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Journal of Magnetism and Magnetic Materials 283 (2004) 1–7

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Exact activation energy of magnetic single domain particles

Daniel Brauna,b,�

aInfineon Technologies, 2070 State Route 52, Hopewell Junction, NY 12533, USAbMRAM Developement Alliance, IBM/Infineon Technologies, IBM SemiconductorResearch and Developement Center,

2070 State Route 52, Hopewell Junction, NY 12533, USA

Received 5 December 2003; received in revised form 27 April 2004

Available online 9 June 2004

Abstract

I present the exact analytical expression for the activation energy as a function of externally applied magnetic fields

for a single-domain magnetic particle with uniaxial anisotropy (Stoner–Wohlfarth model), and investigate the scaling

behavior of the activation energy close to the switching boundary.

r 2004 Elsevier B.V. All rights reserved.

PACS: 75.75.+a; 75.45.+j; 75.10.�b

Keywords: Magnetic particle; Activation energy; Stoner-Wohlfarth model

1. Introduction

A lot of effort has been spent over the last few years to understand the magnetization reversal of smallmagnetic particles [1–9]. At sufficiently low temperatures, macroscopic quantum tunneling has beenobserved [10–13], while for higher temperatures thermally activated behavior may switch the magnetizationof the particle [14]. For the analysis of the experiments one needs to know the activation energy. Moreover,with the advancing developement of integrated magnetoresistive memory devices (MRAM) the dependenceof the energy barrier on the magnetic field has become of crucial technological importance as well. In atypical MRAM array, magnetic memory cells are written by a coincident field technique, where both aselected bitline (BL) and a selected wordline (WL) create magnetic fields, the sum of which are strongenough to switch the memory cell, whereas the fields from either BL or WL alone are not sufficient toswitch the cell. Nevertheless, these fields do destabilize the non-selected cells to some extent, i.e. they reduce

- see front matter r 2004 Elsevier B.V. All rights reserved.

/j.jmmm.2004.05.012

miconductor, MRAM, Bd. John F. Kennedy, 91105 Corbeil Essonne, France. Tel.: +33-1-60-88-55-86; fax: +33-1-60-88-

ddresses: daniel.braun@infineon.com, daniel.braun@altissemiconductor.com (D. Braun).

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D. Braun / Journal of Magnetism and Magnetic Materials 283 (2004) 1–72

the energy barrier against thermally activated switching. Also, even for the selected cells the switching atfinite temperatures happens before the actual zero temperature boundary of stability is reached, again dueto thermal activation during the finite duration of a write pulse [14,15]. To estimate the life time of theinformation in the memory, one needs to know the dependence of the energy barrier on the applied fields asprecisely as possible, as the energy barrier enters the switching rate exponentially.In the study of the switching behavior of small size magnetic particles, the Stoner–Wohlfarth model plays

a central role [16]. It describes a single domain particle with uni-axial anisotropy in an external magneticfield. The single domain approximation greatly simplifies the analysis, and becomes a good approximationif the size of the particle becomes comparable to or smaller than the exchange length, which in memoryelements etched out of a thin magnetic film is typically of the order of 100 nm. The activation energy in theStoner–Wohlfarth model can be calculated trivially if the external field is aligned either parallel to thepreferred axis or perpendicular to it. However, so far no analytical solution has been known for the generalcase of the external field pointing in an arbitrary direction [17]. Given the crucial importance of the fielddependence of the activation energy, an exact analytical solution will be provided in the present paper.

2. The Stoner–Wohlfarth model

The energy of a uniformly magnetized particle with uniaxial symmetry, characterized by the anistropyenergy density K and saturation magnetization Ms depends on its magnetization via the angle W betweenthe magnetization and the preferred axis,

EðWÞ ¼ KVsin2 W� VMsHx cos W

� VMsHy sin W; (1)

where Hx and Hy are the magnetic field components parallel and perpendicular to the preferred axis in aplane containing the magnetization and the preferred axis, respectively, and V denotes the volume of thesample [16]. In the following dimensionless variables will be used by writing the energy in terms of 2KV ,eðWÞ ¼ EðWÞ=ð2KV Þ, and the magnetic field components in terms of the switching field Hc ¼ 2K=Ms ashx ¼ Hx=Hc, hy ¼ Hy=Hc, such that

eðWÞ ¼ 12sin2 W� hx cos W� hy sin W: ð2Þ

For vanishing magnetic fields, this model has a bistable ground state, whereas for very large fields (hx�1 orhy�1) the first term can be neglected, and there is only one minimum in the interval �ppWop, such thatthe magnetization tends to align parallel to the applied field. Thus, for increasing field, one of the originalminima has to disappear, and the fields where this happens mark the boundary of bistability. These fieldsare easily obtained by setting both the first and second derivative of (2) to zero and eliminating W,whereupon the famous Stoner–Wohlfahrt astroid hx ¼ cos3 W, hy ¼ sin3W is obtained [16].

3. Activation energy

The activation energy can in principle be calculated in a straight forward manner by finding the twominima and the two maxima of the energy in the bistability range, determining the metastable of the twominima and the energy barrier to the smaller of the two maxima.1 In the case of hx ¼ 0 or hy ¼ 0 this is

1Note that in principle there are two energy barriers, corresponding to the two escape paths out of the metastable minimum by

clockwise or counterclockwise rotation of the magnetization. In the following the path with the higher activation energy will be

neglected and the activation energy will always be defined as the smaller of the two energy barriers.

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D. Braun / Journal of Magnetism and Magnetic Materials 283 (2004) 1–7 3

straight forward: For hy ¼ 0,

qe

qW¼ cos W sin Wþ hx sin W� hy cos W ¼ 0 ð3Þ

leads to W ¼ 0, W ¼ p, or cosW ¼ �hx. The solution W ¼ 0 is metastable for�1phxp0, stable for hx4 0, andunstable for hxo � 1. Correspondingly, W ¼ p is metastable for 0phxo 1, stable for hxo 0, and unstablefor hx4 1. The solution cosW ¼ �hx leads to two maxima which become complex for jhxj4 1. Thus, theenergy barrier for switching from W ¼ p to 0 is given by EA ¼ eðarccosð�hxÞÞ � eðpÞ ¼ ð1� hxÞ

2=2.Similarly, for hx ¼ 0 one finds the energy barrier EA ¼ ð1� hyÞ

2=2. Therefore, if one of the two fieldcomponents vanishes, the activation energy depends quadratically on the distance from the stabilityboundary.In the general case, where neither hx nor hy vanish, one may substitute sin W ¼ u, cos W ¼ �

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� u2

pinto

Eq. (3). Squaring the equation leads to

� u4 þ 2u3hy þ u2ð1� h2x � h2yÞ

� 2uhy þ h2y ¼ 0: (4)

In order to calculate the energy barrier, one needs to find the roots of this fourth-order polynomial, whichmakes the analysis much more cumbersome than for hxhy ¼ 0. Still, the roots of a fourth-order polynomialcan be obtained analytically. The solutions Wi are conveniently written in terms of the functions

f 1 ¼ h2x þ h2y � 1; (5)

f 2 ¼ 108h2xh2y þ 2f 31; (6)

f 3 ¼ f 2 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 22 � 4f 61

q; (7)

f 4 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ

f 13þ21=3

3

f 21

f1=33

þ1

3

f1=33

21=3� h2y

vuut : (8)

Eq. (4) has four solutions for u, but the ambiguity in the sign of cos W in terms of u leads at this point toeight solutions for W. They differ by three signs m; n;s in various places, and are given by

Wi ¼ s arccos �hx

2þ

m2

f 4 þn2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1�

f 13�21=3f 21

3f1=33

�1

3

f1=33

21=3þ h2x � h2y þ

2mf 4

ð2hx þ f 1hx � h3xÞ

vuut24

35: ð9Þ

The signs will be chosen according to the binary decomposition ðm; n;sÞ ¼ ð�;�;�Þ for W1, ð�;�;þÞ for W2,ð�;þ;�Þ for W3; . . . ; ðþ;þ;þÞ for W8. None of these solutions solves Eq. (3) for all hx; hy. Rather, each ofthem solves the equation only in two quadrants. However, it is possible to construct four uniform solutionsout of the eight partially valid ones, which are continuous (after the identification of p with �p) and solve(3) for all values of hx; hy. The four uniform solutions are

~W1 ¼ yðhyÞW2 þ yð�hyÞW1; (10)

~W2 ¼ yðhxhyÞW5 þ yð�hxhyÞW3; (11)

~W3 ¼ yðhxhyÞW4 þ yð�hxhyÞW6; (12)

~W4 ¼ yðhyÞW8 þ yð�hyÞW7; (13)

where yðxÞ ¼ 1 for x4 0 and zero elsewhere denotes the Heaviside function. Calculating the secondderivative one finds that the first uniform solution is a minimum for hxo 0. It disappears as real solution forhx4 0 outside the astroid. Thus, this is the metastable state for hx4 0. On the other hand, ~W4 remains a

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D. Braun / Journal of Magnetism and Magnetic Materials 283 (2004) 1–74

minimum even outside the astroid for hx4 0, but disappears as real solution outside the astroid for hxo 0.The uniform solutions ~W2 and ~W3 are both maxima inside the astroid, and become complex outside one halfof the astroid (for hyo 0 and hy4 0, respectively). One easily convinces oneself that for hyX0, ~W3 is therelevant maximum (eð ~W3Þpeð ~W2Þ), whereas for hyo 0, escape over the maximum at ~W2 is dominant(eð ~W3Þ4 eð ~W2Þ).The activation energy out of the metastable state is therefore given by

EA ¼ eð ~W3Þ � eð ~W1Þ for hyX0; (14)

EA ¼ eð ~W2Þ � eð ~W1Þ for hyo 0; (15)

Fig. 1 shows a plot of the activation energy as function of hx and hy. In Fig. 2 the contours of constantactivation energy are plotted. This plot can be compared directly to the curves of temperature dependentswitching fields measured in Ref. [18] in the positive half-plane.

Fig. 1. Activation energy as function of hx and hy, out of the state stable for hxo 0. Outside the astroid there is only one stable and no

metastable state, and the activation energy is therefore undefined.

Fig. 2. Contourplot of the activation energy as function of hx and hy, out of the state stable for hxo 0 with 15 equidistant contours

between EA ¼ 0 and 2. The thick line denotes the Stoner–Wohlfarth astroid.

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For hy ¼ 0 both maxima lead to the same activation energy, and EA is an even function of hy, as it shouldbe. In the following the attention will therefore focus solely on the case hyX0, and to the case where theinitial state W ¼ p is metastable, hx4 0, i.e. the first quadrant. In principle, the particle might be excited alsoout of the stable state and end up (for a finite time) in the metastable state, but this process is of much lessimportance, as the activation energy is much larger and it enters exponentially in the thermal switching rate.

4. Scaling

The scaling of the activation energy as function of the distance 1� h from the astroid boundary is ofparticular interest. Here, h is defined by hx ¼ h cos3 x, hy ¼ h sin3 x such that h ¼ 1 always corresponds tothe astroid boundary. It has been shown before [19] that the scaling must be of the form

EA ¼ c3=2ðxÞð1� hÞ3=2 þ c2ðxÞð1� hÞ2

þ c5=2ðxÞð1� hÞ5=2 þ : (16)

All coefficients besides c2ðxÞ vanish for x ¼ 0, but it has been shown numerically [19] and theoretically [20]that already for small values of x the coefficient c3=2 dominates the scaling. This is confirmed and mademore precise by using the exact solution. Fig. 3 shows lnEA as function of lnð1� hÞ. The plot reveals power

Fig. 3. Scaling of the activation energy EA as function of the relative distance h from the astroid boundary (hx ¼ h cos3 x,hy ¼ h sin3 x). The curves correspond to monotonously decreasing and equidistant values of x ¼ x=p from x ¼ 0 (bottom curve), 1

24,

224; . . . up to x ¼ 1

4(top curve). The dashed line corresponds to a power law with exponent 3

2, the dot–dashed line to an exponent 2.

Fig. 4. Scaling exponent a as a function of the astroid parameter x ¼ x=p extracted from a fit of lnEA to a lnð1� hÞ, using the exact

expression for EA, Eq. (15). The three curves correspond to fitting ranges h ¼ 0:93187079 to 0:996837 (top curve), h ¼ 0:99538 to

0:999978 (middle curve), and h ¼ 0:9996837 to 0:999998532 (bottom curve). For all calculations of the activation energy, 80 digits

precision was used.

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Fig. 5. Scaling exponent a as a function of the astroid parameter x ¼ x=p extracted from a fit of lnEA to a lnð1� hÞ þ b ln2ð1� hÞ. The

three curves correspond to the same fitting ranges used in Fig. 4. The right curve corresponds to a fitting range from h ¼ 0:93187079 to0:996837, the left curve to a fitting range from h ¼ 0:9996837 to 0:999998532.

D. Braun / Journal of Magnetism and Magnetic Materials 283 (2004) 1–76

law behavior to a good approximation even rather far away from the astroid boundary, EA / ð1� hÞa withan exponent a ¼ 2 for x ¼ 0, and an exponent close to 3

2for larger values of x.

According to the numerical evaluation of Eq. (15) the exponent is symmetric with respect to x ¼ p=4. Theexact value of the exponent depends on the fitting range and on whether a quadratic term is included in thefit of lnEA as function of lnð1� hÞ. Fig. 4 shows the fitted exponent a as a function of x ¼ x=p for threedifferent fitting ranges, from h ¼ 0:932 to 0:99999853, assuming a pure power law, lnEA ¼ a lnð1� hÞ. Theobserved dependence of the exponent on x is very similar to what was previously calculated numerically[19]. In particular, the exponent appears to become even slightly smaller than 3

2for values of x close to p=4.

However, there is a substantial non-linear part in the scaling behavior, as becomes obvious when fittingto lnEA ¼ a lnð1� hÞ þ b ln2ð1� hÞ. The extracted linear part is plotted in Fig. 5. For small values of x, theexponent can now be substantially larger than 2.

5. Summary

I have derived the exact analytical expression of the activation energy for single domain switching ofsmall magnetic particles in arbitrary magnetic fields (Stoner–Wohlfarth model). The activation energyscales approximately like a power law as a function of the distance of the switching boundary (astroid) upto distances of order unity, but also contains a substantial non-power law term.

Acknowledgements

I am grateful to Daniel Worledge for a useful discussion.

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