Nano-Magnetism of Magnetostriction in Fe35Co65
A. Lisfi, T. Ren, A. Khachaturyan and M. Wuttig
SUPPORTING DATA
I. ANALYSIS OF THE TORQUE DATA OF FE35CO65
Here, we show the results of the more detailed analysis of the torque curves of Fe35Co65. The
black dots, •••••, represent data and the solid red line, ‒‒‒, the curves fitted as a sum of a shape
anisotropy of twofold symmetry, a contribution stemming from the two sets of mutually
orthogonal uniaxial precipitates plus the magnetocrystalline anisotropy of the cubis host crystal.
Fig. S1a: Data taken at 600G. The shape anisotropy of twofold symmetry predominates,
Fig. S1c: Data taken at 1000G. The second torque
featuring a non-monotonous torque becomes noticeable at 10o, 100 o, 190 o and 280 o. It is ascribed to
the uniaxial precipitates. The complicated shape
originates from the instability which occurs as their
magnetization is reversed. Additionally, a torque of
fourfold symmetry emerges, R=0.973.
Fig. S1b: Data taken at 800G. The shape anisotropy still predominates but a second contribution
starts to show, R=0.966.
Fig. S1d: Data taken at 1600G. The fourfold
magnetocrystalline anisotropy of the host now
predominates, R=0.968.
page 1 of 5
Nano-Magnetism of Magnetostriction in Fe35Co65
A. Lisfi, T. Ren, A. Khachaturyan and M. Wuttig
SUPPORTING DATA
The quality of the above analysis can be checked quantitatively by comparing the
experimental value of the torque of twofold symmetry obtained through the curve fitting process
with the demagnetization energy of the Fe35Co65 disk. The deconvolution of the torque yields for
the twofold component the values plotted on the abiszissa of Fig. S2 while the ordinate
represents the the demagnetization anisotropy energy of the sample, 𝐾𝑑 = (𝑁𝑎 −
𝑁𝑏 ) ∫ 𝐻𝑑𝑀, Na , Nb – demagnetization factors along the sample ellipsoid axes, M –
magnetization. At the time of writing this paper the sample does no longer exist because it was
polished to study the domain structure. The most likely source of the demagnetization anisotropy
is a thickness gradient of the very thin disk. Assuming this gradient equals 5% across the sample
its demagnetization factor (𝑁𝑎 − 𝑁𝑏 ) = 0.008 one obtains the plot shown in Fig. S2. Fitting
the first three data points to a straight line results in a slope of 1.14±0.05. In view of the
assumtion about the sample geometry the initial linearity of the plot as well as the correct order
of magnetitude of Kd are relevant. Both give credence to the torque analysis.
Fig. S2: Demagnetization anisotropy energy, Kd, as a function of the torque component of the twofold symmetry obtained from the curve fitting process shown in Fig. S1. The torque error bars are
derived from the sum of the least squares sum, R, of the fit. The error bars of Kd are determined by
the uncertainty of the thickness gradient and more so by the fact that the quantity equals a difference of two comparatively large numbers. The data point at 420JM-3 reflects the onset of saturation.
page 2 of 5
Nano-Magnetism of Magnetostriction in Fe35Co65
A. Lisfi, T. Ren, A. Khachaturyan and M. Wuttig
SUPPORTING DATA
II. MINIMUM SWITCHING FIELD OF UNIAXIAL FERROMAGNET
The switching field of a uniaxial ferromagnet depends on the angle between the ferromagnet’s easy axis and the direction of the switching field.1 It equals the anisotropy field if the externally applied field is directed antiparallel to the magnet’s magnetization but smaller otherwise.
Here, the magnetization of the uniaxial precipitates lies parallel to the [100]bcc or [010]bcc direc⃗⃗⃗⃗𝑐 , in the (001) plane of the sample required
tions. The direction of the minimum critical field, 𝐻
to reverse the magnetization of the uniaxial <100> precipitates is determined by the balance of
the components of the externally applied field directed antiparallel and perpendicular to the precipitate’s axes. It is smallest when the components antiparallel, ⃗⃗⃗⃗⃗⃗
𝐻|| , and perpendicular, ⃗⃗⃗⃗⃗
𝐻⊥ , to
𝜋
them, i.e. if ∡(⃗⃗⃗⃗⃗⃗
𝐻|| , ⃗⃗⃗⃗
𝐻𝑐 ) = ± 4 . Consequently the minimum critical field required to reverse the
magnetization of the <100> precipitates is directed at an angle of ±45 degrees off the <100> axes
of the host crystal. The 45 degree offset of the zero crossings between the intermediate and large
torques of fourfold symmetry therefore indicates that the axes of the precipitates are aligned parallel to <100>.
1
S. Chikazumi, Physics of Magnetism, Oxford University Press, 2nd ed., p.492ff
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Nano-Magnetism of Magnetostriction in Fe35Co65
A. Lisfi, T. Ren, A. Khachaturyan and M. Wuttig
SUPPORTING DATA
III. STRAIN INDUCED MAGNETIC ANISOTROPY
The free energy, 𝐹(𝜀𝑖𝑗 ), can be be presented as a quadratic Taylor series in strain, 𝜀𝑖𝑗 ,
0
(𝑀, 𝜔)𝜀𝑘𝑙 + ½𝐶𝑖𝑗𝑘𝑙 𝜀𝑖𝑗 𝜀𝑘𝑙 .
𝐹(𝜀𝑖𝑗 ) = 𝐾𝑖𝑗𝑘𝑙 𝑛𝑖 𝑛𝑗 𝑛𝑘 𝑛𝑙 − 𝐶𝑖𝑗𝑘𝑙 𝜀𝑘𝑙
(1)
0
(𝑀, 𝜔) is an extrinsic strain caused by orientation variants of precipitates
The quantity 𝜀𝑘𝑙
⃗⃗ ⁄𝑀 represents the direction of
corresponding to the direction of the magnetization, 𝑛
⃗ =𝑀
magnetization and ω designates the volume fraction of precipitates.
In the expansion (1), the zero-th term of free energy expanision vanishes because the
reference state that we consider is an undeformed state, 𝜀𝑖𝑗 = 0, which, prior to formation of
precipitates, is a stress-free stable state. However in the presence of the precipitate phase, ω≠0,
the stress-free state, 𝜎𝑖𝑗 = 0, is not an undeformed state anymore. In other words, at ω≠0 the
free energy is minimized at 𝜀𝑖𝑗 ≠ 0. This can be accounted for only if the linear term of the
expansion of 𝐹(𝜀𝑖𝑗 ) is not ignored.
The stress-free strain, 𝜀̅𝑖𝑗 , can now be obtained by minimizing the free energy (1) with
0
(𝑀, 𝜔), which, when substituted into eq. (1), results in the
respect to 𝜀𝑖𝑗 yielding 𝜀̅𝑖𝑗 = 𝜀𝑘𝑙
strain minimized equilibrium free energy
0
0
(𝑀)𝜀𝑘𝑙
(𝑀).
𝐹(𝜀𝑖𝑗 ) = 𝐾𝑖𝑗𝑘𝑙 𝑛𝑖 𝑛𝑗 𝑛𝑘 𝑛𝑙 − ½𝐶𝑖𝑗𝑘𝑙 𝜀𝑖𝑗
(2)
The negative sign in eq. (2) indicates that the relaxed unconstrained free energy, i.e. the
minimized free energy of the equilibrium state, is less than the energy of the unrelaxed
constrained reference state, which, as follows from (1), equals zero.
Assuming now a Vegard-type law for the relaxed strain of a ferromagnetic precipitate, 𝜀̅𝑖𝑗 =
0
(𝑀, 𝜔) = (𝑖𝑗 + 𝑛𝑖 𝑛𝑗 )ω, where the expression in parenthesis repressnts the
𝜀𝑘𝑙
eigenstrain of the orientation variant of a precipitate with the magnetization along the direction,
n, the relaxed elastic energy follows,
0
0
(𝑀)𝜀𝑘𝑙
(𝑀)=[32 (𝐶11 + 2𝐶12 ) + 2(𝐶11 + 2𝐶12 ) + 2 𝐶𝑖𝑗𝑘𝑙 𝑛𝑖 𝑛𝑗 𝑛𝑘 𝑛𝑙 ]𝜔2 ,
𝐶𝑖𝑗𝑘𝑙 𝜀𝑖𝑗
(3)
which for cubic crystals reads
𝐶𝑖𝑗𝑘𝑙 𝑛𝑖 𝑛𝑗 𝑛𝑘 𝑛𝑙 = 𝐶11 − 2(𝐶11 − 𝐶12 − 𝐶44 )(𝑛12 𝑛22 + 𝑛12 𝑛32 + 𝑛22 𝑛32 ).
(4)
page 4 of 5
Nano-Magnetism of Magnetostriction in Fe35Co65
A. Lisfi, T. Ren, A. Khachaturyan and M. Wuttig
SUPPORTING DATA
Combining next eqs. (2), (3) and (4) and retaining only directional terms we finally obtain for
the magnetocrystalline free energy of the solid containing precipitates
𝐹[𝑛(𝜀)] = [𝐾1 + 2 [𝐶11 − 2(𝐶11 − 𝐶12 − 𝐶44 )]𝜔2 ](𝑛12 𝑛22 + 𝑛12 𝑛32 + 𝑛22 𝑛32 )
meaning that the average magnetocrystalline energy density depends quadratically on the
precipitate concentration:
〈𝐾1 〉 = [𝐾1 + 2 [2(𝐶11 − 𝐶12 − 𝐶44 )]𝜔2 ].
page 5 of 5