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Controllable emergent spatial spin modulation in Sr2IrO4 by in situ shear strain

Symmetric anisotropic interaction can be ferromagnetic and antiferromagnetic at the same time but for different crystallographic axes. We show that inducing competition of anisotropic interactions of orthogonal irreducible representations represents a general route to obtain new exotic magnetic states. We demonstrate it here by observing the emergence of a continuously tunable 12-layer spatial spin modulation when distorting the square lattice planes in the quasi-2D antiferromagnetic Sr2IrO4 under in situ shear strain. This translation-symmetry-breaking phase is a result of an unusual strain activated anisotropic interaction which is at the 4th order and competing with the inherent quadratic anisotropic interaction. Such a mechanism of competing anisotropy is distinct from that among the ferromagnetic, antiferromagnetic, and/or the Dzyaloshinskii-Moriya interactions, and it could be widely applicable and highly controllable in low dimensional magnets.

Exotic spatial spin modulations, including various forms of spin spirals [1,2] and magnetic skyrmions [3][4][5], have been extensively studied for magnetoelectric effects [6], topology [7,8], spintronics [9][10][11][12], etc. Their common origin is competition between ferromagnetic, antiferromagnetic, and/or the Dzyaloshinskii-Moriya (antisymmetric anisotropic) interactions that favor parallel, antiparallel, and orthonormal spin arrangement, respectively. In contrast to these directional interactions, symmetric anisotropic interaction has an axial nature that it could be ferromagnetic for one axial component of the moments and antiferromagnetic for another at the same time, breaking rotational symmetry. Competition of anisotropic interactions of different symmetry channels could be an attractive alternative mechanism for stabilizing exotic magnetic phases. However, exploration of competing anisotropic interaction has been limited without efficient ways to introduce and tune the competition.
The strategy for realizing such competition could be remarkably simple and general.
Since magnetic order often preserves certain rotational/mirror symmetry according to the inherent anisotropic interaction, one may apply shear strain to break the same symmetry to introduce antagonistic anisotropy. This necessarily causes competition of anisotropy with different irreducible point-group representations. Identifying materials for this method is however not trivial. The key ingredients are that the anisotropy must play a vital role in stabilizing the ordering and the magnetoelastic coupling of the shear-strain channel must be strong for inducing sizable anisotropy. As a pseudospin-half Mott insulator [13][14][15][16][17][18][19][20][21][22][23][24][25][26][27], Sr2IrO4 is an ideal system for this purpose. It is known for its analogy with the parent phase of high-Tc cuprates [28][29][30][31][32][33][34][35][36][37], including the quasi-2D tetragonal lattice and an antiferromagnetic (AF) ordering below TN~230K [14,16]. Due to the Ruddlesden-Popper crystal structure, the spontaneous quasi-2D AF order relies on the inter-plane anisotropic interaction to be stabilized, and breaks the D4h point group symmetry with the moments aligned along the a-or b-axis (Fig.   1a), i.e., orthorhombic twin domains of the B1g symmetry (Fig.1a) [38,39]. Because of the strong spin-orbit coupling, the magnetoelastic coupling can be described by the pseudo-Jahn-Teller effect, where the pseudospin-half wavefunction is highly sensitive to the lattice distortion.
Sr2IrO4 thus provide a promising platform for exploring competing anisotropy under shear strain, i.e. strain of the B2g symmetry. Here we report that applying in situ shear strain to the ab-plane in Sr2IrO4 leads to a surprising translational symmetry-breaking modulation along the c-axis consisting of 12 IrO2 planes ( Fig.1a) that has not been observed before. Our study unambiguously shows that the spatial modulation is a result of competing anisotropy because the shearing turns on a hidden quartic anisotropic interaction of B2g symmetry that is orthogonal to the spontaneous quadratic anisotropy of B1g symmetry. Moreover, the competition and the modulation can be continuously and efficiently tuned here by applying a strain of less than 0.05%. Our observation of spatially modulated magnetic phases adds a significant representative to the expanding list of emergent phases due to tuning of interactions of different irreducible point-group representations such as electronic nematicity [40,41], quadrupolar order [42], and unconventional superconductivity [43][44][45].
As can be seen in Fig.1b, a key feature of Sr2IrO4 is the octahedral rotation around the caxis [20], leading to a unit cell expansion from the original Ruddlesden-Popper structure. As a result, the spontaneous AF order fully preserves the translational symmetry and displays spin canting of the planar Ir AF moments within the ab-plane, which orthogonally locks the net moment and the staggered moment of each IrO2 plane together with a fixed chirality and ratio [46][47][48]. Before discussing the shear strain effect, it is important to establish the normal strain effect for comparison. Since the spontaneous orthorhombic distortion is weak [39] and the system can be considered pseudo-tetragonal, applying normal strain, i.e., B1g strain, by external stress has been shown to be highly efficient in detwinning the AF domains [49,50] (Fig.1a). A more pronounced effect occurs when the strain is applied with a magnetic field. Since the interplane anisotropic interaction stabilizes a ↓-↓-↑-↑ (or ↓-↑-↑-↓) stacking sequence of the IrO2 planes where ↑ and ↓ refer to the net moments [16,38], a sufficiently large field polarizes them through a metamagnetic phase transition [20] to a ↑-↑-↑-↑ sequence (Fig.1b), which for simplicity we call the ferromagnetic (FM) state hereafter. Figure 2a shows an example when we applied the field and the in situ B1g strain that stretched the a-axis and contracted the b-axis at 210 K by a piezo actuator, which delivers up to ~0.05% of anisotropic strain [49]. As can be seen, the transition monitored by magnetoconductance (MC) [51][52][53][54] is radically sharpened and shifted to a much lower field with H//b, whereas it is significantly broadened and shifted to much higher field with H//a. Such a strong anisotropic response with high tunability is a result of the uniaxial anisotropy induced by the B1g strain that restricts the metamagnetic transition to a sharp spin-flip or a broad continuous rotation depending on whether the field is parallel or perpendicular to the enforced easy axis [49,50]. The section below the kink shows weak strain-dependence, whereas the section above is significantly extended to a higher field by increasing 2 , giving rise to a prolonged "tail" of an overall "tied-up" shape of the MC curves.
An analysis of the derivative of the MC curve (see supplementary) reveals a fine movement of the kink that it is slightly upshifted first before being slightly downshifted with increasing 2 .
A B2g strain-field phase diagram can now be constructed from the MC results as shown in Fig. 2c.  , indicating translational symmetry-breaking along the caxis with a tripled size. Such a 12-IrO2-plane modulation of the staggered moment corresponds to a 6-IrO2-plane modulation of the net moment according to the out-of-phase octahedral rotation within the structural cell (Fig.1b). We denote this state as the BTS state hereafter. The fielddependence of the satellite peak intensities displayed in Fig.3b exhibits a clear correlation with the integer peaks. It is now apparent that the ↓-↓-↑-↑ state first transforms into the BTS state, and the FM component of the BTS state gradually increases at the expense of the amplitude of the spatial modulation until reaching the FM state. Figure 3d further shows the increase of the satellite peak intensity at 925 Oe as a function of 2 , demonstrating that the B2g strain not only stabilizes the spatial modulation but also efficiently tunes its amplitude. These results confirm the new phase in the phase diagram to be the BTS state.
The emergence of the BTS state corroborates that the B2g strain effect is to compete with the spontaneous anisotropy that favors B1g symmetry rather than changing the easy axis. To unveil this competition, we investigated the minimal free energy model that captures the BTS state. In virtue of the robust intra-plane AF order and the locking between the net moment and staggered moment [47], the stacking of the quasi-2D planes can be modeled as an effective 1D system of the net moments = , where is the moment size and is the unit vector in the ab-plane characterized by angle with respect to the a-axis. The ↓-↓-↑-↑ state, BTS state, and FM state are thus characterized by a 1D wave vector q = 1/4, 1/6, and 0, respectively. The free energy can be written as: where 0 includes inherent interactions that account for the spontaneous ↓-↓-↑-↑ state of B1g symmetry, is the Zeeman energy, 1 and 2 represent interactions induced by the externally applied strain of B1g and B2g symmetry, respectively. At a fixed temperature below TN, is considered as a constant under the perturbations of field and strain, and it is ~0.04 at 210 K according to magnetization. Thus, all -independent terms are neglected, and the ground state is obtained by energy minimization with respect to . By mapping the reported inter-plane anisotropic interactions and the pseudospin-lattice coupling of Sr2IrO4 [39,49] to this effective model, one has (1) Where (1) ≡ 2 − 2 is the quadrupole of B1g symmetry. The -term is the anisotropic exchange interaction of B1g symmetry with an alternating sign that favors a q = 1/4 modulation.
It becomes the ↓-↓-↑-↑ state with the -term and the ′ -term, which have the A1g symmetry and force the moments along the a-or b-axis [38,39] (see supplementary). We constrain , , and ′ by reproducing the metamagnetic transition at zero strain and under the B1g strain by including a uniaxial anisotropy term in 1 [49] (Fig. S3).
Next we turn off 1 and proceed to simulate the B2g strain effect. We first tried B2g symmetry terms of uniaxial and exchange anisotropy. But both fail to describe the experimental results (see supplementary). This inconsistency leads us to consider increasing the order of the anisotropic interaction from quadratic to quartic. The only nontrivial quartic term of B2g symmetry has the form (see Methods) or equivalently where (2) ≡ is the quadrupole of B2g symmetry and is the coupling coefficient. Figure   4a shows the calculated metamagnetic transition, which is characterized by the q = 0 order parameter and turns out to capture the observed characteristics remarkably well, especially the two-section shape of the transition with an extended tail region when H// [110]. Even the fine movement of the kink in the MC curve is mimicked by the strain dependence of the critical field that separates the two sections. More importantly, this quartic anisotropy correctly predicts a finite q = 1/6 modulation within the tail region where the q = 1/6 modulation appears and slowly decreases with increasing field.  Figure 4b shows a representative evolution of the simulated spatial modulation. Initially, the moments cant away from the a/b-axis towards [110] to gain Zeeman energy while staying in pairs to maintain the ↓-↓-↑-↑ state. When the field increases further, the q = 1/6 modulation sets in by "inserting" an unpaired site in between the adjacent pairs and breaking the translation symmetry. The modulation amplitude of this BTS state then slowly decreases with increasing field until both the unpaired and paired moments become fully aligned. The periodic emergence of the unpaired sites is evidently the key since it plays a domain wall-like role in separating the pairs, allowing the -term to save more energies on the paired sites while letting theinteraction to concentrate more on the unpaired site. The full 12-IrO2-layer magnetic structure is also shown for one of the BTS configurations as an example to illustrate the modulation in  4), where the nearest-neighbor interaction is gated by (2) , controlling the sign of the interaction to be positive or negative. It could even be zero whenever is along the a/b-axis, and the interaction would be zero in this case regardless of the direction of −1 and +1 . This character is absent in quadratic interactions and provides more options to find compromised configurations when competing with the -interactions of orthogonal symmetry. We further confirmed this anisotropy competition by simulating a hypothetical situation where the A1g and ′ terms are removed (see supplement).
Overall, we demonstrate that introducing anisotropy that is orthogonal to the spontaneous one could lead to translational symmetry-breaking and spatial spin modulation. The possibility of introducing the competition between the B1g and B2g anisotropy in Sr2IrO4 by in situ shear strain allows continuous tuning of the spatial modulation. This approach to trigger competition and obtain controllable new phases could be applicable to many quasi-2D materials due to the critical role of anisotropy in stabilizing low-dimensional magnetic orders. The emergence of the B2g quartic interaction also highlights the rich unexplored physics of higher-order anisotropy. To our knowledge, quartic anisotropic interaction has not been observed before. Since quartic exchange interactions in Mott insulators arise from the fourth-order perturbation [56], distinct hopping processes must be involved at the same time, such as that between the planes and within the planes. These results should motivate theoretical investigations on extracting the underlying microscopic model as well as exploration of high-order anisotropic interactions in other materials.