±0.01
Bulk
0.38±0.05
510±4
0.37±0.02
-0.20±0.02
Quadrupole
Figure 10 Room-temperature Mössbauer spectra for the
nanorings and nanotubes. The spectra are fitted by a single
sextet (solid lines) with the fitting parameters displayed in
Finally, we can quantitatively explain the
strong shape dependence of the Morin transition
Table 1.
10
temperature if we assume that the surface
magnetic anisotropy constant KS is negative in the
surface planes parallel to the c-axis and positive
Figure 11 Numerically calculated TM as a function of
in the surface planes perpendicular to the c-axis.
KMD/KMD(bulk), where KMD (bulk) is the bulk anisotropy
Indeed a negative value of KS was found in Ni
constant. The calculation is based on a simple model presented
(111) surface [29] while Ks is positive in Co(0001)
in Ref. [31] and on the assumption that KFS remains
surface [30]. For the nanorings, the surface area
unchanged.
for the planes parallel to the c-axis is similar to
that for the planes perpendicular to the c-axis.
Following
this
simple
model,
we
can
Therefore, the total KS will have a small positive
numerically calculate TM as a function of
or negative value due to a partial cancellation of
KMD/KMD(bulk) on the
the KS values (with opposite signs) in different
remains unchanged, where KMD(bulk) is the bulk
surface planes. In contrast, the surface area of a
anisotropy constant. The calculated result is
nanotube for the surface planes parallel to the
shown in Fig. 11. It is apparent that TM is
c-axis is much larger than that for the planes
suppressed to zero when the magnitude of KMD
perpendicular to the c-axis. This implies that the
increases by 2.2%. Near this critical point, TM
total KS in the nanotubes should have a large
decreases rapidly with increasing the magnitude
negative value.
of KMD. For nanotubes, contribution of the surface
assumption that
KFS
For bulk hematite, the Morin transition
anisotropy is substantial and should be added to
temperature is uniquely determined by the total
the total anisotropy constant. Following the
bulk anisotropy constant K at zero temperature
expressions used in Refs.[29, 30], we have
[31]. Contributions to K are mainly dipolar
K MD  K MD (bulk)  2K S // t a  2K S La
anisotropy constant KMD, arising from magnetic
dipolar interaction, and fine structure anisotropy
(magneto-crystalline anisotropy) KFS, arising from
spin-orbit coupling [31]. With KMD = −9.2×10−6
(4)
where KS// and KS is the surface anisotropy

constants
for
the
planes
parallel
and
erg/cm in the bulk
perpendicular to the c-axis, respectively, and Laυ
hematite, the Morin transition temperature was
is the average tube length. Here we have
predicted to be 0.281TN = 270 K (Ref. 31), very
assumed that the surface areas of the inner and
close to the measured bulk value of 258 K (Ref.
outer walls are the same for simplicity. For the
17).
nanotubes, TM is nearly suppressed to zero.
erg/cm and KFS= 9.4×10
3
−6
3
According to Fig. 11, KMD/KMD(bulk) should be
close to 1.022 for the nanotubes with Laυ = 317 nm
and taυ = 32 nm. For the nanorings (short
nanotubes), the zero-field TM is about 211 K. This
implies that TM is totally suppressed by 47 K
compared with the bulk value of 258 K. Since the
finite-size effect can suppress TM by 22 K (see
discussion above), the additional suppression of
TM by 25 K should be due to an increase in KMD by
about 0.58% according to Fig. 11, that is,
KMD/KMD(bulk) = 1.0058 for the short nanotubes
11
with Laυ = 115 nm and taυ = 58 nm. Substituting
lowest 220.8 K for rhombohedra structure, with
these KMD/KMD(bulk), Laυ, and taυ values of both
intermediate values of TM for the other two
nanoring and nanotube samples into Eq. 4, we
structures. In Table 2, we compare some
obtain
parameters for four nanostructures reported in
two
equations
with
two
unknown
variables, KS// and KS. Solving the two equations
for the unknown KS// and KS yields KS// = −0.37
erg/cm2 and KS = 0.42 erg/cm2. The deduced
Ref. [22] and two nanostructures reported here.
The total lattice strain ε is calculated using ε = 2εa
+ εc, where εi is the percentage difference in the
lattice constant of a nanostructure and the bulk. It
is apparent that TM does not correlate with any of
these parameters. For example, the Fe occupancy
magnitudes of the surface anisotropy constants
(0.96) in the ellipsoidal structure is significantly
are in the same order of the experimental values
lower than 1.0, but TM is the highest and close to
found for Ni and Co [KS = -0.22 erg/cm2 for Ni(111)
the bulk value. This implies that the Fe vacancies
and 0.5 erg/cm for Co(0001) [29,30]. Therefore,
should have little effect on the Morin transition.
the observed intriguing experimental results can
We thus believe that the weak shape dependence
be naturally explained by a negative and a
of the Morin transition observed in the previous
positive surface anisotropy constant in the
work [22] should also arise from the opposite
surface planes parallel and perpendicular to the
signs
crystalline c-axis, respectively.
constants.
2
of
the
surface
The
much
magnetic
lower
anisotropic
TM
in
the
rhombohedra structure can be explained as due
Table 2 Some parameters for six nanostructures. The
to a much larger surface area parallel to the c axis
parameters for ellipsoidal, spindle, fattened, and rhombohedra
in the structure, in agreement with the observed
structures are calculated from the data reported in Ref. [22].
HRTEM image [22].
The Fe occupancies for the nanoring and nanotube structures
are inferred from the measured saturation magnetizations (see
5. Conclusions
discussion in the text).
In summary, we have prepared single-crystalline
(%)
Ellipsoida
TM
(K)
251.4
Spindle
c/a
0.028(1)
Fe
occupancy
0.9600(3)
245.4
0.063(1)
0.9921(10)
2.7329(1)
Fattened
231.5
-0.132(2)
0.9787(3)
2.7327(1)
Rhombohedra
220.8
0.053(2)
0.9934(16)
2.7340(1)
Nanoring
211
-0.007(25)
1.0
2.7341(5)
Nanotube
<10
-0.028(48)
1.0
2.7381(9)
hematite nanorings and nanotubes using a
hydrothermal
2.7337(1)
transmission
method.
electron
High-resolution
microscope
and
selected-area electron diffraction confirm that the
axial directions of both nanorings and nanotubes
are parallel to the crystalline c-axis. Magnetic
measurements show that there exists a first-order
Morin transition at about 210 K in the nanoring
crystals while this transition disappears in
nanotube crystals. The current results suggest
that the Morin transition depends very strongly
Now we discuss the shape dependence of
the Morin transition temperature observed in
other nanostructures [22]. It was shown that TM
shifted from highest 251.4 K for ellipsoidal to
on the shape of nanostructures. This strong shape
dependence of the Morin transition can be well
explained by a negative and a positive surface
anisotropy constant in the surface planes parallel
12
and perpendicular to the crystalline c-axis,
respectively.
Acknowledgements
This work was supported by the National Natural
Science Foundation of China (11174165), the
Natural
Science
Foundation
of
Ningbo
(2012A610051), and the K. C. Wong Magna
Foundation. VA acknowledges financial support from
NIH and NIGMS under MBRS-RISE M.S.-to-Ph.D.
Program (R25GM061331).
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