Long-range-order in a magnetic field

Domain re-distribution in the helical and collinear phases

In the previous section, the commensurate structures are revealed to be helical forT<1.46 K and collinear for 1.64<T<2.2 K. Both phases are single-qstructures, meaning that the 12 arms of the〈0.75 0.75 0〉star are independent from each other and correspond to 12 magnetic domains. Since the magnetic susceptibilities are anisotropic for both structures, a strong field-induced domain re-distribution effect is expected.

For the collinear structure, the highest magnetic susceptibility isχ⊥perpendicular to the spins.

Therefore, a magnetic field will favor domains with spins perpendicular to the field direction.

Under a magnetic field along the [001] direction, the favored domains in the MnSc2S4collinear phase are those withqarms in the (hk0) plane, since their spins also lie in the (hk0) plane and are perpendicular to the field. Our neutron diffraction measurement, which is summarized in Fig. 4.14a, confirms this domain re-distribution.

Similar to the collinear structure, the helical structure also possesses the highest magnetic susceptibilityχ⊥in the direction perpendicular to the helical plane. As a result, domains with the helical plane parallel with the magnetic field have a lower susceptibilityχ_∥and are disfavored by the magnetic field. For the helical phase of MnSc2S4, these are domains withq arm inside of the (hk0) plane if the magnetic field is applied along the [001] direction. As is summarized in Fig. 4.14b, such a domain redistribution is confirmed experimentally.

4.3. Long-range magnetic order in MnSc2S4

Figure 4.15 –(a)Field-dependence of the intensity of the (0.75 0.75 0) reflection measured at T =0.1 K. Up-pointing empty (down-pointing filled) triangles belong to the measurements with increasing (decreasing) field.(b)Temperature-dependence of the neutron diffraction intensity of the (0.75 0.75 0) reflection measured for different magnetic fields. (c)Phase diagram from neutron diffraction experiments. Up-pointing (left-pointing) triangles mark the transition positions extracted from the measurements with increasing field (decreasing temperature). HL, CL and IC represent the helical, sinusoidally-modulated collinear, and incommensurate phases, respectively. The IC phase disappears under field cooling. Error bars represent standard deviations (panels a and b) and are smaller than the symbol size.

Field-induced triple-qvortex-like phase

With the domain effect clear, we investigate the temperature and magnetic field dependence of the intensity of the (0.75 0.75 0) reflection on TASP at PSI. A cryomagnet together with a dilution refrigerator was used. PG(002) monochromator and PG(002) analyzer selected the neutron wavelength of 4.22 Å. A PG-filter was installed in between the sample and the analyzer.

A crystal from the batch of ATR198 was mounted with the (hk0) plane horizontal and the field was along the [001] direction.

Fig. 4.15a shows a representative field-dependence of the intensity for the (0.75 0.75 0) reflec-tion measured atT=0.1 K. With increasing field, the intensity first drops to zero at∼2.5 T due to the helical domain re-distribution summarized in Fig. 4.14b, and then increases to about 2/3 of the zero-field value at fields above 3 T. Such a re-appearance of the (0.75 0.75 0) reflection suggests the existence of a field-induced phase transition, which is also evident in a decreasing field as a sudden drop of the intensity at 3 T.

Fig. 4.15b shows a representative temperature-dependence of the intensity of (0.75 0.75 0)

mea-sured for fixed magnetic fields. At 0 T, the intensity increases with decreasingT in the collinear and helical phases and disappears between 1.64 and 1.46 K due to the incommensurate-commensurate phase transition shown in Fig. 4.8. At higher fields, the intensity of (0.75 0.75 0) increases in the collinear phase due to the domain re-distribution presented in Fig. 4.14a.

However, in the helical phase, a non-monotonous field-dependence similar to that shown in Fig. 4.15a is observed.

Both the field- and the temperature-dependence of the (0.75 0.75 0) Bragg peak suggest the existence of a field-induced phase transition, and the phase diagram is summarized in Fig.

4.15c. Such a phase diagram is also compatible with the magnetic susceptibility results shown in section 4.3.1, where a sudden drop ofχ(T) and a kink inM(H) have been observed at the phase boundaries. For the phase diagram shown in Fig. 4.15, it should be noted that the incommensurate peaks at (0.75±0.02 0.75∓0.020) disappear for field cooling, and thus the IC region is mapped out by first zero-field cooling to the designate temperature and then gradually applying a magnetic field.

In order to determine the structure of the field-induced phase, neutron diffraction datasets of 131, 50, and 56 reflections were collected on TriCS for a [001] field of 3.5 T atT=1.38, 1.62, and 1.85 K, respectively. Surprisingly, for all the three temperatures, the data of each arm can be independently fitted by a collinear structure as in the zero-field collinear phase. The only difference comes from the intensity distribution among the arms. At 1.85 K in the collinear phase, only the four arms in the (hk0) plane are observed due to domain population (see Fig.

4.14a). However, at 1.38 K in the field-induced phase, the suppressed arms re-appear, leading to equal intensities for all the 12 arms (see Fig. 4.14c). Such an unusual intensity distribution in the field-induced phase cannot be explained by the domain population of the single-q collinear structure and instead suggests a multi-qsolution where different arms are coupled together.

We propose the field-induced phase to be a triple-qstate with:

M(r)=

3

X

j=1

(m_je^i(qj·r+φj)+c.c.), (4.7)

where the three coplanarqjsatisfyP₃

j=1qj=0 (e.g. q1= (0.75 0.75 0),q2= (0 0.75 0.75), and q₃= (0.75 0 0.75));m_j is the real basis vector perpendicular toq_j that describes a collinear structure;φjis an additional phase factor; and c.c. is the complex conjugate. As is shown in Fig. 4.16, such a triple-qstructure provides a good refinement of the collected data. The four triple-qdomains formed in this way are symmetrically equivalent under the [001] magnetic field, which explains the equal intensity distribution shown in Fig. 4.14c.

Neutron diffraction data were also collected for a 3.5 T magnetic field along the [111] direction on TriCS atT =1.60 K. Different from the situation with a field along the [001] direction, the four triple-qdomains become inequivalent for this field direction and only the domain with q-vectors perpendicular to the [111] direction were observed. Considering that the dataset for

4.3. Long-range magnetic order in MnSc2S4

observed intensity (a. u.)

0 5 10 15 20

calculated intensity (a. u.)

0 5 10 15 20

Figure 4.16 – Comparison between the observed and calculated intensities assuming the triple-qstructure shown in Eqn. (4.7).

each arm can still be fitted independently with the sinusoidally modulated collinear structure, in Fig. 4.17 we compare the measured intensity distribution and the expected distribution for the single-qcollinear structure. If a single-qstructure was realized, the domains withqout of the (110) plane would have the highest intensity, which is completely opposite to what we observed. Thus our neutron diffraction results with field along the [111] direction corroborate a triple-qstructure for the field-induced phase.

Neutron diffraction is not sensitive to the phase factorφj. In Fig. 4.18, we present two possible structures that preserve theC3symmetry along the [111] direction together with the expected structures with a shorterq= (0.25 0.25 0) that might be realized in less frustrated A-site spinels [151]. In these structures, spin components in the (111) plane exhibit a winding behavior around the C₃ axis, resulting in a U(1) vortex state similar to that predicted in frustrated antiferromagnets [184–189]. Since the winding feature persists regardless of the choice ofφj, our experiments actually reveal thatA-site spinels are new systems that realize a U(1) vortex lattice.

a b

[110]

[110]

Triple-q, H₁₁₁ CL, H₁₁₁

Figure 4.17 –(a) Intensity distribution for the 12 arms of the〈0.75 0.75 0〉star measured for a 3.5 T magnetic field along the [111] direction at 1.6 K, which is compatible with the triple-q structure.(b)Expected intensity distribution for the sinusoidally modulated collinear structure, which is opposite to the experimental results.

a b

c d

q = (0.75 0.75 0)

φ = π / 2 q = (0.75 0.75 0)

φ = 0

q = (0.25 0.25 0)

φ = π / 2 q = (0.25 0.25 0)

φ = 0

Figure 4.18 –(a)The triple-qstructure forq=(0.75 0.75 0) withφ1=φ2=φ3=π/2.(b)The triple-qstructure forq=(0.75 0.75 0) withφ1=φ2=φ3=0.(c)Expected triple-qstructure for q=(0.25 0.25 0) withφ1=φ2=φ3=π/2.(d)Expected triple-qstructure forq=(0.25 0.25 0) withφ1=φ2=φ3=0.

Comparison with the skyrmion lattice Theqcombination rule ofP3

j=1q_j=0 observed in the field-induced triple-qvortex-like phase is very similar to that of the skyrmion lattice [190–193]. The difference lies in the basis vectors m_j. For the skyrmion lattice, the basis vectors are two-dimensional, meaning that each arm forms a helical structure. For the vortex phase in MnSc₂S₄, however, the basis vectors are one-dimensional and each arm represents a collinear structure. As is compared in Fig. 4.19, such a difference in basis vectors leads to different topologies for the vortex and skyrmion lattice.

It is also interesting to compare the stability of the observed vortex lattice with the skyrmion lattice. An immediate observation is that both phases require the application of a magnetic

4.3. Long-range magnetic order in MnSc2S4

vortex lattice q = (1/4 1/4 0)

0 S

-S Skyrmion lattice

q = (1/4 1/4 0)

a b

Figure 4.19 –(a)The vortex lattice composed of one-dimensional basis vectors withφ1= φ2=φ3=π/2, similar to that shown in Fig. 4.18c. (b)The skyrmion lattice composed of two-dimensional basis vectors perpendicular to the wave vectorq. The colormap shows the size of the spin componentS_∥along the (111) direction.

field. For the skyrmion lattice, the necessity of the magnetic field has been elegantly explained through a phenomenological approach [191]. Assuming the existence of the fourth order coupling term of the coarse-grained magnetic momentM⁴in the free energy, the applied magnetic field induces a ferromagnetic componentM_f that couples to the order parameter Φ=M−M_f in the form ofΦ³M_f. After Fourier transform, thisΦ³M_f term will have a coef-ficient ofδ(q₁+q₂+q₃), which is just the observedqcombination rule. Therefore, with the replacement of the ferromagnetic momentMby the antiferromagnetic order parameterL, we expect the vortex state to be similarly stabilized by phenomenological high-order couplings.

Despite the similarity of the field influence, the stability as a function of temperature is very different for the vortex and the skyrmion lattices. The skyrmion lattice in three-dimensional crystals usually exists only in a narrow temperature region just below the long-range order transition. While the vortex phase in MnSc₂S₄ is stable even at temperatures below 0.1 K (see Fig. 4.15). However, recent measurements of two-dimensional systems have revealed a broadened region for the skyrmion lattice, which has been theoretically attributed to the effect of a small out-of-plane anisotropy [194, 195]. Thus the broad temperature region of the vortex phase in MnSc₂S₄might be also induced by some effective anisotropy along the [111]

directions.

Finally, it is important to note that the mechanism of the spin twisting in these two lattices are very different: in known skyrmion lattices, it originates from the relativistic Dzyaloshinskii-Moriya interaction [190, 191]; while in the MnSc2S4vortex lattice, it is a pure frustration effect [187, 188], which leads to a much smaller vortex size in contrast with the known skyrmions [190–193].