** 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 peaksP_{1} = (10) and P_{2} = (01) and

*4.3. BRAGG SCATTERING ON ARTIFICIAL SQUARE ICE* 71
the two second order peaks P_{3} = (1¯1) and P_{4} = (¯11). The hysteresis curves of the first
order peaks (figure 4.12c) reveal a small difference in their respective coercive fields of
H_{P}1 = 475±2Oe andH_{P}2 = 455±2Oe. The behaviour of the two second order peaksP_{3}
andP_{4} (see figure 4.12d) are identical within the experimental uncertainty with a coercive
field value of H_{P}3 =H_{P}4 = 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. ϕ_{1}> ϕ_{2} 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 sublatticeS_{1}(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 onS_{2}, thus giving rise to
the small difference in the apparent coercive field. From the coercive field values ofP_{1} 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 (P_{3} and
P4). The reason for this is that these peaks sample both sublattices equally. Consequently,
the field dependence of the dichroic contrast forP_{3} and P_{4} 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 peaksP_{1} to P_{4} 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 ofM_{r}1 and
M_{r}2. 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
F_{n}(q) becomes [158]:
wherej_{0} and j_{1} 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

P_{2}
P_{3}

P_{4}

### 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 P_{4} (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 fromS_{2} (ϕ_{2}). c) XMCD contrast of the first order peaks P_{1} and
P_{2}. The two loops have different coercive fields, H_{P}1 = 475±2 Oe andH_{P}2 = 455±2Oe,
respectively. d) XMCD contrast of the second order peaksP3andP4. Here, the two curves
are identical within the experimental error, the coercive field isH_{P}3 =H_{P}4 = 468±2Oe. [9]