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First-Principles Prediction of Skyrmionic Phase Behavior in GdFe_{2} Films Capped by 4d and 5d Transition Metals

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## Abstract

**:**

## 1. Introduction

## 2. Methods

**q**with a constant angle of $\varphi $, where $\varphi $ is defined as

**q**·

**R**.

**M**${}_{i}$ and

**M**${}_{j}$ at sites

**R**${}_{i}$ and

**R**${}_{j}$, respectively. Here, ${m}_{i}$ is defined as

**M**${}_{i}$/${\mu}_{s}$. Both energy dispersion curves (with and without SOC) are calculated and fitted to extract the parameters for the exchange interactions (${J}_{ij}$) and the DMI (${D}_{ij}$).

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Energy dispersion E(q) of homogeneous cycloidal flat spin spirals in high-symmetry direction $\overline{\Gamma}$-$\overline{\mathrm{K}}$ for (

**a**) GdFe${}_{2}$/Rh and (

**b**) GdFe${}_{2}$/Rh films. Filled and empty symbols represent E(q) with and without SOC, respectively. The energy is given relative to the magnetic ground state. The dispersion is fitted to the Heisenberg model (dotted line) and includes the DMI and MCA (solid line).

**Figure 2.**(

**a**) Total magnetocrystalline anisotropy (MCA) energy and (

**b**) effective Dzyaloshinskii–Moriya interaction (DMI) of GdFe${}_{2}$ with TM capping layer.

**Figure 3.**(

**a**) Side view and top view of GeFe${}_{2}$ film capped by a transition-metal (TM) monolayer. Blue, gray, and red balls represent Gd, Fe, and TM atoms, respectively. TM atoms are on the hollow site of GeFe${}_{2}$; (

**b**) Interface distances between the TM capping layer and GeFe${}_{2}$ after structural optimization; (

**c**) Magnetic moments of TM atoms, induced by GeFe${}_{2}$.

**Figure 4.**Phase diagrams for the (

**a**) GdFe${}_{2}$/Rh and the (

**b**) GdFe${}_{2}$/Ir films at zero temperature. The relative energies of the spin spiral states, skyrmion lattice, and ferromagnetic state are shown. The red, green, and blue colors represent the regime of the spin spiral states, skyrmion lattice, and ferromagnetic state, respectively. (

**c**) Radii of skyrmions in the films of GdFe${}_{2}$/Rh and GdFe${}_{2}$/Ir as a function of the applied magnetic field. (

**d**) Schematic representation of possible spin configurations in a magnetic material with Dzyaloshinsky–Moriya interaction for different values of an external field.

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**MDPI and ACS Style**

Jekal, S.; Danilo, A.; Phuong, D.; Zheng, X.
First-Principles Prediction of Skyrmionic Phase Behavior in GdFe_{2} Films Capped by 4*d* and 5*d* Transition Metals. *Appl. Sci.* **2019**, *9*, 630.
https://doi.org/10.3390/app9040630

**AMA Style**

Jekal S, Danilo A, Phuong D, Zheng X.
First-Principles Prediction of Skyrmionic Phase Behavior in GdFe_{2} Films Capped by 4*d* and 5*d* Transition Metals. *Applied Sciences*. 2019; 9(4):630.
https://doi.org/10.3390/app9040630

**Chicago/Turabian Style**

Jekal, Soyoung, Andreas Danilo, Dao Phuong, and Xiao Zheng.
2019. "First-Principles Prediction of Skyrmionic Phase Behavior in GdFe_{2} Films Capped by 4*d* and 5*d* Transition Metals" *Applied Sciences* 9, no. 4: 630.
https://doi.org/10.3390/app9040630