- Email: [email protected]
Magnetic modelling at nanoscale level
Journal of Materials Processing Technology 153–154 (2004) 785–790
Magnetic modelling at nanoscale level A. Boboc a,b , I.Z. Rahman a,b,∗ , M.A. Rahman a,c a
Material and Surface Science Institute (MSSI), University of Limerick, National Technological Park, Limerick, Ireland b Department of Physics, University of Limerick, Limerick, Ireland c Department Electronic and Computer Engineering, University of Limerick, Limerick, Ireland
Abstract With the development of the information technology in all areas of present life, high-density data storage devices are in great demand. Currently, the bit dimension is below a micron and in the future will approach the nanoscale level. At present, not many theoretical approaches are available to explain the magnetic information storage properties at nanoscale level, as there is a lack of experimental data for example. This is due to difficulty in implementing measurement techniques for magnetic properties at this low scale. In this article, we have presented a short review of the most used methods to represent the physical behaviour at microscale level. Our goal was to apply some of these methods at a nanoscale range, where the physical properties change dramatically and various physical effects cannot be neglected that are not considered at micron scale. Some of the results regarding the simulation of magnetic properties of the nanostructured patterned media and comparison with experimental data are also presented. © 2004 Elsevier B.V. All rights reserved. Keywords: Patterned media; Magnetic recording; Nanomagnetic structures; Magnetic modelling
1. Introduction In the last two decades, the information technology has an exponential development. Newer technologies are being developed and magnetic recording technology has the same trend. But with miniaturisation of electronics and recording devices the technology has arrived at a point where many of the existing models and techniques are no longer applicable. Currently, patterned magnetic media are showing promises for realisation of ultra high-density data storage capacity. One method for increasing the recording data density is to reduce the elementary cell for recording (bit). At the nanosize recording bit level, some effects cannot be neglected and many others need to be taken into account. Magnetic recording media based on patterned nanonetworks [1] or the novel “nanodrive technology” appear to be very promising. The existing micromagnetic modelling system, however, need to be adapted to handle the nanoscale level in order to incorporate the changes in physical behaviour of the system. This paper focuses on description of computational problems related to various aspects that need to be taken into account when developing new models of magnetic materials at nanoscale level. Also, a description of the model to simulate magnetic properties at nanoscale has been presented. Some ∗ Corresponding author. Tel.: +353-61-202205; fax: +353-61-202423. E-mail address: [email protected] (I.Z. Rahman).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.04.108
results that are obtained by simulating magnetic behaviour of nanopatterned magnetic media and compared with experimental results are discussed.
2. Basics for magnetic recording At present the recording media consist of a continuous thin ferromagnetic film supported by a rigid and non-magnetic substrate. This type of media shows many tiny polycrystalline grains with a rather broad distribution in size and shape and a random distribution of crystallisation direction. During writing, write head aligns the otherwise randomly oriented grains into tiny patches. The data is represented by the direction of magnetic moment, area, size and location of this patch. Increasing the areal density of recording media demands the ability to store a magnetic transition or, a bit in the medium at a length as small as possible. To maintain it at a stable state the smallest bit length a, that can be sustained in a given media is given by a=
Mr δ 2πHc
(1)
where Hc is the coercivity of the media, δ the thickness of the magnetic thin film and Mr is the remanence magnetisation. To increase data storage density, bits to be stored in a given length of the medium should allow smaller transition
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length than the exiting media. This leads to a smaller distance between the magnetic reversals that produce strong demagnetising fields on the recorded bit. To enable the stored bit to be stable, the coercive force, Hc , therefore, must be high enough to counteract these demagnetising effects. The remanence and thickness product, Mr δ must also be small to make the transition length a smaller. At the same time high remanence is required to ensure sufficient stray magnetic field in the medium, which is required to have sufficient readout signal. To get high enough remanence, high saturation magnetisation Ms is required that increases the demagnetising field and necessitates still higher coercivity. The magnetic properties of nanometre sized particles or crystallites differ from the properties of their bulk counterparts, as a large fraction of atoms are located on the surface and the interfaces. So, the size dependence of magnetisation, anisotropy, Curie transition, coercivity and remnant magnetisation is expected. This creates a physical obstacle to high-density recording that is the super paramagnetic limit, for a typical magnetic media this is limited to a minimum particle size of about 10 nm. For a grain size below this limit, the orientation of the magnetisation changes because of the thermal fluctuations.
3. Modelling methods Existing micromagnetic techniques are essentially a continuum approximation which allows the calculation of magnetisation structures and magnetisation reversal assuming the magnetisation to be a continuous function of position, and deriving relevant expressions for the important contributions arising from exchange, magnetostatic and anisotropy energies. Currently, atomistic simulation methods can be used to study systems containing hundreds of thousands of atoms, but these systems are still orders of magnitude too small to describe macroscopic behaviour. Continuum methods, typically using finite element methods (FE) [2] fail to adequately describe many important properties because the methods use phenomenology that has little connection to the real physical processes that govern physical and magnetic interactions. Modelling at an intermediate length scale, the mesoscale where, many defects can be included and from which predictive models at the continuum scale can be developed. At this intermediate length-scale it is necessary to model the collective phenomena that include well over a billions of atoms. In order to understand the macroscopic behaviour of materials, studies on size effects are necessary and a comparison between experimental results and computational methodologies need to be made. This is vital in linking disparate length scales and creating a scientifically rigorous understanding of materials performance and behaviour. In magnetism two main groups of methods are used for modelling: • probabilistic methods; • analytical methods.
Table 1 Methods used in micromagnetism Task and methods
References
Discretisation scheme Constant cubic element (FE) Linear cubic (FE) Linear triangular/tetrahedral (FE)
[3] [4] [5]
Stray field calculation Direct evaluation FFT methods Magnetic scalar potential (FE) Magnetic vector potential (FE)
[3,5] [6] [4] [7]
Minimisation Landau–Lifshitz–Gilbert (LLG) equation Gauss–Seidel methods Newton methods Conjugate gradient based method
[5,8,9] [10] [11] [4,5]
3.1. Probabilistic methods One of the probabilistic methods is the statistical Monte Carlo (MC) method. In magnetism, the MC method is implemented by using random numbers to simulate statistical fluctuations in order to generate the correct thermodynamical probability distributions. With this method one may obtain information about complex systems that cannot be solved analytically. The purpose of the MC can be either to compare a specific model with real experiments, or to compare its results with analytical theories starting with the same model but using approximations during analytical treatment. In addition, one is able to obtain microscopic information on the system, which may not be accessible in a real experiment. 3.2. Analytical methods Analytical methods are used for describing the magnetic properties of the materials and there are many approaches, some are shown in Table 1. Each of these methods has advantages and disadvantages and can be applicable for certain classes of magnetic materials and magnetic configurations. The simplest flow diagram to implement a modelling algorithm by involving analytical methods is shown in Fig. 1. Each of these terms contained in the diagram in Fig. 1 can be defined by using various methods as described in Table 1. One of the methods that we have employed in our simulations is the finite elements (FE) method for discretisation and stray field calculation. This is a widely used numerical
Fig. 1. Simplest magnetic modelling algorithm.
A. Boboc et al. / Journal of Materials Processing Technology 153–154 (2004) 785–790
787
method for finding approximate values of solutions involving partial differential equations. The basic idea of the method consists of approximating the partial derivative of a function u(r, t) by a finite difference (FD) quotients x, y, z and t: u(x + x, y, z, t) = u(x, y, z, t) + x
∂u(x, y, z, t) + ··· ∂x (2)
The process of replacing partial derivatives by FD quotients is known as a discretisation process and the associated error is the discretisation error. A partial differential equation can be changed to a system of algebraic equation by replacing the partial derivative in the differential equation with their FE approximations. An iterative process can solve the system of algebraic equations numerically in order to obtain an approximate solution.
4. Modelling method used The software developed in the present investigation is based on the free available object oriented micromagnetic framework (OOMMF) codes [8], a public domain micromagnetics program developed at the National Institute of Standards and Technology, USA. The software has an object oriented programming (OOP) structure, which means that improvements of the code do not affect the previous developments. Maintenance of the code is easy. Graphical interface is in TCL/Tk language and the core part is written in C++ language. The method to solve the magnetisation dynamics in time is carried out by implementing a first order forward Euler method with step size control on the Landau–Lifshitz ordinary differential equation [9]. The magnetisation of the sample is computed as a function of time using damped Landau–Lifshitz–Gilbert equation of motion as follows: M M dM α dM = − |γ| M × H eff + M× dt MS dt
(3)
The first term of Eq. (3) describes the precession of the magnetisation vector M about the effective field and the second term describes its dissipation. The constant γ is the gyromagnetic ratio, Ms is the saturation magnetisation and H eff is the effective field that consists of the superposition of the external field and contributions from anisotropy, exchange and demagnetisation fields. The dissipation or magnetic damping is described by the dimensionless constant, α called the Gilbert damping parameter. Conventional terms H eff ) and enfor the various contributions to the local field (H ergy are included in the calculation. The exchange energy is computed via a six neighbour dot products and the magnetostatic energy is calculated using a fast Fourier transform (FFT) based scalar potential. This code has been used for magnetic simulations of a number of magnetic configu-
Fig. 2. Nanonetwork media: (a) negative structure; (b) positive structure.
rations but primarily for magnetic simulations of patterned magnetic nanonetwork media.
5. Results and discussion Porous anodised alumina (commonly known as nanochannel alumina (NCA)) can be used as a template to fabricate nanostructured materials [1]. Commercially available NCA templates come in threes different nominal pore sizes, e.g. 20, 100 and 200 nm in which nanowires [12] can be grown using electrodeposition or the template can be sputtered with magnetic material to form positive or negative nanonetwork [13], respectively, as described in Fig. 2. In the case of negative nanonetworks, a thin film surrounds the non-magnetic pores and in the case of positive nanonetwork the pores are filled with magnetic material as shown in Fig. 2. Various theoretical studies and the magnetic representation of the patterned magnetic nanostructured networks media based on NCA have been reported [1]. For modelling, structural information was extracted from non-uniformly distributed pores of the NCA. The geometrical information has been obtained by using the images generated from SEM analysis of a fabricated sample in the laboratory as shown in Fig. 3(a). Fig. 3(b) shows the high contrast image in black and white of the SEM pattern. This generated image from the real template provides true information about the pore geometry to be used by the code for the distribution of magnetic regions. 5.1. Magnetic behaviour of the Ni nanowires grown inside pores of NCA templates The measurement of the hysteresis curve (dependence of the magnetisation with the applied magnetic field) is one of the most used techniques for characterisation of the magnetic properties of materials. From the hysteresis curve important parameters such as remanent magnetisation or magnetic coercivity can be recovered. In our laboratory, experiments have been carried out to characterise the magnetic behaviour of Ni nanowires grown inside NCA pores that can be referred as positive nanonetwork media. A vibrating sample magnetometer (VSM) has been employed to trace the hysteresis curve of samples with positive and negative nanostructures. A field is applied to a sample between maximum
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Fig. 3. Nickel nanowires grown using electrodeposition method inside NCA templates: (a) SEM image of the Ni wires; (b) high contrast image in black and white of SEM image. Table 2 Input parameters for nickel Anisotropy constant (K1 ) Crystal planes Exchange coefficient (Ax ) Damping constant (α) Wire lengths range
75 kJ m−3 [1 1 1]/[2 0 0] 0.245 pJ 0.5 10–12 m
and minimum values in steps and for each steps the magnetisation and applied field values are stored in a computer for further analysis and graphics. The simulation of the magnetic behaviour of Ni nanowires inside NCA pores with nominal diameter of 200 and 20 nm (pore distribution obtained from SEM image) is presented in Fig. 4, which has been compared with the experimental results. The basic magnetic parameters to calculate magnetisation of the samples as a function of applied field are shown in Table 2. The external magnetic field was applied in steps between minimum and maximum values of ±550 mT in order to trace the hysteresis loop. For every applied field value, the equilibrium configuration of the system has been obtained after a certain number of computer iterations. The study was carried out for two applied field directions to the wires axis: (a) perpendicular; (b) parallel. Fig. 4(a) and (b) shows the two examples of the hysteresis curve for Ni wires in NCA templates with nominal pore sizes of 200 and
20 nm, respectively. The normalised value of magnetisation (ratio of magnetisation, M of the sample to the saturation magnetisation, Ms ) is plotted as a function of applied field. For comparison, in this case, only applied magnetic field perpendicular to wire axis has been considered. Agreement between the simulation results and the experimental values is very good in case of samples where, nanowires are grown inside 200 nm nominal pore size templates. The match degrades in case of 20 nm nominal pore size templates. This is due to the non-linear effects that are present at a very low scale and non-uniformity of the nanowires that are grown inside the pores. It is reported [14] that the commercially available NCA templates (AnodiscTM 13) with nominal 20 nm pore size show variation of pores on filtration and reverse sides ranging from average size of 29–201 nm, respectively. There is also branching inside the pore. Therefore, for simulation purpose, the distance of separation between the wires and the shape of the wires do not comply with the practical samples. This type of pore distribution causes strong magnetic interaction between the parallel wires due to the high pore density that influences saturation magnetisation and coercivity values. Magnetic properties are strongly influenced by the dimension and crystal properties, which again depend on the physical structure of the templates and the growth mechanism of the wires [15]. For each values of the applied magnetic field, contributions
Fig. 4. Hysteresis curves of Ni nanowires array grown inside NCA template pores: (a) NCA200 nm; (b) NCA20 nm.
A. Boboc et al. / Journal of Materials Processing Technology 153–154 (2004) 785–790
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Fig. 5. Distribution of energies: (a) exchange; (b) demagnetisation.
from the internal energies due to anisotropy, exchange and demagnetisation and Zeeman effect to the effective magnetisation value of the sample can be calculated. This is very useful in order to understand the dominating effect of various energies at a certain value of the applied field or about the effects that can cause a degradation of magnetisation of the sample. Fig. 5 shows an example of the role of various energy effects that occur during a hysteresis loop tracing. For each value of the applied magnetic field a value of the exchange and demagnetisation effect (expressed in energy terms) is calculated and then graphically represented. These energies affect the total energy of the sample and change the magnetic dynamics that influences the overall magnetisation behaviour. 5.2. Variation of coercivity (Hc ) as a function of length of Ni wires inside NCA templates For this study an ideal distribution of pores that are located at the centre of the hexagonal alumina matrix has been considered. Ni nanowires array of diameter 200 nm arranged at 200 nm separation distance are chosen. For calculations magnetic parameters are obtained from Table 2. Simulations were performed for various wire lengths and the external magnetic field was applied parallel to the wires axis direc-
tion. Statistical results are represented in Fig. 6 that shows a strong dependence of the coercivity with the wire length below 10 m. It has been reported that with an aspect ratio of ∼10 (aspect ratio is defined as ratio between wire length and its diameter), individual Ni nanowire behaves as a single domain particle [12], when the magnetic field is applied parallel to the wire axis. Wires with aspect ratio below 8, show rapid decrease in coercivity. Whereas, for higher aspect ratio wires above 10, multi-domain configuration appears, further increase in aspect ratio does not affect coercivity. 5.3. Variation of coercivity (Hc ) with thickness of iron negative nanonetworks on NCA This study has been carried for negative nanonetwork media. These samples are obtained by depositing magnetic materials by sputtering technique on the surface of NCA templates in order to obtain a negative nanonetwork. The depositions are carried by varying various deposition parameters such as deposition time, substrate temperature bias voltage, etc. In this case, we have chosen a batch sample that are grown by varying deposition time in order to vary the thicknesses of the deposited materials and then characterised the magnetic properties using the VSM technique as described in Section 5.1. In this paper, we are presenting the study of a negative iron nanonetworks with nominal pore size of 200 nm template and simulation parameters as shown in Table 3. The external magnetic field is applied in steps between minimum and maximum values ±0.3 T in order to trace the hysteresis loop. For the simulation purpose, ideal structure of the NCA with 200 nm pore size has been considered. The simulations are performed for various thicknesses ranging from Table 3 Input parameters for iron
Fig. 6. Coercivity (Hc ) dependence with Ni nanowire length grown inside NCA templates.
Anisotropy constant (K1 ) Crystal planes Exchange coefficient (Ax ) Damping constant (α) Discretisation cell
45 kJ m−3 [1 0 0]/[0 0 1] 21 pJ 0.5 2 nm × 2 nm × 2 nm
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A. Boboc et al. / Journal of Materials Processing Technology 153–154 (2004) 785–790
looks promising. The study has been performed on a number of magnetic configurations (negative and positive nanonetworks or nanowires). Simulation results and experimental measurements are in good agreement qualitatively as shown in Section 5.1, for example. In some cases availability of more experimental data and considerations of nanoscopic properties are necessary to improve the model in the near future.
Acknowledgements
Fig. 7. Variation of Hc with film thickness for iron nanonetworks.
2 to 30 nm. The results are summarised in Fig. 7 where, the plot of the simulated and the experimental values of the coercivity are shown. It can be observed that the two sets of data are qualitatively in good agreement for nearly all the ranges of thickness and especially at a thickness of 5 and 24 nm. The experimental coercivity values are measured on samples that showed partial oxidation of the iron due to deposition on the alumina matrix [13]. Also, XRD results on these samples show presence of various types of cubic structure of iron oxides depending on the thickness of the deposited layer. It is well known that the iron shows single domain size of about 5 nm. So, when film thickness is in the range of 5 nm, one can expect single domain formation at the surface. With film thickness above 20 nm, the classical effects can play a major role in determining the magnetisation dynamics. At an intermediate scale, 5–20 nm and below 5 nm various other size effects can play a dominant role, e.g. resonance effects, quantum tunnelling and long range exchange interaction [16]. These effects are presently under study with a multi-scale approach such as ab-initio method and macroscopic study using OOMMF package and will be published in future communications.
6. Conclusions Simulation of magnetic properties at nanoscale range using an analytical approach from micromagnetic concept
This project is funded by FP5 Marie Curie Host Development Fellowship award (Contract No. HPMD-CT-200000045). The authors would like to thank K.M. Razeeb and S.A.M. Tofail for providing some of the experimental results.
References [1] S.A.M. Tofail, Z. Rahman, M.A. Rahman, Appl. Org. Met. Chem. 5 (2001) 373–382. [2] J. Fiddler, T. Schrefl, J. Appl. Phys. 33 (2000) 135–156. [3] M.E. Schabes, J. Magn. Mater. 95 (1991) 249–288. [4] A. Bagneres-Viallix, P. Baras, J.B. Albertini, IEEE Trans. Magn. 27 (1991) 3319–3321. [5] W. Chen, D.R. Redkin, T.R. Koehler, IEEE Trans. Magn. 29 (1993) 2124. [6] D.V. Berkov, K. Ramstöck, A. Hubert, Phys. Status Solidi A 137 (1993) 207–225. [7] D. R Fredkin, IEEE Trans. Magn. 23 (1987) 3385–3390. [8] M.J. Donahue, D.G. Porter, NISTIR 6376 NIST, Gaithersburg, MD, 1999. [9] R.D. McMichael, M.J. Donahue, IEEE Trans. Magn. 33 (1997) 4167– 4169. [10] E. Della Torre, IEEE Trans. Magn. 21 (1985) 1423–1425. [11] H.F. Schmidts, J. Magn. Magn. Mater. 130 (1994) 329– 341. [12] J. Sellmyer, M. Zheng, R. Skomski, J. Phys. Condens. Matter 13 (2001) 433–480. [13] S.A.M. Tofail, I.Z. Rahman, M.A. Rahman, K.U. Mahmood, Chem. Monthly 133 (2002) 859–872. [14] K.M. Razeeb, Electrodeposition of magnetic nanowires in NCA templates, PhD Thesis, University of Limerick, Limerick, Ireland, 2003. [15] I.Z. Rahman, K.M. Razeeb, M.A. Rahman, Md. Kamruzzaman, J. Magn. Magn. Mater. 262 (1) (2003) 166–169. [16] R.H. Brown, D.M.C. Nicholson, T.C. Schulthess, X.-D. Wang, J. Appl. Phys. 85 (1999) 4830.
Magnetic modelling at nanoscale level A. Boboc a,b , I.Z. Rahman a,b,∗ , M.A. Rahman a,c a
Material and Surface Science Institute (MSSI), University of Limerick, National Technological Park, Limerick, Ireland b Department of Physics, University of Limerick, Limerick, Ireland c Department Electronic and Computer Engineering, University of Limerick, Limerick, Ireland
Abstract With the development of the information technology in all areas of present life, high-density data storage devices are in great demand. Currently, the bit dimension is below a micron and in the future will approach the nanoscale level. At present, not many theoretical approaches are available to explain the magnetic information storage properties at nanoscale level, as there is a lack of experimental data for example. This is due to difficulty in implementing measurement techniques for magnetic properties at this low scale. In this article, we have presented a short review of the most used methods to represent the physical behaviour at microscale level. Our goal was to apply some of these methods at a nanoscale range, where the physical properties change dramatically and various physical effects cannot be neglected that are not considered at micron scale. Some of the results regarding the simulation of magnetic properties of the nanostructured patterned media and comparison with experimental data are also presented. © 2004 Elsevier B.V. All rights reserved. Keywords: Patterned media; Magnetic recording; Nanomagnetic structures; Magnetic modelling
1. Introduction In the last two decades, the information technology has an exponential development. Newer technologies are being developed and magnetic recording technology has the same trend. But with miniaturisation of electronics and recording devices the technology has arrived at a point where many of the existing models and techniques are no longer applicable. Currently, patterned magnetic media are showing promises for realisation of ultra high-density data storage capacity. One method for increasing the recording data density is to reduce the elementary cell for recording (bit). At the nanosize recording bit level, some effects cannot be neglected and many others need to be taken into account. Magnetic recording media based on patterned nanonetworks [1] or the novel “nanodrive technology” appear to be very promising. The existing micromagnetic modelling system, however, need to be adapted to handle the nanoscale level in order to incorporate the changes in physical behaviour of the system. This paper focuses on description of computational problems related to various aspects that need to be taken into account when developing new models of magnetic materials at nanoscale level. Also, a description of the model to simulate magnetic properties at nanoscale has been presented. Some ∗ Corresponding author. Tel.: +353-61-202205; fax: +353-61-202423. E-mail address: [email protected] (I.Z. Rahman).
0924-0136/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jmatprotec.2004.04.108
results that are obtained by simulating magnetic behaviour of nanopatterned magnetic media and compared with experimental results are discussed.
2. Basics for magnetic recording At present the recording media consist of a continuous thin ferromagnetic film supported by a rigid and non-magnetic substrate. This type of media shows many tiny polycrystalline grains with a rather broad distribution in size and shape and a random distribution of crystallisation direction. During writing, write head aligns the otherwise randomly oriented grains into tiny patches. The data is represented by the direction of magnetic moment, area, size and location of this patch. Increasing the areal density of recording media demands the ability to store a magnetic transition or, a bit in the medium at a length as small as possible. To maintain it at a stable state the smallest bit length a, that can be sustained in a given media is given by a=
Mr δ 2πHc
(1)
where Hc is the coercivity of the media, δ the thickness of the magnetic thin film and Mr is the remanence magnetisation. To increase data storage density, bits to be stored in a given length of the medium should allow smaller transition
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length than the exiting media. This leads to a smaller distance between the magnetic reversals that produce strong demagnetising fields on the recorded bit. To enable the stored bit to be stable, the coercive force, Hc , therefore, must be high enough to counteract these demagnetising effects. The remanence and thickness product, Mr δ must also be small to make the transition length a smaller. At the same time high remanence is required to ensure sufficient stray magnetic field in the medium, which is required to have sufficient readout signal. To get high enough remanence, high saturation magnetisation Ms is required that increases the demagnetising field and necessitates still higher coercivity. The magnetic properties of nanometre sized particles or crystallites differ from the properties of their bulk counterparts, as a large fraction of atoms are located on the surface and the interfaces. So, the size dependence of magnetisation, anisotropy, Curie transition, coercivity and remnant magnetisation is expected. This creates a physical obstacle to high-density recording that is the super paramagnetic limit, for a typical magnetic media this is limited to a minimum particle size of about 10 nm. For a grain size below this limit, the orientation of the magnetisation changes because of the thermal fluctuations.
3. Modelling methods Existing micromagnetic techniques are essentially a continuum approximation which allows the calculation of magnetisation structures and magnetisation reversal assuming the magnetisation to be a continuous function of position, and deriving relevant expressions for the important contributions arising from exchange, magnetostatic and anisotropy energies. Currently, atomistic simulation methods can be used to study systems containing hundreds of thousands of atoms, but these systems are still orders of magnitude too small to describe macroscopic behaviour. Continuum methods, typically using finite element methods (FE) [2] fail to adequately describe many important properties because the methods use phenomenology that has little connection to the real physical processes that govern physical and magnetic interactions. Modelling at an intermediate length scale, the mesoscale where, many defects can be included and from which predictive models at the continuum scale can be developed. At this intermediate length-scale it is necessary to model the collective phenomena that include well over a billions of atoms. In order to understand the macroscopic behaviour of materials, studies on size effects are necessary and a comparison between experimental results and computational methodologies need to be made. This is vital in linking disparate length scales and creating a scientifically rigorous understanding of materials performance and behaviour. In magnetism two main groups of methods are used for modelling: • probabilistic methods; • analytical methods.
Table 1 Methods used in micromagnetism Task and methods
References
Discretisation scheme Constant cubic element (FE) Linear cubic (FE) Linear triangular/tetrahedral (FE)
[3] [4] [5]
Stray field calculation Direct evaluation FFT methods Magnetic scalar potential (FE) Magnetic vector potential (FE)
[3,5] [6] [4] [7]
Minimisation Landau–Lifshitz–Gilbert (LLG) equation Gauss–Seidel methods Newton methods Conjugate gradient based method
[5,8,9] [10] [11] [4,5]
3.1. Probabilistic methods One of the probabilistic methods is the statistical Monte Carlo (MC) method. In magnetism, the MC method is implemented by using random numbers to simulate statistical fluctuations in order to generate the correct thermodynamical probability distributions. With this method one may obtain information about complex systems that cannot be solved analytically. The purpose of the MC can be either to compare a specific model with real experiments, or to compare its results with analytical theories starting with the same model but using approximations during analytical treatment. In addition, one is able to obtain microscopic information on the system, which may not be accessible in a real experiment. 3.2. Analytical methods Analytical methods are used for describing the magnetic properties of the materials and there are many approaches, some are shown in Table 1. Each of these methods has advantages and disadvantages and can be applicable for certain classes of magnetic materials and magnetic configurations. The simplest flow diagram to implement a modelling algorithm by involving analytical methods is shown in Fig. 1. Each of these terms contained in the diagram in Fig. 1 can be defined by using various methods as described in Table 1. One of the methods that we have employed in our simulations is the finite elements (FE) method for discretisation and stray field calculation. This is a widely used numerical
Fig. 1. Simplest magnetic modelling algorithm.
A. Boboc et al. / Journal of Materials Processing Technology 153–154 (2004) 785–790
787
method for finding approximate values of solutions involving partial differential equations. The basic idea of the method consists of approximating the partial derivative of a function u(r, t) by a finite difference (FD) quotients x, y, z and t: u(x + x, y, z, t) = u(x, y, z, t) + x
∂u(x, y, z, t) + ··· ∂x (2)
The process of replacing partial derivatives by FD quotients is known as a discretisation process and the associated error is the discretisation error. A partial differential equation can be changed to a system of algebraic equation by replacing the partial derivative in the differential equation with their FE approximations. An iterative process can solve the system of algebraic equations numerically in order to obtain an approximate solution.
4. Modelling method used The software developed in the present investigation is based on the free available object oriented micromagnetic framework (OOMMF) codes [8], a public domain micromagnetics program developed at the National Institute of Standards and Technology, USA. The software has an object oriented programming (OOP) structure, which means that improvements of the code do not affect the previous developments. Maintenance of the code is easy. Graphical interface is in TCL/Tk language and the core part is written in C++ language. The method to solve the magnetisation dynamics in time is carried out by implementing a first order forward Euler method with step size control on the Landau–Lifshitz ordinary differential equation [9]. The magnetisation of the sample is computed as a function of time using damped Landau–Lifshitz–Gilbert equation of motion as follows: M M dM α dM = − |γ| M × H eff + M× dt MS dt
(3)
The first term of Eq. (3) describes the precession of the magnetisation vector M about the effective field and the second term describes its dissipation. The constant γ is the gyromagnetic ratio, Ms is the saturation magnetisation and H eff is the effective field that consists of the superposition of the external field and contributions from anisotropy, exchange and demagnetisation fields. The dissipation or magnetic damping is described by the dimensionless constant, α called the Gilbert damping parameter. Conventional terms H eff ) and enfor the various contributions to the local field (H ergy are included in the calculation. The exchange energy is computed via a six neighbour dot products and the magnetostatic energy is calculated using a fast Fourier transform (FFT) based scalar potential. This code has been used for magnetic simulations of a number of magnetic configu-
Fig. 2. Nanonetwork media: (a) negative structure; (b) positive structure.
rations but primarily for magnetic simulations of patterned magnetic nanonetwork media.
5. Results and discussion Porous anodised alumina (commonly known as nanochannel alumina (NCA)) can be used as a template to fabricate nanostructured materials [1]. Commercially available NCA templates come in threes different nominal pore sizes, e.g. 20, 100 and 200 nm in which nanowires [12] can be grown using electrodeposition or the template can be sputtered with magnetic material to form positive or negative nanonetwork [13], respectively, as described in Fig. 2. In the case of negative nanonetworks, a thin film surrounds the non-magnetic pores and in the case of positive nanonetwork the pores are filled with magnetic material as shown in Fig. 2. Various theoretical studies and the magnetic representation of the patterned magnetic nanostructured networks media based on NCA have been reported [1]. For modelling, structural information was extracted from non-uniformly distributed pores of the NCA. The geometrical information has been obtained by using the images generated from SEM analysis of a fabricated sample in the laboratory as shown in Fig. 3(a). Fig. 3(b) shows the high contrast image in black and white of the SEM pattern. This generated image from the real template provides true information about the pore geometry to be used by the code for the distribution of magnetic regions. 5.1. Magnetic behaviour of the Ni nanowires grown inside pores of NCA templates The measurement of the hysteresis curve (dependence of the magnetisation with the applied magnetic field) is one of the most used techniques for characterisation of the magnetic properties of materials. From the hysteresis curve important parameters such as remanent magnetisation or magnetic coercivity can be recovered. In our laboratory, experiments have been carried out to characterise the magnetic behaviour of Ni nanowires grown inside NCA pores that can be referred as positive nanonetwork media. A vibrating sample magnetometer (VSM) has been employed to trace the hysteresis curve of samples with positive and negative nanostructures. A field is applied to a sample between maximum
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Fig. 3. Nickel nanowires grown using electrodeposition method inside NCA templates: (a) SEM image of the Ni wires; (b) high contrast image in black and white of SEM image. Table 2 Input parameters for nickel Anisotropy constant (K1 ) Crystal planes Exchange coefficient (Ax ) Damping constant (α) Wire lengths range
75 kJ m−3 [1 1 1]/[2 0 0] 0.245 pJ 0.5 10–12 m
and minimum values in steps and for each steps the magnetisation and applied field values are stored in a computer for further analysis and graphics. The simulation of the magnetic behaviour of Ni nanowires inside NCA pores with nominal diameter of 200 and 20 nm (pore distribution obtained from SEM image) is presented in Fig. 4, which has been compared with the experimental results. The basic magnetic parameters to calculate magnetisation of the samples as a function of applied field are shown in Table 2. The external magnetic field was applied in steps between minimum and maximum values of ±550 mT in order to trace the hysteresis loop. For every applied field value, the equilibrium configuration of the system has been obtained after a certain number of computer iterations. The study was carried out for two applied field directions to the wires axis: (a) perpendicular; (b) parallel. Fig. 4(a) and (b) shows the two examples of the hysteresis curve for Ni wires in NCA templates with nominal pore sizes of 200 and
20 nm, respectively. The normalised value of magnetisation (ratio of magnetisation, M of the sample to the saturation magnetisation, Ms ) is plotted as a function of applied field. For comparison, in this case, only applied magnetic field perpendicular to wire axis has been considered. Agreement between the simulation results and the experimental values is very good in case of samples where, nanowires are grown inside 200 nm nominal pore size templates. The match degrades in case of 20 nm nominal pore size templates. This is due to the non-linear effects that are present at a very low scale and non-uniformity of the nanowires that are grown inside the pores. It is reported [14] that the commercially available NCA templates (AnodiscTM 13) with nominal 20 nm pore size show variation of pores on filtration and reverse sides ranging from average size of 29–201 nm, respectively. There is also branching inside the pore. Therefore, for simulation purpose, the distance of separation between the wires and the shape of the wires do not comply with the practical samples. This type of pore distribution causes strong magnetic interaction between the parallel wires due to the high pore density that influences saturation magnetisation and coercivity values. Magnetic properties are strongly influenced by the dimension and crystal properties, which again depend on the physical structure of the templates and the growth mechanism of the wires [15]. For each values of the applied magnetic field, contributions
Fig. 4. Hysteresis curves of Ni nanowires array grown inside NCA template pores: (a) NCA200 nm; (b) NCA20 nm.
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Fig. 5. Distribution of energies: (a) exchange; (b) demagnetisation.
from the internal energies due to anisotropy, exchange and demagnetisation and Zeeman effect to the effective magnetisation value of the sample can be calculated. This is very useful in order to understand the dominating effect of various energies at a certain value of the applied field or about the effects that can cause a degradation of magnetisation of the sample. Fig. 5 shows an example of the role of various energy effects that occur during a hysteresis loop tracing. For each value of the applied magnetic field a value of the exchange and demagnetisation effect (expressed in energy terms) is calculated and then graphically represented. These energies affect the total energy of the sample and change the magnetic dynamics that influences the overall magnetisation behaviour. 5.2. Variation of coercivity (Hc ) as a function of length of Ni wires inside NCA templates For this study an ideal distribution of pores that are located at the centre of the hexagonal alumina matrix has been considered. Ni nanowires array of diameter 200 nm arranged at 200 nm separation distance are chosen. For calculations magnetic parameters are obtained from Table 2. Simulations were performed for various wire lengths and the external magnetic field was applied parallel to the wires axis direc-
tion. Statistical results are represented in Fig. 6 that shows a strong dependence of the coercivity with the wire length below 10 m. It has been reported that with an aspect ratio of ∼10 (aspect ratio is defined as ratio between wire length and its diameter), individual Ni nanowire behaves as a single domain particle [12], when the magnetic field is applied parallel to the wire axis. Wires with aspect ratio below 8, show rapid decrease in coercivity. Whereas, for higher aspect ratio wires above 10, multi-domain configuration appears, further increase in aspect ratio does not affect coercivity. 5.3. Variation of coercivity (Hc ) with thickness of iron negative nanonetworks on NCA This study has been carried for negative nanonetwork media. These samples are obtained by depositing magnetic materials by sputtering technique on the surface of NCA templates in order to obtain a negative nanonetwork. The depositions are carried by varying various deposition parameters such as deposition time, substrate temperature bias voltage, etc. In this case, we have chosen a batch sample that are grown by varying deposition time in order to vary the thicknesses of the deposited materials and then characterised the magnetic properties using the VSM technique as described in Section 5.1. In this paper, we are presenting the study of a negative iron nanonetworks with nominal pore size of 200 nm template and simulation parameters as shown in Table 3. The external magnetic field is applied in steps between minimum and maximum values ±0.3 T in order to trace the hysteresis loop. For the simulation purpose, ideal structure of the NCA with 200 nm pore size has been considered. The simulations are performed for various thicknesses ranging from Table 3 Input parameters for iron
Fig. 6. Coercivity (Hc ) dependence with Ni nanowire length grown inside NCA templates.
Anisotropy constant (K1 ) Crystal planes Exchange coefficient (Ax ) Damping constant (α) Discretisation cell
45 kJ m−3 [1 0 0]/[0 0 1] 21 pJ 0.5 2 nm × 2 nm × 2 nm
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looks promising. The study has been performed on a number of magnetic configurations (negative and positive nanonetworks or nanowires). Simulation results and experimental measurements are in good agreement qualitatively as shown in Section 5.1, for example. In some cases availability of more experimental data and considerations of nanoscopic properties are necessary to improve the model in the near future.
Acknowledgements
Fig. 7. Variation of Hc with film thickness for iron nanonetworks.
2 to 30 nm. The results are summarised in Fig. 7 where, the plot of the simulated and the experimental values of the coercivity are shown. It can be observed that the two sets of data are qualitatively in good agreement for nearly all the ranges of thickness and especially at a thickness of 5 and 24 nm. The experimental coercivity values are measured on samples that showed partial oxidation of the iron due to deposition on the alumina matrix [13]. Also, XRD results on these samples show presence of various types of cubic structure of iron oxides depending on the thickness of the deposited layer. It is well known that the iron shows single domain size of about 5 nm. So, when film thickness is in the range of 5 nm, one can expect single domain formation at the surface. With film thickness above 20 nm, the classical effects can play a major role in determining the magnetisation dynamics. At an intermediate scale, 5–20 nm and below 5 nm various other size effects can play a dominant role, e.g. resonance effects, quantum tunnelling and long range exchange interaction [16]. These effects are presently under study with a multi-scale approach such as ab-initio method and macroscopic study using OOMMF package and will be published in future communications.
6. Conclusions Simulation of magnetic properties at nanoscale range using an analytical approach from micromagnetic concept
This project is funded by FP5 Marie Curie Host Development Fellowship award (Contract No. HPMD-CT-200000045). The authors would like to thank K.M. Razeeb and S.A.M. Tofail for providing some of the experimental results.
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