Determination of the size of
Fe nano-grains in Ag
J. Balogh, D. Kaptás, L. F. Kiss, and I. Vincze
Research Institute for Solid State Physics and Optics,
P.O. Box 49, H-1525 Budapest, Hungary
A. Kovács
The Institute of Scientific and Industrial Research,
8-1 Mihogaoka, Ibaraki 567-0047, Osaka, Japan
M. Csontos and G. Mihály
Department of Physics, Budapest University of Technology and Economics,
1521 Budapest, P.O. Box 91, Hungary
E-mail: baloghj@szfki.hu
http://www.szfki.hu
When discussing the physical properties of different nano-materials the determination of
the size is a basic task. There are several image forming and diffraction methods that can be used in
the case of nano-size objects (e.g. transmission electron microscopy, atomic force microscopy, x-ray
diffraction line-broadening and small angle scattering), but in the case of magnetic elements, size
determination from the magnetic properties is also of great interest. It is an especially important
issue when the standard methods are hindered for some reason, as it is the case for very small Fe
grains (up to a few hundred atoms) in Ag. (See: Y. Xu et al. J. Appl. Phys. 76 (5), 2969 (1994), M.
Csontos et al., Phys. Rev. B 73, 184412 (2006))
If the size of a magnetic particle is smaller than the domain wall width then it will form a
single domain, and its volume can be estimated from the appearing superparamagnetic properties.
(For a review see: J. L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. 98, 283 1997.)
According to the model of Néel for interaction free particles with uniaxial anisotropy and a supposed
uniform rotation of the magnetisation: the blocking temperature (TB,) and the external field and
temperature dependence of the magnetisation (M) above TB are determined by the magnetic
anisotropy energy (K) and the volume (V), or in proportion to it the magnetic moment (), of the
particle. The experimentally measured TB should also depend on the ratio of the characteristic time
scale of the measuring technique (tm) and the inverse jump frequency (t0 ).
KV
kTB 
ln t m / t0 
1
 H 
M T , H   M 0 L
 where La   ctha  
 kT 
a
(1)
(2)
Our aim is to compare the grain size determined by Mössbauer spectroscopy to
those measured by two other powerful methods, bulk magnetization and magnetoresistance
measurements. The Fe-Ag system studied is a good choice, since the average grain size can be
easily varied in the most interesting 1 to 10 nm range. The understandig of the giant
magnetoresistance (GMR) behavior gives a further motivation to these studies.
Forming Fe nano-grains in the immiscible Fe-Ag system
Discontinuous multilayers
(by vacuum evaporation)
Granular alloys
(co-evaporation, co-sputtering)
Fe28Ag72
10x[2.6nm Ag / 0.7nm Fe]
High resolution transmission electron
microscopy images of two typical samples
TB (i.e. the average grain size)
can be tuned by:
the Fe and the Ag layer thickness
(On the role of the Ag layer thickness see:
J. Balogh et al., Appl. Phys. Lett. 87, 102501 (2005))
the Fe concentration
Size determination from the bulk magnetisation
The bulk magnetization of the thin films could
be measured by a superconducting quantum
interference device (SQUID) type
magnetometer.
EXAMPLE:
Two multilayer samples:
A: Si/ 75x[5.4 nm Ag/0.2 nm 57Fe]
B: Si/ 75x[2.6 nm Ag/0.2 nm 57Fe]
0.6
A
100
0.2
0
200
0.0
2
1
B
5K
50K
100K
100
0
0
0 100 200 300
T [K]
0.1
0.2
0H/T [T/K]
Applied Physics Letters 87, 102501 (2005)
M [emu/gFe]
M [emu/gFe]
0.4
200
The temperature dependence of the
magnetization measured by the SQUID in an
applied field of 1 mT after zero-field cooling
[0.2
(ZFC) and field cooling (FC) in 1 mT is
[0.2
shown in the left panels in red and black,
respectively. Samples A and B show magnetic
F
irreversibility, typical of superparamagnetic
systems, with a TB of 12 K and 40 K.
A characteristic property of non-interacting SPM
particles is the scaling of the magnetization
5K
curves measured at different temperatures
when
50K
100K
they are plotted as a function of H/T, i.e., the
applied field divided by temperature. The
right panels show that this scaling can be
observed above TB for both samples.
Fitting by eq. (2) yields 200 and 600 B for the
average cluster moment (about 90 and 270 Fe
atoms which means about 1.2 and 1.8 nm grain
diameters supposing sperical particles), in good
agreement with the observed variation of TB.
Size determination from external field induced hyperfine field
(see also: P. H. Christensen, S. Mørup, and J. W. Niemantsverdriet, J. Phys. Chem. 89, 4898 1985.)
Hflow=col(A)*col(E)+col(Bh)*SQRT(1-((col(A)/col(Bh))^2)*(1-col(E
Si/ [5.4 nm Ag/0.2 nm 57Fe]75 (sample A in the SQUID measurements)
0T
T=50K
<Bobs>+ Bext 
[ Tesla
Bext ]
38B obs
1T
36
2T
34
36
34
In large external
fields:
32
BA
ext / kT  1
32
3T
30
30
B
 0 2kT 4
1]

B obs  B extBext
 B[ 0T
(a) Bext 
0
5T
7T
-10
 B 0 L  38   B ext
 kT 
5
0
-5
velocity [mm/s]
10
2
4
l(Bobs-Bext)l [ T ]
35
30
25
0.0
0.5
1.0
1/Bext [1/T]
(b)
Phys. Rev. B 76,052408 (2007)
These fits yield, =420(12)B, B0=37.0(1)T and =307(12)B, B0=32.7(1)T
for the high and low field components. The grain diameters calculated from
The magnetically split components the two components are 1.6 and 1.4 nm (supposing spherical particles
and 2.2B atomic moments), slightly larger than the 1.2 nm value which
of the Mössbauer spectra were
fitted by two broad sextets and the was estimated from the SQUID measurement. The difference hardly
remaining parts of the spectra were exceeds the accuracy of determining an average size by x-ray diffraction
or electron microscopy methods when they can be applied succesfully.
described by two singlets, not
shown here.
Size determination from the static hyperfine fieldshflow
Hfhigh
Hflow
A: Si/ [5.4 nm Ag/0.2 nm 57Fe]75
Hfhigh
57
L1T150Khf
B: Si/ [2.6 nm Ag/0.2 nm Fe]75
(These samples were studied by SQUID)
L1T150KHf1
C0414T5Hf
Spectra measured at 4.2 K in
For both samples, the spectra exhibit
broad but
C0414T5Hf1
Hflow=col(A)*col(E)+col(Bh)*SQRT(1-((col(A)/col(Bh))^2)*(1-col(E)^2))
definitely structured lines, which allow a
separation into two components
y=mx+C described by
<Bobs>+ Bext[26A
[ Tesla
two
HF
distributions.
Since the hyperfine field
l(Bobs
-Bext
)l [ T ]
Ag / 2A ]Fe(57)]
38
38
C0414T50K_Hfhigh:
is aligned opposite to the
magnetization, the
35
C=36.98558
saturation of B+=Bobs+Bext
, i.e., the sum of the
[54A 36
Ag / 2A Fe(57)]
36
m=-6.55329
30 measured HF and the external
field, indicates
34
34
25 the ferromagnetic alignment of the magnetic
32
32
field in accordance
20 moments along the applied
L1T150K_Hfhigh:
A
B
with the disappearance C=34.34835
of the 2nd and 5th
30
30
15
m=-6.84411
spectral
proves that the
0 2 4
0 2 4
0.0
0.5 lines.
1.0 This
1.5 study
1/Bext spectrum
[1/T]
Bext [ T ]
observed
features belong to static
C0414T50K_Hflow:
properties.
(b)
(a)
C=32.69002
Phys. Rev. B 76, 052408 (2007)
m=-7.91993
The relative fraction of the two components varies in
L1T150K_hflow:
accordance with the grain size determined
from the
C=30.5062
SQUID measurements (D=1.2 and 1.8 nm) if, in a simple
m=-9.46667
model, they are associated with Fe atoms at the surface
and in the volume of the grains. The assignment of the
low field (blue) component to surface atoms is in
accordance with theoretical calculations. ((R. N. Nogueira
and H. M. Petrilli, Phys. Rev. B 60, 4120 (1999), C. O.
Rodriguez et al., Phys. Rev. B 63, 184413 (2001))
perpendicular external field
B
A
o
o
75
o
0T
75
1T
3T
5T
-5
0
5
-5 0
velocity [mm/s]
o
5
low field component:
53% and 70%
Fe grain size from magnetoresistance measurements
The giant magnetoresistance (GMR) in multilayer structures of alternating ferromagnetic and nonmagnetic
layers and granular composites have been explained by elastic scattering of the conduction electrons on
magnetic moments of differently aligned magnetic entities. In superparamagnetic granular alloys this
consideration (Gittleman et al. Phys. Rev. B 5, 3609 (1972)) leads to a magnetoresistance proportional to
the square of the magnetization. Deviations from this proportionality can be a result of a grain size
distribution, i.e. a distribution of the magnetization of the particles. (For review see: X. Batlle and A. Labarta,
J. Phys. D: Appl. Phys. 35 (2002) R15–R42.) Since the resistivity can be influenced by as little as a few
ppm of magnetic impurities, it is an effective method to search for the extremely small grains.
sample: Si/ [2.6 nm Ag/0.2 nm Fe]75
(nominally equal to sample B, but with natural Fe)
B=2T
0.00
[(B) - (0)] / (0)
4T
-0.02
-0.04
12 T
T = 290K
-0.06
0
1
2
2
4
Deviation from the simple ΔR(B)  M2
quadratic behavior signifies the
presence of small clusters, which
barely contribute to the magnetization
but dominate the scattering in high
fields.
3
2
M [10 (emu/g) ]
Phys. Rev. B 73, 184412 (2006)
Grain size from the temperature dependence in large field
Separation of the phonon and the magnetic scattering
(a)
2.0
B = 0T
3 /T
1.8
T 
 ph T   a1   

 T , B    0   ph T    magn T , B 
1.6
1.4
 / 0
1.2
1.0
(b)
2.0
B = 12T
1.8
1.6
1.4
1.2
1.0
0
50
100
150
200
250
300
Temperature (K)
B = 12T
0
In zero magnetic field, well above the blocking temperature (40 K) the magnetic
moments of all the grains are fully disordered and we can assume that the
temperature dependence arises solely from the phonon contribution. The phonon
term is linear above the Debye temperature (210 K) and the strength of phonon
scattering (a1) can be determined from the high temperature slope. The curve
calculated according to the formula above for phonon scattering is shown by the
dashed line. The remaining part (dotted line) is attributed to the magnetic
scattering. Since the phonon term is magnetic field independent, the calculated
ρph(T) curve can be used to separate the magnetic scattering contribution in the
B = 12 T measurement.
Grain size from ρmagn(T, B =12 T).
0.4
magn (T,B = 12T) / 0

x 2 dx
e x 1
We assume that in high field the magnetic scattering of the spin-polarized
electrons is proportional to the spin disorder of the small clusters and this gives
rise to the strong temperature dependence of ρmagn(T,B=12T). The spin disorder
for a characteristic moment S can be expressed by the Brillouin-function:
0.3
0.2



1
2
 2S  1g B B  1  g B B 
  cth

2 k BT

 2  2 k BT  
 magn T , B   a2 S  S z   a2  S   S  cth
0.1

0.0
0
100
200
Temperature (K)
300
The left figure shows the resistivity change attributed to magnetic scattering in
B = 12 T. The fitted curve shown by the solid line belongs to S = 16.6 B.
Phys. Rev. B 73, 184412 (2006)
Magnetoresistance curves - large grains and small clusters
The SQUID magnetization measurements of this sample indicated the presence of large grains with 
 500 B average moment, while the temperature dependence of the resistivity in high magnetic field
has shown the presence of small clusters with S  17 B. The magnetoresistance curves can be
described with these two characteristic magnetic moments.
[(B) - (0)] / (0)
0.00 (a)
(b)

 B 
 B 
2
 b1 L2 
  b2 L
 BS B, T   b3 BS B, T 

 kT 
 kT 
(c)
-0.04
-0.08
experimental
calculated
2
L
L*BS
-0.12
290K
-0.16
-10
-5
0
5
180K
10 -10
-5
0
5
10 -10
120K
-5
0
5
10
Magnetic field (T)
The magnetoresistance curves are well
described by electron scattering from grain to
grain, between a grain and a cluster and from
cluster to cluster with amplitudes b1, b2, and
b3, respectively. (L and BS are the Langevin
and the Brillouin functions.) Since the volume
fraction of the clusters is small (see below), b3
is negligible. At low temperature scattering
between grains and clusters is responsible for
the non-saturating magnetoresistance.
Phys. Rev. B 73, 184412 (2006)
-5
4.2K
Bext=0T
4.2K
Bext=3T
4.2K
Bext=7T
0
velocity [mm/s]
The Mössbauer spectra clearly show that the vast majority of the magnetic
moments can be ferromagnetically aligned along the external field direction at
4.2 K. (See also results for samples A and B shown previously). With
considerations to the statistical errors, the ratio of Fe atoms with paramagnetic
or superparamagnetic moments can be estimated as less than 2%.
5
Comparison of the three methods
The cluster moments, determined from the external field induced hyperfine fields above TB, are significantly
larger than those determined from the SQUID measurements. The evaluation based on the description of the
magnetically split sharp features of the spectra obviously overestimates the grain size, because it does not
take into account those small grains that do not exhibit well resolved peaks in the field range. In the studied
few nm grain size range, however, the corresponding difference of the calculated grain diameters hardly
exceeds the accuracy of determining an average size by x-ray diffraction or electron microscopy methods.
The static hyperfine field distributions evaluated from measurements at 4.2 K in various applied fields are also
found to reflect the grain-size difference. The relative fraction of the observed two components varies in
accordance with the grain size determined from the SQUID measurements if, in a simple model, they are
associated with Fe atoms at the surface and in the volume of the grains. Since the ground state static HF is
not influenced by magnetic or exchange interactions between the grains, it is an important check of the grain
size determined from the dynamic properties.
The magnetoresistance measurements can reveal the presence of tiny clusters that give negligible
contribution to the magnetization. The magnetic field and the temperature dependence of the resistivity could
be described in consistancy with the SQUID and the Mössbauer results.
Mössbauer spectroscopy can supply unique information:
- in multicomponent systems (e.g. Fe-Co-Ag, Fe-Ni-Ag)
- in heterogeneous systems (e.g. bimodal size distribution, heterostructures)
- when magnetic interactions influence the dynamic behaviour (e.g. high
concentration of the magnetic elements).