Domain structures and switching mechanisms in patterned magnetic elements

ELSEVIER

Journal of Magnetism and Magnetic Materials 175 (1997) 193 204

Journalof magnetism ~ l ~ and magnetic ~ l ~ materials

Domain structures and switching mechanisms in patterned magnetic elements Thomas

S c h r e f l a'*, J o s e f F i d l e r a, K . J . K i r k bx, J . N . C h a p m a n a

alnstitute of Applied and Technical Physics, Vienna University of Technology, A-1040 Wien, Austria bDepartment of Physics and Astronomy, University of Glasgow, Glasgow, G12 8QQ, UK CDepartment of Electronics and Electrical Engineering, University of Glasgow, Glasgow, G12 8QQ, UK

Abstract Domain formation and magnetisation reversal in lithographically fabricated magnetic elements 10(~300 nm wide and 1.5~4.0 lam long have been investigated using Lorentz imaging and finite element micromagnetics. The numerical integration of the Gilbert equation of motion resolves magnetisation processes in time and in space. The calculated domain patterns are in qualitative agreement with magnetic images obtained from Lorentz electron microscopy. NiFe elements show a small scale domain structure in the remanent state which may be attributed to a transverse anisotropy. In bars with one pointed end, the formation of the domains starts from the flat ends. Narrow elements with a width smaller than 200 nm remain in a nearly single domain state. Pointed ends suppress the formation of domains in NiFe elements and increase the switching field by about a factor of two in Co elements. PACS: 75.60

Keywords: Micromagnetics; Finite element techniques; Lorentz electron microscopy; Domain wall motion; Magnetisation reversal; Magnetic nano-elements

1. Introduction The development of high resolution magnetic imaging and the decreasing costs of powerful computational resources have led to a rapid growth of experimental and computational micromagnetics [1]. D o m a i n formation and the specific domain structure will be important in the future applications of patterned media such as quantum magnetic * Corresponding author. Tel.: + 43-1-58801-5618; fax: + 431-5868814; e-mail: [email protected].

disks [2] and magnetic sensors [-3]. The influence of shape on domain formation was previously studied in magnetoresistive permalloy elements with a width of several micrometers. Domain observations using the longitudinal magneto-optical Kerr effect [-4] showed that pointed ends suppress the formation of end domains and thus increase the stability of the high-remanence state which is desired for sensor operation. As the size of the elements decreases into the nanometer regime, a detailed comparison of observed and simulated domain processes becomes possible assuming the

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72 Schrefl et al. / Journal of Magnetism and Magneti~ Materials 175 (1997) 193 204

very same dimensions of the particle in the experiments and in the micromagnetic simulations. These co-operative studies considerably improve the basic understanding of magnetisation processes in magnetic nano-elements. Smyth and co-workers [-5] prepared arrays of permalloy particles and investigated the effect of particle size and aspect ratio on the hysteresis experimentally and numerically. They reported an increase in the coercive field as the particle width decreases below 300 nm. Gabois and Zhu [-6] demonstrated numerically that edge roughness reduces the switching field of nano-scale Ni bars. This work shows the effects of bar width and of tip shape on domain formation and magnetisation reversal in acicular NiFe and Co elements using Lorentz imaging and finite element micromagnetics. The continuum theory of micromagnetism [7] provides the basis for the theoretical treatment of magnetisation reversal. Stable equilibrium magnetic states follow from the minimisation of the total magnetic Gibb's free energy with respect to the direction of the magnetisation vector. The magnetisation is assumed to be a continuous function of space with constant magnitude. The path the magnetisation moves from its initial state towards equilibrium is given by the Gilbert equation of motion [--8]. Rigorous solutions of the micromagnetic equations exist only for ideally shaped particles. In nonellipsoidal particles the demagnetising field becomes inhomogeneous leading to a highly non-linear system of partial differential equations. A semi-analytic treatment is possible for the domain wall dynamics in thin films infinitely extended in the direction normal to the wall [9 I. In ultra-thin films an approximate solution for the equilibrium domain structure is given by the analytic many soliton solutions of the imaginary time sine-Gordon equation [10]. However, the calculation of domain patterns and of magnetisation reversal for realistic particle shapes and dimensions requires numerical techniques. Yan and Jurisch [11] derived possible domain patterns in NiFe films minimising the total energy on a 2-dimensional rectangular computational grid. Koehler and Fredkin used the finite element method to calculate the static equilibrium states of 2 : 1 [-12] and 8 : 1 [-13] aspect ratio permalloy thin films at different

applied fields during magnetisation reversal. They reported an end domain in the remanent state of elongated bars. The end domains reduce the magnetostatic energy and are the starting points for magnetisation reversal. McMicheal and Donahue [14] calculated the wall structure of head-to-head domain walls which occur during magnetisation reversal in acicular magnetic elements. In this paper, we compare experimentally observed magnetic images with micromagnetic simulations using the very same particle shapes and dimensions. By Lorentz imaging in the transmission electron microscope (TEM) we have observed details of the magnetisation reversal mechanisms in acicular elements 10~300 nm wide and 1.5-4.0 lam long. The time evolution of magnetic domain patterns is calculated using a hybrid finite element/boundary element technique. The numerical solution of the Gilbert equation of motion reveals the static equilibrium domain structures as well as the transient magnetic states during magnetisation reversal. Comparison of the observed and calculated domain patterns helps to understand how bar width and tip shape influence the domain structure and the switching mechanism. In Sections 2 and 3, details are given of the fabrication process and the TEM techniques used for the magnetic experiment, followed by our experimental results in Section 4. Section 5 describes the micromagnetic and numerical background. Section 6 presents micromagnetic simulations of domain formation in NiFe elements and of magnetisation reversal in Co elements.

2. Fabrication of the elements

The magnetic elements were fabricated from randomly oriented polycrystalline films of NisoFe20 (NiFe) and Co 20-50 nm thick by electron beam lithography and lift-off patterning. Electron transparent substrates were used, consisting of a 35-50 nm thick Si3N4 membrane window of area 100 x 100 lam2, supported on 400 ~tm thick Si [,15]. After patterning, the elements could be examined in the TEM with no further substrate thinning. The patterning process is described in more detail in Refs. [15, 16]. First the substrate was coated in polymethyl-methacrylate (PMMA) electron beam

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195

resist: a double layer was used in order to produce an undercut profile, and the total thickness was 80-100 rim, allowing films up to 50 nm thick to be lifted off. Then the P M M A was exposed in a LeicaCambridge BeamWriter using a 100 keV electron beam with a spot size of 12 nm and an exposure dose ~ 15 C m - 2 . After developing the pattern in I P A - M i B K (3 : 1) solution, the films were deposited by thermal evaporation at deposition rates of ~0.1 nm s-1. The P M M A was removed by immersing the substrates in acetone to lift off the excess magnetic material and finally the samples were carbon coated to prevent the Si3N4 from charging in the TEM. Bright field T E M images of three NiFe elements are shown in Fig. 1.

3. Magnetic imaging in the TEM Both the Fresnel and the Foucault modes of Lorentz microscopy [17] were useful for determining the magnetic state of the elements. Fresnel imaging was most commonly used for in-situ magnetising experiments. In this mode the domain walls appear as narrow dark and bright bands on a neutral background. The Foucault mode reveals the direction of magnetisation within the domains, relative to some mapping direction, by contrasting bright and dark shades, and in addition shows any stray fields outside the elements. Two T E M s were used, both specially modified for magnetic imaging, in particular by having the specimen located in field-free space. The J E O L 2000FX has a non-immersion objective pole-piece [18], and the Philips CM20 (with a field emission gun) has two additional lenses which are used for imaging when the standard objective lens is switched off for field-free domain structure observations [19]. The Philips CM20 is especially useful for magnetising experiments since it is possible to activate the objective lens to provide a known vertical field in the sample region which will have a variable component in the plane of the elements if the sample is tilted. In this way the elements can be taken through a magnetisation cycle without altering the field in the microscope and therefore the imaging conditions [-20], and changes in the magnetic structure can be observed as they happen.

/

200nm

Fig. 1. Bright field T E M images of 200 nm wide acicular elements fabricated by lift-off:

4. Experimental observations Initial experiments were carried out to study how the magnetic properties vary with the width and length of elements [16]. Elements with widths 100, 200 and 300 nm and lengths between 600 nm and 4.2 tam were made from NiFe (21 + 1) nm and Co (27 _ 2) nm thick. Each element had one end either curved or pointed and the other end flat. Fig. 2 shows Foucault images and schematics of the domain structure in Co and NiFe elements in the remanent state after a large field was applied

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7( Schrefl et al. / Journal q/Magnetism and Magnetic" Materials 175 (1997) 193 204

(b)

7

l

S ,I.--

(d) Fig. 2. Foucault images (with mapping directions shown by arrows) and schematics of the domain structure in the remanent state in 300 nm wide elements of Co (a), (b) and NiFe (c), (d).

along the long axis. In each of the Co elements (Fig. 2a) a single domain occupied most of the volume but very close to the flat end there was a complex small-scale domain structure leading to a degree of flux closure, shown in more detail in Fig. 2b. Strong stray field contrast can be seen outside the elements. In 200 and 300 nm wide NiFe elements, a flux closure structure of small domains extended throughout the element and there was a complete absence of stray field contrast (Fig. 2c). The 100 nm wide NiFe elements appeared to have a single domain occupying the bulk of the element in a similar way to the Co elements. The flux closure structure in the wider NiFe elements is shown schematically in Fig. 2d. Such a density of domain walls was unexpected in NiFe elements of this size, and it is most likely that the observed structure, in which the largest domains support transverse magnetisation, was a result of transverse anisotropy created by stress release in the patterned

(b) Fig. 3. Co element switching by means of a travelling "reversing" structure: (a) Eoucault image with mapping direction shown, note that the magnetisation in the element and the stray field point in opposite directions on either side of the reversing structure (fringes in the stray field are an artefact of the imaging process), (b) schematic of magnetisation distribution.

film which may have been off the ideal non-magnetostrictive composition [21, 22]. In the magnetising experiments, field was applied in the easy axis direction, parallel to the long axis of the elements. Switching fields in the Co elements increased markedly with decreasing element width but were essentially independent of element length. The reversal mechanism was found to involve the propagation of a small 'reversing' domain structure along the length of the element, always starting from the flat end where it branched off from the pre-existing end domains. Fig. 3 shows a Foucault image of a Co element in the process of switching, together with a schematic of the magnetisation

1~ Schrefl et al. /Journal qf Magnetism and Magnetic: Materials 175 ('1997) 193 204

197

AI. A

0

1

2

L

(e)

I, W Fig. 5. G e o m e t r y of elements with zero, one and two flat ends: L = 2.5 or 1.6 I-tm, W = 200 nm, P = 500 nm.

(e)

(f) m

,1,

.

~

t

$

t

~

m

t

(g) Fig. 4. D o m a i n structures in a 300 n m wide N i F e element during m a g n e t i s a t i o n reversal: (a)Hd) F o u c a u l t images ( m a p p i n g direction shown), (e) (g) schematics.

distribution in the reversing structure. It is likely that the initial formation of the reversing structure is more difficult in narrower elements thus accounting for the increase in switching field with decreasing width. The reversal process in NiFe elements was observed directly in real time. Fig. 4a Fig. 4d show Foucault images of stages in the reversal process with Fig. 4e-Fig. 4g showing schematics of the

domain structures involved. In the first image the domain structure develops from one end of the element. Development proceeds over a very small field range, from one or both ends of the element (Fig. 4b), until the pattern occupies the entire element volume. At zero field there is complete flux closure as already described (Fig. 4c and Fig. 4f), then increasing the reverse field leads to growth of the favourably oriented domains (Fig. 4d and Fig. 4g). Beyond this stage, domains begin to disappear, the ends of the element generally being the last to be affected. Since the ends of the elements played such an important role in magnetisation reversal, we investigated 200 nm wide elements with either two flat ends, one flat end and one pointed end, or two pointed ends [23,24], as seen earlier in Fig. 1. Fig. 5 gives the element geometries. In the remanent state the rectangular elements were predominantly magnetised along the long axis but had small flux closure domains at the ends. However if the element had two pointed ends these end domains were not present at remanence and the element appeared to be single domain.

198

T. Schrefl et al. / Journal of Magnetism and Magnetic" Materials 175 (1997) 193 204 700 600

~500 ----- 400 .~ 300

5 200 o~ 100

i

! 0

1 number of flat ends

2

Fig. 6. Dependence of the switching field of elements on the number of flat ends. During magnetisation reversal, elements with flat ends first underwent a reversible phase during which new domain walls branched off from the end domains; then the magnetisation flipped irreversibly at the switching field. This mechanism was not possible in the elements with two pointed ends. Instead they reversed suddenly after a gradual intensification of magnetisation ripple, especially near to regions A and A'. The switching field was strongly affected by the number of flat ends, as shown in Fig. 6. Elements with two pointed ends have approximately twice the coercivity of elements with at least one flat end, because the partial flux closure structure enables switching to occur at a lower field than is possible for single domain elements. Having two flat ends rather than one slightly reduces the coercivity since there is twice as much chance for reversal to begin at a particular field.

5. Mieromagnetic and computational background The magnetic behaviour in acicular magnetic elements results from the correlation between the intrinsic magnetic properties, the size and shape of the elements, and the dynamics of domain formation. The theoretical treatment of magnetisation reversal dynamics requires the solution of the Gilbert equation [8] ~J ~t

-

17lJ × Heft

+

~ ./sOt

~J

(1)

which describes the physical path of the magnetic polarisation J towards equilibrium. Here 7 is the gyromagnetic ratio of the free electron spin and ~ is the Gilbert damping constant. The effective field Heff, given by the negative variational derivative of the total magnetic Gibb's free energy, is the sum of the exchange field, the anisotropy field, the external field, and the demagnetising field [7]. The demagnetising field Hd arises from the divergence of the magnetic polarisation within the magnet and on its boundary. It follows from a magnetic scalar potential U by Ha = - V U , where U obeys Poisson's equation within the magnetic region, ~'~m, and Laplace's equation outside the magnetic particle, Qc: vZU(r) _ VJ(r) Ito V2U(r) = 0

forreQm,

(2)

for r ~ f2e.

(3)

On the boundary of the magnetic particle with unit normal n, the boundary conditions e in = e °ut,

( V U in - - V U ° U t ) ' n

-

j.n

(4)

tto hold. Eqs. (2) and (4) can be effectively solved by applying a hybrid finite element/boundary element method. Fredkin and Koehler [253 proposed to split the magnetic scalar potential into U = U1 + U2, where U1 accounts for the divergence of J and U2 is required to meet the boundary condition at the surface of the particle. The potential U1 is evaluated by a standard finite element formulation, whereas the potential U2 is derived using boundary elements. The magnetic bar, which may be of arbitrary shape, is divided into tetrahedral finite elements. Fig. 7 shows the finite element mesh of an acicular magnetic particle with one pointed end. Within a tetrahedron the direction cosines of the magnetisation are interpolated by linear functions whereas quadratic functions represent the magnetic potentials. The use of an individual order of interpolation for the different components, popular in computational fluid dynamics, is called the mixed finite element method [263. It adapts the finite element subspaces to the physical nature of the problem. The demagnetising field arising from magnetic volume charges varies linearly with the magnetic

Z Schrefl et al./Journal of Magnetism and Magnetic Materials 175 (1997) 193-204

199

to a fully populated matrix involving the nodes at the surface of the particle. In order to keep the system matrix sparse, Uz is kept constant during the Newton iterations and is updated after each time step. This so-called incomplete Newton method [28] allows larger time steps as compared to fully explicit methods for the time integration of the Gilbert equation.

6. Numerical results

•

magnetic polarization:

131,132, 133 q = J'~13) scalar potential: U 1 and U 2

Fig. 7. Mixed finite elements. The bar is divided into tetrahedrons. Only the triangles at the surface are visible. The tetrahedton on the right shows the interpolation scheme. Dots: Nodes for the linear interpolation of the direction cosines of the magnetisation. Crosses: Nodes for the quadratic interpolation of the magnetic potentials.

polarisation. In order to avoid a loss accuracy with differentiation, the interpolation order of the scalar potential should be higher than that of the magnetic polarisation. The Galerkin method transforms the Gilbert Eq. (1) and Poisson's equation for U1 into a system of algebraic differential equations. Its time integration is performed using backward differentiation formulas with variable step size and variable order [273. At each time step the direction cosines of the magnetisation and the potential U1 follow from the solution of a non-linear system of equations. The boundary element discretisation of the integral equation giving U2 leads

The observed remanent domain patterns in the 300 nm wide NiFe elements, presented in Fig. 2c and in Fig. 4c are comparable to the domain structure of the Landau Lifshitz type, indicating a transverse anisotropy. The ratio, Ku/A, of the uniaxial anisotropy constant Ku and the exchange constant A can be derived from the experimentally observed domain width D and the bar width W [29]: Ku

W 2

A - 64 O~w.

(5)

According to Eq. (5) different combinations of Ku and A are possible. However, numerical tests showed that the magnetisation becomes nonuniform in the direction normal to the film plane if an exchange constant smaller than A = 10 -11 J/m is assumed, leading to so-called chess board domains [30]. Table 1 shows the intrinsic magnetic properties and the dimensions of the acicular elements used for the calculations. The polycrystalline microstrueture was not taken into account in the micromagnetic simulations. Whereas the NiFe elements show a transverse anisotropy, the anisotropy direction is assumed parallel to the long axis in the Co bars. In order to calculate the remanent domain patterns, a procedure similar to the experiments was applied numerically. First, the samples were saturated along the long-axis. Then, the Gilbert equation of motion was solved for zero applied field to describe the time evolution of the magnetic domain structure. The grey-scale plots of the domain patterns, given in Fig. 8, show the influence of the bar width on the formation of domains in NiFe elements. The magnetisation rotates out of the long

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200

Table 1 Shape, dimensions, and intrinsic magnetic properties of the NiFe and Co elements used for the micromagnetic simulations. The length L. the width W and the length of the pointed end P refer to Fig. 5, giving the geometry of the bars. The thickness of all elements was t = 21 nm../~, Ku, and A denote the spontaneous magnetic polarisation, the uniaxial anisotropy constant, and the exchange constant, respectively Pointed ends

L (bLm)

W (pro)

P (~_tm)

Material

.1~ (T)

Ku (kJ/m 3)

A (pJ:m)

1 2 3 4 5 6

1.6 1.6 1.6 1.6 1.6 1.6

0.1 0.2 0.2 0.2 0.2 0.2

0 0 0.5 0.5 0 0.5

NilCe NiFe NiFe NiFe Co Co

1 1 1 1 1.76 1.76

38 (transverse} 38 (transverse) 38 (transverse) 38 (transverse) 450 (longitudinal) 450 (longitudinal)

10 10 10 10 13 13

0.8 ns

1.5 ns

0 0 1 2 0 2

0.1 ns

2 0 ns

1.0

¢0 .N L-

-6 0.5

I 3

Q.,. .13 0")

E

i 0.0 t0

Fig. 8. Influence of bar width on the remanent magnetic state: Time evolution of magnetic domain patterns for two NiFe elements with a width of 100 and 200 n m (damping constant

5

10 time (ns)

15

20

Fig. 9. Magnetic polarisation parallel to the long axis as a function of time for a NiFe element with two flat ends, bar width, W = 200 nm: damping constant. :~ - 1. The numbers refer to the domain patterns given in Fig. 10.

~=1).

axes near the corners of the wide and the narrow bar. These rotations of the magnetisation significantly suppress the magnetic surface charges at the blunt ends and thus decrease the magnetostatic energy [31]. When the expense of exchange energy in the 18@' walls between the two corner domains becomes too high, symmetry breaks, leading to a single end domain at each of the blunt ends. The

nearly uniform magnetic state remains stable in the 100nm wide bar, whereas a multidomaim state forms in the 200 nm wide bar. Fig. 9 gives the total magnetic polarisation parallel to the long axis as a function of time for a NiFe bar with two flat end and a width of W = 200 nm. The Gilbert damping constant was :~ = 1. Fig. 10 shows the corresponding domain patterns. This sequence of domain patterns may be compared with the in-situ domain observations during the

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T. Schrefl et al. / Journal (?,/'Magnetism and Magnetic Materials' 175 (1997) 193 204

1

2

3

4

5

0.2 ns

0.6 ns

1.0 ns

Fig. 10. Domain patterns as a function of time for a NiFe element with two flat ends, bar width, W - 200 nm; damping constant, :~ = 1.

Fig. 11. Influence of damping on domain formation: NiFe element, two flat ends, bar width. W = 200 nm. The domain patterns refer to a Gilbert damping constant of ~ = 0.05 (left-hand side) and ~ = 1 (right-hand side).

reversal of the NiFe elements given in Fig. 4a-Fig. 4c. The formation of domains starts at the corners. Then the domain structure develops from the two blunt ends and proceeds towards the center of the bar. This process leads to a zigzag pattern showing domains with the magnetisation to the left and to the right, and triangular closure domains with the magnetisation parallel to its original direction. This intermediate d o m a i n structure corresponds to the schematics of Fig. 4e. After the two patterns merge in the middle of the bar the total magnetic polarisation parallel to the long axis remains constant for several nanoseconds. This waiting period is associated with the formation of closure domains with the magnetisation antiparallel to the saturation direction (transition from the d o m a i n structures of Fig. 4e to the d o m a i n structure of Fig. 4f). The micromagnetic simulation closely resembles the development of the domains observed in the experiment. The remaining discrepancies in the remanent d o m a i n structure may be attributed to neglecting the polycrystalline microstructure and to the rather crude assumption of the intrinsic magnetic properties.

In the micromagnetic simulations the Gilbert d a m p i n g constant determines the time scale. F o r the sake of numerical stability a d a m p i n g constant c~ = 1 was used for most of the calculations. Fig. 11 shows the influence of d a m p i n g on the time evolution of the domains. With decreasing d a m p i n g constant the d o m a i n wall mobility increases [9]. Thus the d o m a i n structure develops much faster for a smaller Gilbert d a m p i n g constant. The domain patterns, given in Fig. 12, show that a pointed end suppresses the formation of domains. Fig. 12 compares the time evolution of d o m a i n patterns for NiFe elements with one and two pointed ends. The bar width and the Gilbert d a m p ing constant were W = 200 nm and ~ = 0.05, respectively. In the element with one pointed end, the formation of the multidomain structure starts from the flat end. The element with two pointed ends remains in a single d o m a i n state. However, near the corners the magnetisation slightly rotates out of the saturation direction up to an angle of a b o u t 10. In the rightmost pattern given in Fig. 12 the colour m a p has been changed, in order to resolve these deviations of the magnetisation from the long axis.

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Z Schrefi et al. /"Journal of Magnetism and Magnetic Materials 175 (1997) 193 204

-6 0 ._o

g fi

3

-2

-800

a

i

i

[

,

-600 -400 -2;0 external field (kA/m)

,

m

0

Fig. 13. Numerically calculated demagnetisation curves of Co clements with two pointed ends and two flat ends, respectively. The numbers refer to the domain patterns given in Fig. 14. The arrows indicate the direction of uniaxial anisotropy.

0.15

0.3

0.5 1 time (ns)

1

Fig. 12. Influence of tip shape on domain formation. Left: Time evolution of domain patterns in NiFe elements with one pointed end. Right: Virtually single domain state in a NiFe element with two pointed ends. In the right most picture the colour map was changed, in order to visualise the rotations of the magnetisation out of the long axis with angles up to 10.

Experimentally, the switching field of acicular Co elements nearly doubles in the bar with two pointed ends as compared to switching field of the element with no pointed ends. The very same dependence of the switching fields of Co elements on the tip shape was found in the micromagnetic simulations. Fig. 13 gives the numerically calculated demagnetisation curves for Co elements with two and zero pointed ends, respectively. The demagnetisation curves follow from the time integration of the Gilbert equation using a time dependent external field. The assumption of a uniaxial anisotropy parallel to the long axis, an unrealistic high sweeping rate of the external field, and overdamping (:~ = 1) drastically overestimates the switching field in the micromagnetic simulations as compared to the experimental results. However, the calculations qualitatively agree with the experiments and thus help to

understand the switching mechanism. The influence of tip shape on the switching field can be explained by the effect of corners on the demagnetising field. Fig. 14 gives the domain patterns at different stages during magnetisation reversal. The nucleation of reversed domains starts near the corners of the elements where a transverse demagnetising field occurs. Near the apex of the pointed ends, the demagnetising field arising from the magnetic surface charges at the two inclined faces compensate resulting in a zero field component parallel to the short axis. However, a transverse component of the demagnetising field occurs at the corners forming the pointed end. These transverse fields near the regions denoted by A and A' in Fig. 5 cause magnetisation ripple favouring the nucleation of reversed domains. The high transverse component of the demagnetising field at flat ends initiates the formation of end domains in zero applied field. As a consequence, Co elements with at least one flat end show a reduced switching field. Fig. 14 shows that reversed domains nucleate at the corners and expand into the interior of the particle. During magnetisation reversal, head to head domain walls form and move towards the center of the bar. The character of the calculated domain wall corresponds with the domain walls found in Lorentz electron microscopy studies of magnetisation reversal in Co bars (Fig. 3).

71 Sehrefl et al. / Journal of Magnetism and Magnetic Materials 175 (1997) 193-204

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2

3 1

2

3

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the nucleation of reversed domains. Micromagnetic models based on the Gilbert equation of motion resolve magnetisation processes in space and in time. In addition to the hysteresis properties, the micromagnetic simulations predict where the reversed domains nucleate and how they expand with time.

Acknowledgements We thank the following for their collaboration on this work: B. Khamsehpour and C.D.W. Wilkinson for fabrication of elements, M. Rtihrig and P. Aitchison for microscopy, and A. Hubert for useful discussions on the experimental results. We acknowledge the support of UK EPSRC and of the Austrian Science Foundation (Grant No. P10511NAW).

References

Fig. 14. Influence of tip shape on magnetisation reversal of Co elements: domain patterns at different points (see Fig. 13) of the demagnetisation curve.

7. Conclusion The comparison of high resolution magnetic imaging using Lorentz electron microscopy and dynamic micromagnetic modelling provides a precise understanding of domain formation and of the switching mechanism in patterned magnetic elements. Small scale domains in the remanent state of NiFe elements can be explained by a uniaxial anisotropy parallel to the short axis. Micromagnetic simulations successfully reproduce the influence of bar width and tip shape on the domain structure. The dependence of the switching field of Co elements on the shape of the ends has to be attributed to the demagnetising field at the corners causing

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