BRAGG SCATTERING ON ARTIFICIAL SQUARE ICE 69

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Resonant scattering on artificial spin ice

4.3. BRAGG SCATTERING ON ARTIFICIAL SQUARE ICE 69

200 300 400 500 600 700 800 900

10−4

Figure 4.10: Evolution of the magnetic Bragg peaks intensities. a) Miller indices of the magnetic Bragg peaks. b) Evolution of the normalized intensity of the magnetic Bragg peaks as a function of the magnetic field. c) Evolution of the intensity for the same peaks derived from the amount of reversed spins.

-1 0 1

a a)

b)

c)

d)

[10]

[01]

Figure 4.11: Magnetic dichroism of the scattering pattern recorded at opposite saturation. The scattering patterns exhibits only Bragg peaks at the location of the non-resonant (structural) Bragg peaks. To follow the evolution of the magnetic contribution to the scattering, we used the XMCD.

4.3.3 Magnetization reversal

Once the sample is saturated, the purely magnetic Bragg peaks have disappeared (figure 4.9d). The magnetic and structural unit cells are identical (figure 4.11a), and the scattering pattern exhibits Bragg peaks only at the positions corresponding to this periodicity. No magnetic peaks appear while moving further along the hysteresis loop. This is a stronger confirmation that no large domain of type I vertices are formed. Indeed, no such domains have been observed with PEEM and MFM which only probe small areas and have therefore a limited field of view. On the other hand, scattering probes the entire sample and indicates that no significant fraction of ordered regions exist. This is coherent with what has been described in the literature. As we discussed in section 1.4.2, the nanomagnets have a distribution of coercivities related to the defects induced by the lithography process, which is also referred as quenched disorder [56, 157]. This disorder prevents the system from reaching low-energy magnetic states by means of a demagnetization process [157].

In order to extract the magnetic component of the Bragg peaks, we recorded scattering patterns with right- and left-handed polarizations. As discussed in section 3.1.1, subtract-ing one from the other (equation (3.12)) yields the magnetic component. The dichroic contrast characterizing the remanent state after saturation along the beam axis are shown in figure 4.11. When all magnetic moments points towards the direction of the beam (y in figure 4.7), the observed dichroic contrast corresponds to its largest positive value which is shown by the blue colour of the peaks (figure 4.11b). The lowest negative value is observed after saturation along the negativey-axis, illustrated by a red colour of the peaks (figure 4.11d).

Applying a reverse magnetic field following saturation leads to changes in the dichroic scattering pattern, with particularly strong changes occurring for applied field values around the coercive field where the magnetization reversal takes place. As already ob-served by Morgan and co-workers [86], the dichroic contrast does not change equally for all Bragg peaks, but can differ from peak to peak. We show in the following that is via these differences that a detailed insight can be obtained into the reversal process.

In figure 4.12 we show the hysteretic behaviour of the dichroic intensity of four Bragg peaks (see figure 4.12a, c and d): the two first order peaksP1 = (10) and P2 = (01) and

4.3. BRAGG SCATTERING ON ARTIFICIAL SQUARE ICE 71 the two second order peaks P3 = (1¯1) and P4 = (¯11). The hysteresis curves of the first order peaks (figure 4.12c) reveal a small difference in their respective coercive fields of HP1 = 475±2Oe andHP2 = 455±2Oe. The behaviour of the two second order peaksP3 andP4 (see figure 4.12d) are identical within the experimental uncertainty with a coercive field value of HP3 =HP4 = 468±2 Oe. Note that this value is close to the mean value of the coercive fields of the two first order peaks.

The observation of different coercive fields for the two first order peaks (P1 andP2) can be understood by assuming a slight misalignment between the sample axis and the direction of the applied magnetic field, i.e. ϕ1> ϕ2 as illustrated in figure 4.12b. Discussion on how P1 and P2 are sampling the different sublattices is performed in the next section. Here we focus on the consequence of such a misalignment, which increases the applied effective field along the long axis of the nanomagnets for the sublatticeS1(in white), while it is decreased for the sublatticeS2 (in gray). Consequently, the magnetic moments on sublatticeS1 will tend to switch at a lower external field value than the moments onS2, thus giving rise to the small difference in the apparent coercive field. From the coercive field values ofP1 and P2, we can estimate the angle of misalignment, which we find to be smaller than 1.5. The observation of different coercive fields for the two sublattices is also in line with the absence of such a difference in the hysteresis curves of the two second order Bragg peaks (P3 and P4). The reason for this is that these peaks sample both sublattices equally. Consequently, the field dependence of the dichroic contrast forP3 and P4 is directly proportional to the total magnetization of the array.

4.3.4 Determination of the nanomagnet contributions to the remanent magnetization

Method

The dichroic contrast of the Bragg peaks gives statistical information about the configura-tion of the magnetic moments and we show in this secconfigura-tion how one can derive separately the remanent magnetizationMr1 andMr2 of the two sublatticesS1 andS2from the intensities of the Bragg peaksP1 to P4 indicated in figure 4.12a. The procedure is as follow. First, we assume a distribution of the magnetic moments for which we simulate the scattering pattern. We then compare experiment and simulation characterized by values ofMr1 and Mr2. We repeat the simulation until the best agreement is reached.

To evaluate equation (4.4), we need first a representation of the form factor of the nanomagnets. Here we approximate the shape of the nanomagnets with a cylinder of height h and an elliptical cross-section with semiaxes a and b. The set of form factors Fn(q) becomes [158]: wherej0 and j1 are spherical Bessel functions. As indicated in figure 4.8a, the structural unit cell contains two structurally identical nanomagnets, but with different orientations.

Their respective form factorsF1(q) and F2(q) are therefore given by the following sets of vectors that define their spatial orientation:

P1

P2 P3

P4

S

1

S

2

H, y-axis

φ

1

φ

2

a) b)

c)

d)

-750 -500 -250 0 250 500 750

XMCDe[a.u.]

Magneticefielde[Oe]

-750 -500 -250 0 250 500 750

XMCDe[a.u.]

Magneticefielde[Oe]

P1

P2

P3

P4

SP

Figure 4.12: XMCD contrast of the selected Bragg peaks at remanence as a function of applied field. a) Scattering pattern indicated the selected Bragg peaks P1

to P4 (SP indicates the specular reflection). b) Illustration of the sample misalignment with respect to the direction of the applied field. Note that due to the misalignment the nanomagnets of the sublatticeS1 have a large angle relative (ϕ1) to the applied field direction than the ones fromS22). c) XMCD contrast of the first order peaks P1 and P2. The two loops have different coercive fields, HP1 = 475±2 Oe andHP2 = 455±2Oe, respectively. d) XMCD contrast of the second order peaksP3andP4. Here, the two curves are identical within the experimental error, the coercive field isHP3 =HP4 = 468±2Oe. [9]

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