1. 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

2. (1) where(2) 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 (T B,) and the external field and temperature dependence of the magnetisation (M) above T B 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 T B should also depend on the ratio of the characteristic time scale of the measuring technique (t m ) and the inverse jump frequency (t 0 ). 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.

3. Forming Fe nano-grains in the immiscible Fe-Ag system Granular alloys (co-evaporation, co-sputtering) Discontinuous multilayers (by vacuum evaporation) 10x[2.6nm Ag / 0.7nm Fe] T B (i.e. the average grain size) can be tuned by: the Fe concentrationthe 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)) Fe 28 Ag 72 High resolution transmission electron microscopy images of two typical samples

4. Size determination from the bulk magnetisation EXAMPLE: Two multilayer samples: A: Si/ 75x[5.4 nm Ag/0.2 nm 57 Fe] B: Si/ 75x[2.6 nm Ag/0.2 nm 57 Fe] Applied Physics Letters 87, 102501 (2005) The temperature dependence of the magnetization measured by the SQUID in an applied field of 1 mT after zero-field cooling (ZFC) and field cooling (FC) in 1 mT is shown in the left panels in red and black, respectively. Samples A and B show magnetic irreversibility, typical of superparamagnetic systems, with a T B of 12 K and 40 K. A characteristic property of non-interacting SPM particles is the scaling of the magnetization curves measured at different temperatures when 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 T B 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 T B. The bulk magnetization of the thin films could be measured by a superconducting quantum interference device (SQUID) type magnetometer.

5. The magnetically split components of the Mössbauer spectra were fitted by two broad sextets and the remaining parts of the spectra were described by two singlets, not shown here. Si/ [5.4 nm Ag/0.2 nm 57 Fe] 75 (sample A in the SQUID measurements) Size determination from external field induced hyperfine field In large external fields: (see also: P. H. Christensen, S. Mørup, and J. W. Niemantsverdriet, J. Phys. Chem. 89, 4898 1985.) These fits yield,  =420(12)  B, B 0 =37.0(1)T and  =307(12)  B, B 0 =32.7(1)T for the high and low field components. The grain diameters calculated from 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 was estimated from the SQUID measurement. The difference hardly exceeds the accuracy of determining an average size by x-ray diffraction or electron microscopy methods when they can be applied succesfully. Phys. Rev. B 76,052408 (2007)

6. Size determination from the static hyperfine fields Phys. Rev. B 76, 052408 (2007) A: Si/ [5.4 nm Ag/0.2 nm 57 Fe] 75 B: Si/ [2.6 nm Ag/0.2 nm 57 Fe] 75 (These samples were studied by SQUID) For both samples, the spectra exhibit broad but definitely structured lines, which allow a separation into two components described by two HF distributions. Since the hyperfine field is aligned opposite to the magnetization, the saturation of B + =B obs +B ext, i.e., the sum of the measured HF and the external field, indicates the ferromagnetic alignment of the magnetic moments along the applied field in accordance with the disappearance of the 2nd and 5th spectral lines. This study proves that the observed spectrum features belong to static properties. The relative fraction of the two components varies in accordance with the grain size determined from the SQUID measurements (D=1.2 and 1.8 nm) if, in a simple 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)) low field component: 53% and 70% Spectra measured at 4.2 K in perpendicular external field

7. 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) Deviation from the simple ΔR(B)  M 2 quadratic behavior signifies the presence of small clusters, which barely contribute to the magnetization but dominate the scattering in high fields. Phys. Rev. B 73, 184412 (2006)

8. Grain size from the temperature dependence in large field 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 (a 1 ) 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 ρ ph (T) curve can be used to separate the magnetic scattering contribution in the B = 12 T measurement. Separation of the phonon and the magnetic scattering 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 ρ magn (T,B=12T). The spin disorder for a characteristic moment S can be expressed by the Brillouin-function: Grain size from ρ magn (T, B =12 T). 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)

9. 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. 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 b 1, b 2, and b 3, respectively. (L and B S are the Langevin and the Brillouin functions.) Since the volume fraction of the clusters is small (see below), b 3 is negligible. At low temperature scattering between grains and clusters is responsible for the non-saturating magnetoresistance. Phys. Rev. B 73, 184412 (2006) 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%.

10. Comparison of the three methods The cluster moments, determined from the external field induced hyperfine fields above T B, 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. 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). 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.