Oct 15, 1999 - gular, triangular, square and pentagonal geometries, corresponding, respectively, to rotational symmetries of order 2, 3, 4 and 5. The parent ...
EUROPHYSICS LETTERS
15 October 1999
Europhys. Lett., 48 (2), pp. 221-227 (1999)
Designing nanostructured magnetic materials by symmetry R. P. Cowburn, D. K. Koltsov, A. O. Adeyeye and M. E. Welland Nanoscale Science Group, Department of Engineering, University of Cambridge Trumpington Street, Cambridge CB2 1PZ, UK (received 24 June 1999; accepted 30 August 1999) PACS. 75.30Gw – Magnetic anisotropy. PACS. 75.50Tt – Fine-particle systems. PACS. 78.20Ls – Magnetooptical effects.
Abstract. – We have investigated experimentally the influence of the geometric shape of deep sub-micron nanomagnets on their magnetic properties. We have used high-resolution electron beam lithography to make arrays of nanomagnets in the size range 40-500 nm which had rectangular, triangular, square and pentagonal geometries, corresponding, respectively, to rotational symmetries of order 2, 3, 4 and 5. The parent material was Supermalloy (Ni80 Fe14 Mo5 ). We find that an enormously wide range of magnetic properties, including some not found in conventional unstructured materials, can be obtained by using different geometries. We demonstrate that this is because the geometric shape imposes a strong anisotropy field of related symmetry order on the nanomagnet via a recently discovered phenomenon called configurational anisotropy. We show that the coercive field and remanence of these structures is determined directly by this anisotropy.
The volume of information which can be stored on a computer hard disk has risen by a factor of 107 in the past 40 years and looks set to continue rising at an exponential rate in coming decades [1]. Today’s conventional magnetic materials will be unable to meet the demanding performance requirements of tomorrow’s magnetic data storage industry. One option currently being considered is an application of nanotechnology to make nanometre scale magnetic particles called nanomagnets. These, by virtue of their extremely small size, possess very different magnetic properties from their parent bulk material. The rapidly growing field of nanomagnetism [2] may provide among other things advanced replacements for hard disk media [3] and a new generation of high speed, low power, non-volatile computer memory chips [4]. The most important property of a naturally occurring magnetic element or alloy is its anisotropy. This refers to the presence of preferred magnetisation directions within the material and is ultimately responsible for determining the way in which a magnetic material behaves and the technological applications for which it is suitable. In nanomagnets, the anisotropy depends not only on the properties of the parent material, but also on the shape of the nanomagnet. One of the most attractive features of nanostructured magnetic materials therefore is that their magnetic properties can be engineered by the choice of the shape of the constituent nanomagnets. c EDP Sciences °
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Fig. 1. – A collection of Scanning Electron Micrographs of some of the different nanostructured magnetic materials made during this study.
In this letter we present the first detailed experimental survey of the influence of the symmetry of the geometric shape of nanomagnets on their magnetic properties. We have made small arrays (25–100 µm2 ) of nanomagnets on the surface of a piece of silicon using electron beam lithography. The constituent nanomagnets were in the size range 40–500 nm and in the thickness range 3–10 nm and had rectangular, triangular, square and pentagonal geometries which, respectively, correspond to rotational symmetries of order 2, 3, 4 and 5. The parent material was Supermalloy (Ni80 Fe14 Mo5 ), which was chosen because it is intrinsically almost isotropic (stray anisotropy field were found to be less than 4 Oe) and so any anisotropy in the nanomagnets must come from their shape. We show experimentally that the rotational symmetry order of the nanomagnets plays a key role in determining magnetic properties. Consequently, we are able to make nanostructured magnetic materials with an enormously wide range of properties simply by varying the symmetry of the constituent nanomagnets. A standard lift-off based electron beam lithography process has been used which we have already described elsewhere [5]. The sample substrate was (100) oriented single crystal silicon. The spacing between each nanomagnet was always at least equal to the diameter of the nanomagnet, and for the smallest structures was as large as 3 times the diameter. The surface roughness of the nanomagnets was less than 0.5 nm and the microstructure showed ∼ 10 nm size randomly oriented grains. The top surface of each nanomagnet was covered with a 5 nm thick layer of gold to prevent oxidation. The integrity of the geometric shape was found to be preserved in structures as small as 40 nm in size. Figure 1 shows Scanning Electron Microscopy images of some of the structures. In order to determine the magnetic properties of these different nanostructured materials, we have measured their hysteresis loops (M -H loops) using a high-sensitivity magneto-optical method [6]. The silicon surface can be viewed under an optical microscope, while a laser spot (size ∼ 5 µm) is moved over the surface until focused on top of one of the arrays of nanomagnets. The reflected laser beam is polarisation analysed in order to access the longitudinal Kerr effect, which serves as a probe of the component of magnetisation lying in the optical plane of incidence. This magnetisation is then recorded while a 27 Hz alternating magnetic field of up to 1000 Oe strength is applied in the plane of the sample. All measurements were performed at room temperature. Figure 2 shows some of the hysteresis loops measured from nanomagnets of different size, thickness and geometric shape. One sees immediately that the loops are very different from
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Fig. 2. – Hysteresis loops measured from different nanostructured magnetic materials. A shows for comparison the loop from unstructured Supermalloy. The y-axis of each graph is magnetisation, normalised to the saturation value. The applied field in the schematic pictures of the nanostructures is assumed to point up the page. The nanostructures had thicknesses of A: 6 nm, B: 10 nm, C: 5 nm, D: 3 nm, E: 5 nm, F: 3 nm. Fig. 3. – Experimentally measured coercivities as a function of nanomagnet size for different thicknesses (◦ : 3 nm, • : 5 nm, ¤ : 7.5 nm, ¥ : 10 nm) and nanomagnet geometries (A: 2:1 aspect ratio rectangles, B: equilateral triangles, C: squares, D: pentagons).
each other and from that obtained from the conventional unstructured material. The properties covered range from those usually associated with moderately “hard” magnetic materials with a large coercivity suitable for data storage (e.g., the small rectangles of fig. 2B) down to those
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associated with very “soft” magnetic materials with a high relative permeability suitable for field sensing (e.g., the pentagons of fig. 2F with an effective relative permeability of 3000 and zero hysteresis). These changes in properties are a direct result of varying the size, thickness and, most importantly, the symmetry of the nanomagnets in the arrays. In order to quantify this important effect, we have measured coercivity from the hysteresis loops as a function of size, thickness and symmetry order of the nanomagnets. Figure 3 shows the results, where, in order to be able to compare different geometries, we express the size of the nanomagnets by the square root of their area. The behaviour of the 2-fold nanomagnets is very straightforward and is due to a well understood phenomenon called shape anisotropy [7]. The magnetisation prefers to “streamline” with the longest axis of the nanomagnet in order to minimise the surface area of the pole faces. The field driving this effect is the demagnetising field, which scales approximately with the ratio t/a where t is the thickness of the nanomagnet and a is its size and so one sees in fig. 3A a coercivity which rises with decreasing size. The effects of the 3-, 4- and 5-fold symmetries are, in contrast, not so easily understood. This is because the demagnetising field of any structure is described by a second-rank Cartesian tensor and so can only exhibit uniaxial (2-fold) symmetry. There is therefore no shape anisotropy present, at least in the conventional sense, in the plane of these higher-order symmetry structures. Nevertheless, the square nanomagnets clearly experience some reasonably strong impediment to changing the direction of the magnetisation and so there must be some anisotropy present. The fact that this anisotropy is clearly weaker in the triangular magnets shows that although it is not classical shape anisotropy, it is still linked to the shape of the nanomagnet. We recently reported the prediction [8] and experimental observation [5] of a new type of anisotropy in square nanomagnets called configurational anisotropy. This comes about because of the very small deviations from uniform magnetisation which occur in nearly all nanomagnets. To date it has not been made clear to what extent this new anisotropy is important in defining the magnetic properties of nanomagnets. In order to test the hypothesis that the varied behaviour which one sees in fig. 2 and fig. 3 is due to configurational anisotropy, we have performed a direct measurement of the anisotropy in the nanomagnets using a technique which we call Modulated Field Magneto-optical Anisometry [5, 9]. A strong static magnetic field H (= 350 Oe) is applied in the plane of the nanomagnets and a weaker oscillating field Ht (14 Oe amplitude) is applied in the plane of the nanomagnets, perpendicularly to H. The amplitude of the resulting oscillation of the magnetisation is recorded by the same magneto-optical technique used to obtain the hysteresis loops of fig. 2 and is directly related to the amplitude and symmetry of any anisotropy present in the nanomagnet. We have measured experimentally the anisotropy for triangular, square and pentagonal nanomagnets in the size range 50–500 nm at a thickness of 5 nm and present the results in fig. 4. In these polar plots, the angle gives the in-plane direction φ within the nanomagnet, the radius gives the radius of the nanomagnet in that direction and the colour gives the experimentally measured 2 quantity Ha ≡ M1s ∂∂φE2 for a nanomagnet of that size where Ms is the saturation magnetisation (800 emu cm−3 measured by B-H looping of our unstructured films) and E(φ) is the average magnetic energy density of the nanomagnet when magnetised in the direction φ [10]. Although Ha is not the usual definition of anisotropy field, it can be shown that any oscillations in Ha have the same symmetry order as the underlying anisotropy and an amplitude equal to the magnitude of the anisotropy field. Figure 4 shows experimental data from 22 different arrays of nanomagnets (8 sizes of triangles, 8 sizes of squares and 6 sizes of pentagons), each measured in either 19 or 37 different directions φ (0–180◦ in 10◦ steps for triangles and squares, 0–180◦ in 5◦ steps for pentagons) making a total of 526 measurements. It is immediately clear from fig. 4 that there are strong anisotropy fields present in all of the
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Fig. 4. – The experimentally measured anisotropy field inside 5 nm thick nanomagnets of triangular (A), square (B) and pentagonal (C) symmetry. These are colour polar plots where the direction gives the in-plane direction in the nanomagnet, the radius gives the nanomagnet radius in that direction and the colour gives the experimentally measured anisotropy field of a nanomagnet of that size. High field values correspond to easy anisotropy axes and low values to hard anisotropy axes. The lateral scale of the figure is such that the square data runs from edge length 50 nm to 500 nm. Data was recorded across a 180◦ range and then plotted twice to fill 360◦ . Fig. 5. – The strength of the dominant anisotropy term shown in fig. 4 expressed as an anisotropy field (A) and as an anisotropy energy per nanomagnet (B) for nanomagnets of different size and symmetry (M triangles, ¤ squares, ¨ pentagons).
nanomagnets studied. The triangular nanomagnets exhibit anisotropy with 6-fold symmetry, the square nanomagnets show a 4-fold symmetric anisotropy and the pentagonal nanomagnets posses a remarkable 10-fold anisotropy. Frequency doubling occurs in the triangular and pentagonal structures because energy is always quadratic in the magnetisation and so odd symmetry orders cannot be supported. We have applied a Fourier analysis to the plots of fig. 4 in order to obtain the magnitude of the anisotropy fields as a function of nanomagnet size and symmetry and show the results in two different forms in fig. 5. It is simple to show by differentiating twice that the anisotropy energy U and the anisotropy field Ha of a nanomagnet are related by U=
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where n is the symmetry order of the anisotropy and V is the volume of the nanomagnet. In fig. 5A we plot the anisotropy fields directly, whereas in fig. 5B we have plotted the anisotropy energy of a single nanomagnet (in units of kT , where k is the Boltzmann constant and T is 298 K) using eq. (1). An important element in our understanding of the influence of symmetry
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Table I. – Experimentally determined parameters for the straight line law of eq. (2) which describes the dependence of configurational anisotropy energy on element size. Shape
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order on magnetic properties is provided by the n2 term in this equation. One sees that whereas the anisotropy fields show an initial rise with increasing lateral size which then either falls again (squares and triangles) or forms a plateau (pentagons), the anisotropy energy can be described by the straight line relationship √ U = C( Area − l0 ) , kT
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where the experimentally determined parameters C and l0 are given in table I. The table also shows that the quantity Cn2 is found to be approximately constant across all nanostructures (at one thickness). This thus gives us a useful phenomenological tool for rapidly assessing how magnetic properties are influenced by size and symmetry. The anisotropy energy is particularly interesting because of a phenomenon called superparamagnetism [11], which is the process by which anisotropy energy barriers can be overcome by the kT thermal energy fluctuations in nanometre scale magnets. A barrier can be overcome on the timescale of our measurements once it is less than a few kT in height. This means that as the anisotropy energy approaches kT , one would expect the coercivity to fall rapidly to zero. According to fig. 5B this should occur once the element size is less than 100–150 nm, the squares being last to fall as the size is reduced. Conversely, once the anisotropy energy is significantly greater than kT , the coercivity should approximately follow the anisotropy field. This explains the difference in behaviour which one observes in fig. 3 between the squares on the one hand and the triangles and pentagons on the other. The anisotropy field of the squares (fig. 5A) shows a peak as the element size is reduced and this peak is reflected directly in the square coercivity data (fig. 3C). The pentagon anisotropy field shows no peak, and this is also reflected directly in the coercivity data (fig. 3D). Finally, the triangle anisotropy field does show a peak just like the square, but because the anisotropy energies are lower in the triangle, thermal activation sets in at a larger size and prevents the peak from being seen in the coercivity data (fig. 3B). We can thus explain the experimentally determined coercivity data as being due to a combination of configurational anisotropy and thermal activation. In conclusion, we have performed an experimental survey of the influence of the symmetry of the shape of nanomagnets on their magnetic properties. We find that symmetry plays a vital role, allowing magnetic properties to be controlled over a very wide range. We have shown that the key effect linking symmetry to the magnetic properties is configurational anisotropy, although both the anisotropy energy and the anisotropy field need to be considered. This opens up the possibility of designing new nanostructured magnetic materials where the magnetic properties can be tailored to a particular application with a very high degree of precision. *** This work was supported by the Royal Society, St. NANOWIRES ESPRIT.
John’s College, Cambridge and
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