# Interplay between Single-Ion and Two-Ion Anisotropies in Frustrated 2D Semiconductors and Tuning of Magnetic Structures Topology

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- (i)
- strong magnetic frustration arising from competing nearest-neighbor (${J}^{1iso}$) ferromagnetic (FM) and third-nearest neighbor (${J}^{3iso}$) anti-ferromagnetic (AFM) exchange interactions;
- (ii)
- very weak easy-plane single-ion anisotropy against highly anisotropic symmetric exchange, both driven by spin-orbit coupling (SOC).

**J**${}_{ij}$ and

**A**${}_{i}$ are tensors describing the exchange interaction and single-ion anisotropy, respectively [53]. It is convenient to decompose the exchange coupling tensor into an isotropic part ${J}_{ij}^{iso}=\frac{1}{3}\mathrm{Tr}{\mathbf{J}}_{ij}$ and an anisotropic symmetric part ${\mathbf{J}}_{ij}^{S}=\frac{1}{2}({\mathbf{J}}_{ij}+{\mathbf{J}}_{ij}^{\mathrm{T}})-{J}_{ij}^{iso}\mathbf{I}$, herein also referred to as two-ion anisotropy. Due to the inversion symmetry of the lattice (with ${D}_{3d}$ point group), the DM interaction, which corresponds to the antisymmetric exchange term ${\mathbf{J}}_{ij}^{A}=\frac{1}{2}({\mathbf{J}}_{ij}-{\mathbf{J}}_{ij}^{\mathrm{T}})$, is identically zero. Reference magnetic parameters for NiI${}_{2}$ monolayer, evaluated from density-functional theory (DFT) calculations, are given in Table 1. The interactions were estimated by means of the four-state energy mapping method [53], performing PBE+U+SOC calculations [54,55] (U = 1.8 eV, J = 0.8 eV) via the VASP code [56,57]. We used a $5\times 4\times 1$ supercell to estimate the SIA and first nearest-neighbor interactions, and a $6\times 3\times 1$ supercell for the estimate of the third nearest-neighbor interaction. Supercells were built from the periodic repetition of the optimized NiI${}_{2}$ monolayer unit cell, with the lattice parameter of about $3.96$ Å and a vacuum distance of $\simeq 17.5$ Å between periodic copies of the free-standing layer along the c axis. The matrix elements of the two-ion anisotropy tensor (${\mathbf{J}}^{S}$ or TIA) between nearest-neighbour spins, are expressed in a local cartesian $\{x,y,z\}$ basis, where x is parallel to the Ni-Ni bonding vector, and to the lattice vector

**a**. Further details can be found in the Methods section of Ref. [40].

**q**minimizing the isotropic exchange interaction in the momentum space, J(

**q**). Using the magnetic parameters listed in Table 1, the propagation vector for the isotropic model is given by $q=2{cos}^{-1}[(1+\sqrt{1-2{J}^{1iso}/{J}^{3iso}})/4]$ [26,33], resulting in ${L}_{m.u.c.}\simeq 8$. The results presented are thus obtained by means of calculations performed on supercells with lateral size $L=3{L}_{m.u.c.}=24$.

## 3. Results

$-0.75\lesssim {A}_{zz}/{\left|J\right|}^{1iso}\lesssim -0.30$ | ${\left|Q\right|}_{m.u.c}=0\to 6\to 3\to 0$ |

$-0.30\lesssim {A}_{zz}/{\left|J\right|}^{1iso}\lesssim 0.30$ | ${\left|Q\right|}_{m.u.c}=6\to 3\to 0$ |

$0.30\lesssim {A}_{zz}/{\left|J\right|}^{1iso}\lesssim 0.50$ | ${\left|Q\right|}_{m.u.c}=6\to 3\to 1.5\to 0$ |

$0.50\lesssim {A}_{zz}/{\left|J\right|}^{1iso}\lesssim 0.65$ | ${\left|Q\right|}_{m.u.c}=0\to 6\to 3\to 0$ |

$0.65\lesssim {A}_{zz}/{\left|J\right|}^{1iso}\lesssim 0.75$ | ${\left|Q\right|}_{m.u.c}=0\to 3\to 0$ |

#### 3.1. Topologically Trivial Spin Orderings

**q**states (A and B phases) appear for strong easy-axis and easy-plane SIA, whose spin configurations are shown in Figure 2a,d, respectively. No finite topological charge appears in such phases, as clearly shown in Figure 2b,e. From Figure 2c,f, the propagation vector is ${\mathbf{q}}_{1}=(-2\delta ,\delta )$, with $q/2=\delta \simeq 1/8$ in reduced coordinates, being mostly determined by frustrated isotropic exchange. Due to the three-fold rotational symmetry of the triangular lattice, the single-$\mathbf{q}$ helices propagating along the x-axis with ${\mathbf{q}}_{1}$ are equivalent (and, hence, energetically degenerate) to helices propagating along the symmetry-equivalent directions rotated by $+{120}^{\circ}$, with ${\mathbf{q}}_{2}=(\delta ,-2\delta )$, and $-{120}^{\circ}$, with ${\mathbf{q}}_{3}=(\delta ,\delta )$. The plane of spins rotation is selected by both SIA and TIA, which thus determine the nature of the helical states. The inspection of both spin structures and spin structure factors reveals that: (i) phase (A) consists of a proper-screw spiral propagating along the Cartesian x-axis, i.e., along the Ni-Ni bond direction, with spins rotating in the perpendicular $yz$-plane; (ii) phase (B) is a tilted cycloid, i.e., a helix where spins rotate in a plane containing the propagation vector (parallel to x) but tilted around it, causing the spins to acquire a non-zero z-component highlighted by the peaks of ${S}_{z}\left(\mathbf{q}\right)$ (cfr Figure 2f). The modeling of the single-$\mathbf{q}$ proper-screw and cycloidal helices is presented in Appendix A.

#### 3.2. Topological Spin Orderings

**q**states, with

**q**${}_{1}=(-2\delta ,\delta )$,

**q**${}_{2}=(\delta ,-2\delta )$,

**q**${}_{3}=(\delta ,\delta )$ and $\delta \simeq 1/8$ [Figure 3c,f, Figure 4c,f and Figure 5c], characterized by atomic scale skyrmionic or meronic structures composed by nano-sized topological objects with a radius counting ≃4 spins, and thus a diameter of few units of the lattice parameter ${a}_{0}$.

#### 3.2.1. Topological Lattice with ${\left|Q\right|}_{m.u.c}=6$

**q**state characterized by a hexagonal lattice formed by six vortices (V, $m=+1$) with downward central spin, hosting at the center of each hexagon an anti-bi-vortex (A2V, $m=-2$) with upward core, as shown in Figure 3a. Such spin texture defines a homogeneous topological charge density, as depicted in Figure 3b, hence uniform scalar chirality, and gives rise to a topological charge of six per magnetic unit cell. This topological lattice can be regarded as the periodic repetition of topological objects, consisting of two vortices and one anti-bi-vortex (schematically depicted in Figure 7g), all contributing to defining a global $\left|Q\right|=2$. As evident from the ${s}_{z}$ profile reported in Figure 7a, the central A2V does not display a uniform, unitary polarity along its surrounding perimeter (highlighted as a purple circle traced in Figure 7b): the upward spin of the core (${s}_{z}=+1$) is not reversed downward (${s}_{z}=-1$) at all points of its finite edge. Therefore, it brings a fractionalized topological charge ${Q}_{A2V}=-2\xb7{p}_{A2V}$, with $0<{p}_{A2V}<+1$. The missing Q fraction is carried by the two vortices, which, similarly, bring a fractionalized charge ${Q}_{V}=+1\xb7{p}_{V}$, with $-1<{p}_{V}<0$. Accordingly, $\left|Q\right|=|{Q}_{A2V}+2{Q}_{V}|=2$. The minimal magnetic cell accommodating this topological lattice consists of the composition of three repeated objects; hence ${\left|Q\right|}_{m.u.c}=6$. The topological lattice can therefore be interpreted as a fractionalized anti-bi-skyrmion (A2SK) lattice, where the fractionalization of the topological charge can be ascribed to the A2SKs close packing, leading to the incomplete spin wrapping highlighted in Figure 7 [10]. A similar realization of a fractionalized skyrmion lattice has been recently reported in MnSc${}_{2}$S${}_{4}$, where an applied field has been experimentally shown to stabilize a lattice of fractionalized Bloch-type skyrmions and incipient merons [59].

#### 3.2.2. Topological Lattice with ${\left|Q\right|}_{m.u.c}=3$

#### 3.2.3. Topological Lattice with ${\left|Q\right|}_{m.u.c}=1.5$

**q**) ferromagnetic H-phase upon ${B}_{z}$ (Figure 1c). The competition of the intermediate-strong easy-plane anisotropy with the intermediate-strong applied field causes the transformation of the Bloch-like skyrmion into a meron with halved topological charge. As shown by the spin texture in Figure 5a and the ${s}_{z}$ profile of the vortex core in Figure 8j,k, the latter loses the spin component along the z-direction, while it remains aligned to the field in the surrounding vortices and anti-bi-vortices, which occupy alternatively the center of the triangles formed by the merons: spins at the edge of the topological vortex are directed perpendicularly with respect to its core, halving the topological charge of the Bloch-like skyrmion lattice of the E-phase, ${\left|Q\right|}_{m.u.c}=3\to 1.5$.

#### 3.3. Field-Induced Phase Transition from Two-Ion Anisotropy Tuning

**q**state (Figure 2j,g, respectively).

**q**state; $|{Q}_{m.u.c}|=0\to 3\to 0$ as a function of ${B}_{z}$. In particular, such single-

**q**state is found to be a proper-screw spin-spiral. The magnetic phases induced by the applied fields are closely related to those observed when ${J}_{yz}>0.14{J}^{1iso}$. Therefore, a Bloch skyrmion lattice can still be stabilized by a magnetic field applied on a topologically trivial helical state, when the non-collinearity brought in by the two-ion anisotropy is not sufficiently strong to drive a spontaneous high-Q topological lattice.

## 4. Discussion

**q**) states, ranging from topological (phases C, D, E, F, G) to trivial (H-phase) spin configurations, exhibit a common planar spin texture. To better appreciate this feature, we show in Figure 7c and Figure 8c,f,i,l,o the color-gradient plots of the in-plane spin components highlighting the azimuthal angle $\theta \in [{0}^{\circ},{360}^{\circ})$, defined by the ${s}_{x}$ and ${s}_{y}$ components of the spin vector $\mathit{s}$, and which is related to the in-plane spins rotational direction.

**q**) phases can be in fact seen as a transformation of the initial C-phase via major modifications of the ${s}_{z}$ component of spins; the in-plane directions of the spins remaining unchanged. Moreover, even though single-$\mathbf{q}$ states are found to be energetically more stable than triple-$\mathbf{q}$ ones when ${A}_{zz}\gtrsim 2.5\left|{J}_{yz}\right|$ and ${A}_{zz}\lesssim -1.5\left|{J}_{yz}\right|$ (or also when ${J}_{yz}\le 0.14{J}^{1iso}$), an applied out-of-plane magnetic field can still induce a transition to a topological skyrmion lattice, whose in-plane spin configuration appears to be determined by the two-ion anisotropy.

## 5. Conclusions

**q**) and triple-(

**q**) states to topological triple-(

**q**) states. The strong magnetic frustration, arising from the competing isotropic nearest-neighbour FM and third nearest-neighbour AFM exchange interactions, promotes the onset of short-period helimagnetic configurations, whereas the strong two-ion anisotropy combined with the geometrical frustration of the underlying triangular lattice favours the stabilization of triple-$\mathbf{q}$ states. At zero magnetic field, these result in a lattice of vortices and anti-(bi)-vortices carrying a total topological charge per magnetic unit cell ${\left|Q\right|}_{m.u.c}=6$, which is robust within a wide range of single-ion anisotropy strength. Such topological lattice can be interpreted as a fractionalized anti-bi-skyrmion lattice where each anti-bi-skyrmion is surrounded by six vortices arising from the overlap with neighbouring anti-bi-skyrmions, causing a fractionalization of the topological charge of individual anti-bi-vortices. Both the single-ion anisotropy and the applied field act primarily on the out-of-plane component of the spin configurations, hence on the polarity p of the vortical states, modulating both the size of anti-bi-vortices and the fractionalization of their charges. As a general trend, on the one hand, easy-axis anisotropy is found to increase the localization of anti-bi-vortices, thus reducing the “spilling” of topological charge to surrounding vortices; on the other hand, a strong easy-plane anisotropy, when combined with applied field, would eventually favour a crossover to meronic lattices, i.e., lattices formed by topological objects with half-integer polarity. Nevertheless, an out-of-plane magnetic field, sustained by the strong anisotropic exchange, is always found to trigger a topological transition to a Bloch-like skyrmion lattice, when applied either on a single- or a triple-$\mathbf{q}$ state. Interestingly, the in-plane components of the spin texture are found to be extremely robust across almost the whole phase space explored in this work, with exceptions only for extreme values of SIA and fields.

## Author Contributions

**q**), and contributed to the analysis. D.A. and P.B. wrote the manuscript. S.P. supervised the project. All the authors discussed the results. All authors have read and agreed to the published version of the manuscript.

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

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**Figure 1.**Single-ion anisotropy and field induced phase transitions. (

**a**) Low temperature ($T=1$ K) phase diagram in the SIA-magnetic field ${A}_{zz}-{B}_{z}$ parameters space; both ${A}_{zz}$ and ${B}_{z}$ are expressed in units of the nearest neighbour isotropic exchange interaction ${\left|J\right|}^{1iso}$. Color-map indicates the absolute value of the topological charge per magnetic unit cell ${\left|Q\right|}_{m.u.c.}$. (

**b**) Evolution of the out-of-plane magnetization ${M}_{z}$ as a function of the magnetic field ${B}_{z}$ for increasing SIA, easy-axis (${A}_{zz}/{\left|J\right|}^{1iso}<0$, top) and easy-plane (${A}_{zz}/{\left|J\right|}^{1iso}>0$, bottom), respectively. (

**c**) Schematic ${A}_{zz}-{B}_{z}$ phase diagram identifying regions occupied by the different spin configurations, labelled with capital letters. Phase boundaries must be regarded only as semi-quantitative, as the field-induced topological phase transitions go through almost continuous transformations of the spin texture and metastable spin states with fractionalized topological charge, preventing the identification of accurate and sharp phase boundaries. On the right-side of the y-axis, ${A}_{zz}/\left|{J}_{yz}\right|$ ratio is also reported. The horizontal solid line separates the easy-axis (above) and easy-plane (below) SIA regions.

**Figure 2.**Trivial spin states with ${Q}_{m.u.c}=0$. Snapshots at $T=1$ K from Monte-Carlo simulations of real-space spin configurations (

**a**,

**d**,

**g**,

**j**) and corresponding topological charge densities ${\mathsf{\Omega}}_{i}$ (

**b**,

**e**,

**h**,

**k**), in sequence: black arrows represent in-plane $\{{s}_{x},{s}_{y}\}$ components of spins; colormap indicates the out-of-plane ${s}_{z}$ spins component in the first snapshot and ${\mathsf{\Omega}}_{i}$ in the second one. The associated spin structure factor S(

**q**) is also shown (

**c**,

**f**,

**i**,

**l**); its decomposition in the Cartesian components ${S}_{x}$(

**q**), ${S}_{y}$(

**q**) and ${S}_{z}$(

**q**) is shown for the spin spirals configurations (first and second panels). Capital letters on each spin configuration refer to the labels used in the phase diagram Figure 1c to name each spin configuration: A-phase refers to a single-(

**q**) spiral (proper screw), B-phase refers to a single-(

**q**) spiral (tilted cycloid), H-phase refers to a triple-(

**q**) trivial state with FM component at $\mathsf{\Gamma}$-(

**q**$=0$), I-phase refers to a FM state.

**Figure 3.**Topological lattices with ${\left|Q\right|}_{m.u.c}=6$. Snapshots at $T=1$ K from Monte-Carlo simulations of real-space spin configurations (

**a**,

**d**) and corresponding topological charge densities ${\mathsf{\Omega}}_{i}$ (

**b**,

**e**), in sequence: black arrows represent in-plane $\{{s}_{x},{s}_{y}\}$ components of spins; colormap indicates the out-of-plane ${s}_{z}$ spins component in the first snapshot and ${\mathsf{\Omega}}_{i}$ in the second one. The associated spin structure factor S(

**q**) is also shown (

**c**,

**f**). Capital letters on each spin configuration refer to the labels used in the phase diagram Figure 1c to name each spin configuration: C-phase refers to a triple-(

**q**) state, composed by periodic repetition of two vortices (V, $m=+1$) and an anti-bi-vortex (A2V, $m=-2$) with total $Q=2$; D-phase refers to a triple-(

**q**) state, consisting of a quasi-ideal anti-bi-skyrmions (A2SK) lattice.

**Figure 4.**Topological lattices with ${\left|Q\right|}_{m.u.c}=3$. Similarly to Figure 3, snapshots at $T=1$ K of real-space spin configurations (

**a**,

**d**) and topological charge densities ${\mathsf{\Omega}}_{i}$ (

**b**,

**e**), for the field-induced triple-(

**q**) states: E-phase, consisting of a Bloch-type skyrmions (SK) lattice; F-phase consisting of a anti-bi-merons (A2M) lattice. The peak at $\mathbf{q}=0$ in the S(

**q**) reflects the ferromagnetic component induced by the applied ${B}_{z}$ (

**c**,

**f**).

**Figure 5.**Topological lattice with ${\left|Q\right|}_{m.u.c}=1.5$. Similarly to Figure 3, snapshots at $T=1$ K of the real-space spin configuration (

**a**) and topological charge density ${\mathsf{\Omega}}_{i}$ (

**b**), for the G-phase, i.e., a triple-(

**q**) state, consisting of a lattice of merons from the action of large easy-plane anisotropy and ${B}_{z}$, as from the $\mathbf{q}=0$-peak in the S(

**q**) (

**c**), on the SK-lattice of the E-phase.

**Figure 6.**Evolution of $\left|Q\right|$ per magnetic unit cell (${L}_{m.u.c.}=8{a}_{0}$), as a function of the magnetic field ${B}_{z}$ for different off-diagonal terms, while keeping fixed diagonal exchange parameters and easy-plane anisotropy to values reported in Table 1. The color gradient of the solid points evolves with the ${A}_{zz}/\left|{J}_{yz}\right|$ ratio. At zero and low ${B}_{z}$ field, the system stabilizes a proper screw like the A-phase for ${J}_{yz}/{J}^{1iso}\le 0.14$, and the topological lattice of the C-phase for ${J}_{yz}/{J}^{1iso}>0.14$. At intermediate ${B}_{z}$, the system undergoes a topological phase transition to a Bloch-type skyrmion lattice, E-phase. At high ${B}_{z}$, the transition to the ferromagnetic state, I-phase, takes place passing trough the trivial triple-$\mathbf{q}$ state, H-phase.

**Figure 7.**Profile of the out-of-plane component of spins and in-plane spin texture of the C-phase. (

**a**) ${s}_{z}$ component of spins for each magnetic site from the MC simulations, when moving from the upward (${s}_{z}=+1$) core (spin site n. 0) of the topological object (A2V) along the radial directions (${0}^{\circ},\phantom{\rule{3.33333pt}{0ex}}{30}^{\circ},\phantom{\rule{3.33333pt}{0ex}}{60}^{\circ}\phantom{\rule{3.33333pt}{0ex}}\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}{90}^{\circ}$, purple lines) toward the next-neighbor A2V-cores. This defines the ${s}_{z}$ profile, which can be regarded to as a discretized polarity p associated with the anti-bi-vortex, when moving from its core to its finite edge, defined by the distance with respect to the nearest-neighbor downward cores (${s}_{z}=-1$) of the surrounding vortices; the A2V-radius counts $\simeq 4$ spins (

**b**). (

**c**) In-plane components of spins; arrows are colored with the azimuthal angle $\theta \in [{0}^{\circ},{360}^{\circ})$, defined by the ${s}_{x}$ and ${s}_{y}$ components of the spin vector $\mathit{s}$ and highlighting the in-plane spins rotational direction. A schematic representation of the two vortices and anti-bi-vortex together with a color wheel in (

**g**) further helps the visualization of the planar orientation of spins. (

**d**) Lateral view of the local eigenvectors for each $M-X-M-X$ spin-ligand plaquette on the triangular M-net to help visualization of the non-coplanarity and non-collinearity in the exchange-tensor principal axes. (

**e**) In-plane components of the red eigenvector pointing along the $X-X$ direction for each six nearest-neighbor magnetic $M-M$ pair. Specific case here concerns the two-ion anisotropy (${\mathit{J}}^{S}$) estimated in monolayer NiI${}_{2}$ [40] (Table 1). Spins on the nearest-neighbour M sites of the central site orient according to the in-plane projection of the noncoplanar principal axes, as from the zoom on the spin texture of the A2V-core obtained by the MC simulations (

**f**).

**Figure 8.**Profile of the out-of-plane component of spins (${s}_{z}$) and planar spin texture with associated azimuthal angle defined by the $\{{s}_{x},{s}_{y}\}$ components, as in Figure 7a–c, for the A2SK-type lattice of the D-phase (

**a**–

**c**), the A2M-lattice of the F-phase (

**d**–

**f**), the Bloch-type SK-lattice of the E-phase (

**g**–

**i**), the meron lattice of the G-phase (

**j**–

**l**), and the trivial triple-

**q**state of the H-phase (

**m**–

**o**).

**Table 1.**Magnetic exchange coupling parameters and single-ion anisotropy, in terms of energy units (meV), for the NiI${}_{2}$ monolayer. In the adopted convention, negative (positive) values of the exchange parameters refer to FM (AFM) magnetic interaction, while the positive (negative) value of SIA (${A}_{zz}$) indicates easy-plane (easy-axis) anisotropy.

${\mathit{J}}^{1\mathit{iso}}$ | ${\mathit{J}}_{\mathit{xx}}^{\mathit{S}}$ | ${\mathit{J}}_{\mathit{yy}}^{\mathit{S}}$ | ${\mathit{J}}_{\mathit{zz}}^{\mathit{S}}$ | ${\mathit{J}}_{\mathit{yz}}^{\mathit{S}}$ | ${\mathit{J}}_{\mathit{xz}}^{\mathit{S}}$ | ${\mathit{J}}_{\mathit{xy}}^{\mathit{S}}$ | ${\mathit{J}}^{3\mathit{iso}}$ | ${\mathit{A}}_{\mathit{zz}}$ |
---|---|---|---|---|---|---|---|---|

−7.0 | −1.0 | +1.4 | −0.3 | −1.4 | 0.0 | 0.0 | +5.8 | +0.6 |

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**MDPI and ACS Style**

Amoroso, D.; Barone, P.; Picozzi, S.
Interplay between Single-Ion and Two-Ion Anisotropies in Frustrated 2D Semiconductors and Tuning of Magnetic Structures Topology. *Nanomaterials* **2021**, *11*, 1873.
https://doi.org/10.3390/nano11081873

**AMA Style**

Amoroso D, Barone P, Picozzi S.
Interplay between Single-Ion and Two-Ion Anisotropies in Frustrated 2D Semiconductors and Tuning of Magnetic Structures Topology. *Nanomaterials*. 2021; 11(8):1873.
https://doi.org/10.3390/nano11081873

**Chicago/Turabian Style**

Amoroso, Danila, Paolo Barone, and Silvia Picozzi.
2021. "Interplay between Single-Ion and Two-Ion Anisotropies in Frustrated 2D Semiconductors and Tuning of Magnetic Structures Topology" *Nanomaterials* 11, no. 8: 1873.
https://doi.org/10.3390/nano11081873