Short-Period Skyrmion Crystals in Itinerant Body-Centered Tetragonal Magnets

Abstract

In this study, we investigate the stability of a magnetic skyrmion crystal with short-period magnetic modulations in a centrosymmetric body-centered tetragonal system. By performing the simulated annealing for the spin model, incorporating the effects of the biquadratic interaction and high-harmonic wave–vector interaction in momentum space, we find that the double-Q square skyrmion crystal consisting of two spin density waves is stabilized in an external magnetic field. We also show that double-Q states appear in both low- and high-field regions; the low-field spin configuration is characterized by an anisotropic double-Q modulation consisting of a superposition of the spiral wave and sinusoidal wave, while the high-field spin configuration is characterized by an isotropic double-Q modulation consisting of a superposition of two sinusoidal waves. Furthermore, we show that the obtained multiple-Q instabilities can be realized for various ordering wave vectors. The results provide the possibility of realizing the short-period skyrmion crystals under the body-centered tetragonal lattice structure.

1. Introduction

Magnetic skyrmions with topologically nontrivial swirling spin textures have attracted great interest for years in both theory and experiment [1,2,3,4,5,6,7]. Many interesting phenomena arise owing to their unconventional topological properties ascertained through the spin Berry phase mechanism, such as the topological Hall effect [8,9,10,11,12,13,14,15,16,17,18,19,20,21], the topological Nernst effect [22,23,24,25], the skyrmion Hall effect [26,27,28,29,30,31,32,33,34,35,36], the magnetoelectric effect [37,38,39,40,41,42,43,44,45,46], and nonreciprocal transport [47,48,49,50,51,52,53,54]. Exploring a periodic alignment of the skyrmion, i.e., the skyrmion crystal (SkX), is a central topic in modern condensed matter physics.

One of the guidelines for exploring the SkX in materials is to take into account the effect of the Dzyaloshinskii–Moriya (DM) interaction in noncentrosymmetric crystals [3,55,56,57]. The competition between the ferromagnetic exchange interaction and the DM interaction in an external magnetic field leads to the emergence of the SkX in chiral [4,5,6,37,58,59,60,61,62,63], polar [64,65,66], and other noncentrosymmetric crystals [67,68,69,70], including the two-dimensional van der Waals systems [71,72,73,74]. Meanwhile, recent studies revealed another guideline to engineering the SkX based on the Fermi surface nesting [75,76,77]. Since the nesting property can be found irrespective of the spatial inversion symmetry, such a mechanism might account for the emergence of the SkX in both centrosymmetric magnets, such as NiMnIn [78], NiMnGa [78], ${\mathrm{Gd}}_2$${\mathrm{PdSi}}_3$ [79,80,81,82,83,84,85,86], ${\mathrm{Gd}}_3$${\mathrm{Ru}}_4$${\mathrm{Al}}_{12}$ [87,88,89,90], ${\mathrm{GdRu}}_2$${\mathrm{Si}}_2$ [91,92,93,94,95,96,97,98,99], and ${\mathrm{GdRu}}_2$${\mathrm{Ge}}_2$ [100], and noncentrosymmetric magnets, such as EuPtSi [101,102,103,104,105,106,107,108,109,110,111] and ${\mathrm{EuNiGe}}_3$ [112,113,114,115]. In particular, the latter mechanism often induces the short-period SkX with large multiple-Q ordering wave vectors; the typical magnetic modulation of the spiral pitch is around a few nanometers in the latter mechanism, while it is around a few hundreds of nanometers in the former mechanism [116]. Since a short-period SkX tends to produce a large emergent magnetic field, as found in the similar short-periodic magnetic textures [10,11,117,118], engineering the short-period (high-density) SkX is promising for future spintronics applications, such as energy-efficient devices using high-density topological objects [119].

In the present study, we investigate the stabilization condition of the short-period SkX by focusing on the bond-centered structure in centrosymmetric tetragonal systems. We introduce an effective spin model consisting of minimal ingredients to stabilize the SkX and construct the magnetic phase diagram by performing the simulated annealing. We show that a positive biquadratic interaction and a high-harmonic wave–vector interaction play an important role in stabilizing the SkX. The SkX phase appears in the intermediate-field region, which turns into a topologically trivial double-Q state with anisotropic (isotropic) modulations at two ordering wave vectors when the strength of the magnetic field is smaller (larger). We also discuss the phase diagram obtained by a different set of ordering wave vectors. As a result, we show that the difference in the ordering wave vectors does not alter the stability of the short-period SkX qualitatively. The present results indicate that the stability of the short-period SkX on the body-centered tetragonal lattice is similar to that on the primitive tetragonal lattice, irrespective of the magnitude of the ordering wave vectors [120,121,122].

The rest of this paper is organized as follows. In Section 2, we introduce a fundamental spin model with momentum-resolved spin interactions in order to capture the essence of the SkX. We also show the numerical method based on the simulated annealing from high temperatures. Then, we discuss when and how the SkX is stabilized while showing the magnetic phase diagram in the model in Section 3. We also detail the magnetic structures for other double-Q states. In Section 4, we discuss the results when the ordering wave vectors are varied. We conclude with the results of this paper in Section 5.

2. Model and Method

We investigate the instability toward the short-period SkX by analyzing a spin model with the momentum-resolved interaction on a three-dimensional body-centered tetragonal lattice under the space group $I4/mmm$. The primitive translational vector is given by ${\mathit{a}}_1=(a,0,0)$, ${\mathit{a}}_2=(0,a,0)$, and ${\mathit{a}}_3=(a/2,a/2,c/2)$; we set $a=c=1$ for simplicity. Especially, We focus on the ground-state spin configuration in the thermodynamic limit. For that purpose, we consider the following spin model, which is given by [123]

$$\begin{array}{cc}\hfill \mathcal{H}=& -2{\displaystyle \sum _{\nu }}{J}_{\nu }({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})+{\displaystyle \frac{2}{N}}{\displaystyle \sum _{\nu }}{K}_{\nu }{({\mathit{S}}_{{\mathit{Q}}_{\nu }}·{\mathit{S}}_{-{\mathit{Q}}_{\nu }})}^2-H{\displaystyle \sum _i}{S}_i^z,\hfill \end{array}$$

(1)

where ${\mathit{S}}_i=(S_i^x,S_i^y,S_i^z)$ denotes the classical spin at site i, where the spin length is set to $|{\mathit{S}}_i|=1$, and ${\mathit{S}}_{{\mathit{Q}}_{\nu }}=(S_{{\mathit{Q}}_{\nu }}^x,S_{{\mathit{Q}}_{\nu }}^y,S_{{\mathit{Q}}_{\nu }}^z)$ denotes the ${\mathit{Q}}_{\nu }$ component of ${\mathit{S}}_i$; ${\mathit{Q}}_{\nu }$ is the ordering wave vector and $\nu$ is its index.

The spin model consists of three terms. The first term stands for the momentum-resolved bilinear exchange interaction with the coupling constant $J_{\nu }$. One of the origins of this interaction is the Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction [124,125,126]; for example, the lowest-order perturbative expansions in terms of the Kondo coupling in the Kondo lattice model leads to this bilinear exchange interaction. The schematic Feynman diagram corresponding to the lowest-order contribution in the original Kondo lattice model is shown in Figure 1a. We suppose that the interactions at particular wave vectors give the dominant contributions to the internal energy with the nesting of the Fermi surfaces in mind. We set the dominant interactions $J_1\equiv J=1$ at ${\mathit{Q}}_1=(6\pi /5,0,0)$. From the fourfold rotational symmetry, $J_2$ at ${\mathit{Q}}_2=(0,6\pi /5,0)$ is equivalent to $J_1$. In other words, we consider the situation where the Fermi surface nesting occurs at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$. It is noted that such large ordering wave vectors as $|{\mathit{Q}}_1| >\pi$ are owed to the body-centered structure. Furthermore, we take into account the interactions at the high-harmonic wave vectors, ${\mathit{Q}}_3={\mathit{Q}}_1+{\mathit{Q}}_2$ and ${\mathit{Q}}_4=-{\mathit{Q}}_1+{\mathit{Q}}_2$, as $J_3=J_4\equiv J^{\prime }$ since such high-harmonic wave–vector interactions tend to favor the multiple-Q states including the SkX [96,127]. We ignore the interactions at other wave vectors in the Brillouin zone, since they do not affect the energy of the magnetic phases that are characterized by ${\mathit{Q}}_1$–${\mathit{Q}}_4$. We show that a similar result can be obtained for other ordering wave vectors in Section 4. In addition, in order to focus on the role of the biquadratic interaction and high-harmonic wave–vector interaction on the stabilization of the SkX, we neglect the effect of magnetic anisotropy, such as the DM interaction, Ising-type anisotropy, and bond-dependent anisotropy for simplicity [128], which tends to favor the SkX [129,130].

The second term in Equation (1) stands for the biquadratic exchange interaction with the coupling constant $K_{\nu }$ [$K_1=K_2\equiv K$ and $K_3=K_4\equiv {(J^{\prime }/J)}^{2}K$], where N is the total number of spins in the system. The interaction arises from the higher-order contribution of the RKKY interaction in the perturbative expansion in the Kondo lattice model [131,132], which leads to unconventional multiple-Q states such as the vortex crystal [133] and hedgehog crystal [134]. We show the schematic Feynman diagram corresponding to the biquadratic interaction in Figure 1b. It is noted that the nature of this momentum-resolved biquadratic interaction is different from that of the real-space biquadratic interaction, although both of them tend to prefer the stabilization of the SkX [135,136]. In order to systematically examine the effect of K, we deal with it as a variable. The third term in Equation (1) stands for the Zeeman coupling under an external magnetic field along the z direction.

The model in Equation (1) describes the ground-state spin configuration in an efficient way, since it only includes the dominant interactions at particular ordering wave vectors. On the other hand, the simplification of the model makes the following analyses difficult. One is the stability against defects and surface roughness in real space. Since the model only includes the momentum-resolved interaction, it is difficult to analyze such effects; the analysis based on the spin model with the real-space interaction is desired. The other is the effect of other degrees of freedom, such as the phonon, which often plays an important role in discussing the stability of magnetic states through the spin–lattice coupling. Although the above effects might affect the magnetic stability in the following section, we suppose that the above effects are omitted by considering the clean sample and a negligible spin–lattice coupling.

The magnetic instability in the model in Equation (1) is investigated by performing the simulated annealing by changing K and H for fixed $J=1$ and $J^{\prime }=0.4$. Starting from a random spin configuration at the temperature $T_0=1$–10, we gradually reduce the temperature with a rate $T_{n+1}=0.999999T_n$ in each Monte Carlo sweep up to the final temperature $T=0.001$, where $T_n$ is the nth-step temperature. In each Monte Carlo sweep, we locally update localized spins based on the Metropolis algorithm. At the final temperature, we perform ${10}^5$–${10}^6$ Monte Carlo sweeps for measurements. We set the lattice size as $N={10}^3$ under the periodic boundary condition. Although the effect of thermal fluctuations can be investigated by the steepest descent method [137], we focus on the stability of the ground-state spin configuration; typically, the magnetic states are stable up to the order of $T/J=0.1$–1.

The spin configurations obtained at the final temperature are classified by the spin structure factor and scalar spin chirality. In other words, different magnetic phases in the phase diagram are characterized by different peak profiles of the spin structure factor and scalar spin chirality. The spin structure factor is given by

$$\begin{array}{cc}\hfill S_s^{\eta \eta }\left(\mathit{q}\right)& ={\displaystyle \frac{1}{N}}{\displaystyle \sum _{i,j}}{S}_i^{\eta }{S}_j^{\eta }{e}^{i\mathit{q}·({\mathit{r}}_i-{\mathit{r}}_j)},\hfill \end{array}$$

(2)

for $\eta =x,y,z$. ${\mathit{r}}_i$ is the position vector at site i and $\mathit{q}$ is the wave vector in the first Brillouin zone. By using $S_s^{\eta \eta }\left(\mathit{q}\right)$, the ${\mathit{Q}}_{\nu }$ component of the magnetic moments is given by

$$m_{{\mathit{Q}}_{\nu }}^{\eta }=\sqrt{{\displaystyle \frac{S_s^{\eta \eta }\left({\mathit{Q}}_{\nu }\right)}{N}}},$$

(3)

where the in-plane spin component of $m_{{\mathit{Q}}_{\nu }}^{\eta }$ is defined as ${\left(m_{{\mathit{Q}}_{\nu }}^{xy}\right)}^2={\left(m_{{\mathit{Q}}_{\nu }}^x\right)}^2+{\left(m_{{\mathit{Q}}_{\nu }}^y\right)}^2$. The uniform magnetization along the field direction is given by $M=(1/N){\sum }_i{S}_i^z$.

In the chirality sector, the scalar spin chirality is given by

$$\begin{array}{cc}\hfill {\chi }^{\mathrm{sc}}& ={\displaystyle \frac{1}{N}}{\displaystyle \sum _i}{\displaystyle \sum _{\delta ,{\delta }^{\prime }=\pm 1}}\delta {\delta }^{\prime }{\mathit{S}}_i·({\mathit{S}}_{i+\delta {\widehat{d}}_1}\times {\mathit{S}}_{i+{\delta }^{\prime }{\widehat{d}}_2}),\hfill \end{array}$$

(4)

where ${\widehat{d}}_1$ (${\widehat{d}}_2$) represents a shift by a in the $\left[100\right]$ ($\left[010\right]$) direction. Nonzero ${\chi }^{\mathrm{sc}}$ often leads to the emergence of the SkX.

3. Results

Figure 2 shows the magnetic phase diagram by changing the biquadratic interaction K and the magnetic field H at $J^{\prime }=0.4$. There are four magnetic phases denoted as 1Q, 2$Q^{\prime }$, 2Q, and SkX, except for the fully polarized state appearing in the high-field region for $H\gtrsim 2$; 1Q stands for the single-Q state, while 2Q (2$Q^{\prime }$) stands for the double-Q state with the same (different) intensities at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ in the spin structure factor. The 1Q, 2Q, and 2$Q^{\prime }$ states do not exhibit the scalar spin chirality, while the SkX does. Although the 2Q and 2$Q^{\prime }$ states are topologically trivial, their emergence is important in accounting for the experimental phase diagram in ${\mathrm{GdRu}}_2$${\mathrm{Si}}_2$ [91,92,93].

For $K=0$, the 1Q state appears irrespective of H. The spin configuration in this state is characterized by the conical spiral structure, whose spiral plane lies on the $xy$ plane perpendicular to the magnetic field. When K is introduced, the 1Q state in the low-field region is replaced by the 2$Q^{\prime }$ state, as shown in Figure 2. This state is characterized by the double-Q superposition of the $xy$-plane cycoloidal spiral wave at ${\mathit{Q}}_1$ and the z-component sinusoidal wave at ${\mathit{Q}}_2$, where the magnitude of the ${\mathit{Q}}_1$ component is much larger than that of the ${\mathit{Q}}_2$ component, as shown in Figure 3b. Such a superposition is seen by the real-space spin configuration in the leftmost and middle left panels of Figure 4a. In contrast to the 1Q state, the spin configuration of the $2Q^{\prime }$ state is noncoplanar, even without the uniform magnetization. Accordingly, the 2$Q^{\prime }$ state accompanies the density waves in terms of the scalar spin chirality, as shown in the middle right and rightmost panels in Figure 4a [138]. It is noted that the uniform component of the scalar spin chirality is zero; the 2$Q^{\prime }$ state is topologically trivial. With the increase in H, the ${\mathit{Q}}_2$ component of the magnetic moments becomes smaller and vanishes, meaning that the transition to the 1Q state has occured, as shown in Figure 3b. This transition is of second order with the continuous change in the magnetization, as shown in Figure 3a.

The SkX and the 2Q state emerge for large K, as shown in Figure 2. When H increases in the 2$Q^{\prime }$ state for $K\gtrsim 0.27$, the SkX is realized for $0.6\lesssim H\lesssim 0.8$ in the case of $K=0.3$ in Figure 3c,d. The transition between the 2$Q^{\prime }$ and the SkX is of first order, with the jump of the magnetization M owed to the different topological properties between the 2$Q^{\prime }$ and the SkX, as shown in Figure 3c; it is noted that no SkX appears at the zero field in the present model in contrast to the real-space spin model with a vanishingly small DM interaction [136]. In the SkX, both the $xy$ and z components of magnetic moments have equal intensities at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$ [Figure 3d], which indicates that the fourfold rotational symmetry is kept. The real-space spin configuration of the SkX is shown in the leftmost and middle left panels of Figure 4b. The skyrmion cores denoted as $S_i^z=-1$ are located at the $z=0$ plane; the skyrmion cores form the square lattice. Owing to the short-period magnetic modulations, the spins surrounding the skyrmion core point along the positive z direction. Since the vorticity around the skyrmion core is $-1$, this skyrmion is categorized into the anti-skyrmion with the positive skyrmion number of $+1$. Indeed, the real-space scalar spin chirality takes positive values, as shown in the middle right and rightmost panels of Figure 4b. It is noted that this SkX with the skyrmion number of $+1$ is energetically degenerate with that of the skyrmion number of $-1$, owing to the $xy$-spin rotational symmetry of the model in Equation (1). Such a degeneracy can be lifted when the effect of the magnetic anisotropy arising from the spin–orbit coupling is taken into account [139]. The SkX turns into the 2Q state with the jump of M and ${\chi }^{\mathrm{sc}}$ by increasing H, as shown in Figure 3c.

The 2Q state appears in the high-field region for large K, as shown in Figure 2. The transition from the SkX to the 2Q state is discontinuous where the magnetization shows the jump, as shown in Figure 3c. The spin configuration of this state consists of the superposition of two sinusoidal waves at ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$, where the sinusoidal spin directions are on the $xy$ plane and they are also perpendicular to each other. This state is also topologically trivial without the uniform component of the scalar spin chirality; the scalar spin chirality is locally induced, as shown in the middle right and rightmost panels of Figure 4.

Finally, let us comment on the role of the high-harmonic wave–vector interactions. When $J^{\prime }$ is turned off, the region where the SkX is stabilized almost vanishes. Instead, the regions of the $2Q^{\prime }$ and $2Q$ are extended. Thus, the interplay between the biquadratic interaction and high-harmonic wave–vector interaction plays an important role in realizing the SkX; a similar tendency is also found in the model on the primitive tetragonal lattice [96].

4. Other Ordering Wave Vectors

The results in the previous section indicate the emergence of the short-period SkX on the body-centered tetragonal lattice. Such a stability feature is unchanged for different ordering wave vectors. To demonstrate this, we consider the situation where the ordering wave vectors are located at ${\mathit{Q}}_1^{\prime }=(7\pi /5,0,\pi )$ and ${\mathit{Q}}_2^{\prime }=(0,7\pi /5,\pi )$ instead of ${\mathit{Q}}_1$ and ${\mathit{Q}}_2$; we consider the short-period modulation for the z direction as well as the in-plane direction. We adopt the same model parameters and set the high-harmonic wave vectors, ${\mathit{Q}}_3^{\prime }={\mathit{Q}}_1^{\prime }+{\mathit{Q}}_2^{\prime }$ and ${\mathit{Q}}_4^{\prime }=-{\mathit{Q}}_1^{\prime }+{\mathit{Q}}_2^{\prime }$.

Figure 5 shows the phase diagram at $T=0.001$ in the plane of K and H. Compared to the phase diagram in Figure 2, one finds that the overall phase boundaries among the 1Q, 2Q, 2$Q^{\prime }$, and SkX phases are almost unchanged. Owing to the staggered modulated tendency, the SkX exhibits the uniform component of the scalar spin chirality. Thus, the momentum-resolved spin model in Equation (1) captures the instability toward the SkX, irrespective of the magnitude of the ordering wave vectors.

Additionally, the spin configurations in each phase look slightly different, owing to the different ordering wave vectors. We show the real-space spin and scalar spin chirality configurations of the 2$Q^{\prime }$, SkX, and 2Q states in Figure 6a–c, respectively. In the end, the emergence of the three double-Q states (2$Q^{\prime }$, SkX, and 2Q) is expected in itinerant magnets, irrespective of the ordering wave vectors, where K plays an important role.

5. Conclusions

In conclusion, we have investigated the stability of the SkX on the body-centered tetragonal lattice. In order to focus on the instability toward the short-period SkX, we examined the momentum-resolved spin model with large ordering wave vectors. The simulated annealing analyses revealed the emergence of the SkX as a consequence of the synergy between the biquadratic interaction and high-harmonic wave–vector interaction in the intermediate-field region, irrespective of the position of the ordering wave vectors. We have also shown that two types of double-Q states appear in the low- and high-field regions. The present model will serve as a more realistic fundamental model to examine the SkX-hosting materials in the body-centered tetragonal system in ${\mathrm{GdRu}}_2$${\mathrm{Si}}_2$ [91,92,93,94,95,96,97,98,99] and ${\mathrm{GdRu}}_2$${\mathrm{Ge}}_2$ [100], since some model calculations have been performed only for a simple primitive square lattice [120,121,122]. In particular, three magnetic phases observed in ${\mathrm{GdRu}}_2$${\mathrm{Si}}_2$ [91,92,93] correspond to the 2Q′, the SkX, and the 2Q state from the low-field region.