First-principles prediction of sub-10-nm skyrmions in Pd/Fe bilayers on Rh(111)

We show that stable skyrmions with diameters of a few nanometers can emerge in atomic Pd/Fe bilayers on the Rh(111) surface. Based on density functional theory we calculate the exchange and the Dzyaloshinskii-Moriya interaction as well as the magnetocrystalline anisotropy energy. The latter two terms are driven by spin-orbit coupling and significantly reduced compared to Pd/Fe bilayers on Ir(111) as expected since Rh and Ir are isoelectronic $4d$ and $5d$ transition metals. However, there is still a spin spiral ground state at zero magnetic field. Atomistic spin dynamics simulations show that a skyrmion phase occurs at small magnetic fields of $\sim 1$ T. Skyrmion diameters amount to 2–8 nm and skyrmion lifetimes are up to 1 h at temperatures of 25–45 K.

Figure 1

(a) Energy dispersion $E$(q) of homogeneous cycloidal flat spin spirals for fcc-Pd/Fe/Rh(111) without (green dots) and with SOC (red dots) in $\overline{\mathrm{\Gamma }}\text{−}\overline{K}$ high-symmetry direction for both senses of rotation. The energy is given relative to the ferromagnetic state. The dispersion is fitted to the Heisenberg model (green line) and includes the DMI and MAE (red line). The inset is the energy dispersion along the $\overline{\mathrm{\Gamma }}\text{−}\overline{M}$ direction. (b) Total and element resolved contributions of $\mathrm{\Delta }{E}_{\mathrm{SOC}}\left(\mathbf{q}\right)$ upon including SOC.

Figure 2

Zero-temperature phase diagram for (a) fcc and (b) hcp Pd overlayer stacking on Fe/Rh(111) obtained based on DFT parameters for the magnetic interactions. The energies of the FM, skyrmion lattice (SkX), and spin spiral (SS) states are shown relative to the homogeneous spin spiral (zero line). Blue, red, and green colors represent the regime of the SS, SkX, and FM ground state, respectively.

Figure 3

(a) Radii of skyrmions in Pd/Fe/Rh(111) obtained for DFT parameters as a function of the applied magnetic field. The radii of the skyrmions were computed as in Ref. [39]. The inset shows the radii variations as a function of $K$ and $D$ for a fixed $B=4.0$ T. (b) The energy barriers of isolated skyrmion collapse based on DFT parameters are shown as a function of applied magnetic field for Pd/Fe/Rh(111). For comparison the radii and energy barriers for Pd/Fe/Ir(111) are also shown [37].

Figure 4

Lifetime of skyrmion collapse in (top) fcc-Pd/Fe/Rh(111) and (bottom) hcp-Pd/Fe/Rh(111) obtained in harmonic transition state theory based on DFT parameters as a function of magnetic field and temperature.