Magnetism and exchange-bias effect at the MnN/Fe interface

Based on ab initio calculations and spin dynamics simulations, we perform a detailed study on the magnetic properties of bulk MnN and the MnN/Fe interface. We determine the spin model parameters for the $\theta$-phase of bulk MnN, and we find that the competition between the nearest and the next-nearest-neighbor interactions leads to antiferromagnetic ordering of the Mn spins, in agreement with previous theoretical and experimental results. At the MnN/Fe interface, a sizable Dzyaloshinskii-Moriya interaction appears leading to a stable exchange-bias effect. We study the dependences of the exchange-bias effect on the thicknesses of the ferromagnetic and the antiferromagnetic layers, and we compare them to experimentally obtained results [Meinert et al., Phys. Rev. B 92, 144408 (2015)].

Figure 1

Calculated isotropic exchange interaction parameters, $J_{ij}^{\mathrm{iso}}$, and biquadratic couplings, $B_{ij}$, as a function of distance between Mn atoms.

Figure 2

Calculated magnetization curve of the two sublattices of Mn in bulk MnN as a function of the temperature. The inset illustrates the magnetic ground state of MnN. Violet spheres represent the N atoms, while red and blue arrows correspond to the spin moments of ${\mathrm{Mn}}_1$ and ${\mathrm{Mn}}_2$ atoms, respectively.

Figure 3

Three investigated stacking structures of the Mn/Fe interface: (a) Fe above hollow position, (b) Fe above Mn, and (c) Fe above N. Fe, N, and Mn atoms are represented by blue, green, and red spheres respectively.

Figure 4

Calculated layer-resolved spin magnetic moments $\mu \left(l\right)$, on-site, and two-site anisotropies, $K\left(l\right)$ and $\mathrm{\Delta }J\left(l\right)$, as a function of the atomic layers labeled by $l$. Negative and positive signs of $K\left(l\right)$ and of $\mathrm{\Delta }J\left(l\right)$ correspond to out-of-plane and in-plane preferred directions of the spin moments, respectively. The blue vertical line denotes the Mn layer at the interface.

Figure 5

Calculated isotropic exchange interactions between the Fe atoms in the first Fe layer ($l=1$) and the Mn atoms, $J_{{\mathrm{Fe}}_1\text{−}\mathrm{Mn}}^{\mathrm{iso}}$, and between the Fe atoms in the second Fe layer ($l=2$) and the Mn atoms, $J_{{\mathrm{Fe}}_2\text{−}\mathrm{Mn}}^{\mathrm{iso}}$, as a function of the distance between the Fe and Mn atoms.

Figure 6

Magnitude of the DM vectors $|D_{ij}|$ and biquadratic couplings $B_{ij}$ between Fe and Mn spin moments, selecting the Fe atoms from the first ($l=1$) and second ($l=2$) layer from the interface as in Fig. 5. Note that $|D_{ij}|$ is also shown for the third ($l=3$) Fe layer.

Figure 7

Simulated magnetic state of the MnN/Fe system near the interface after the field-cooling process in an external cooling field, $H_{\text{cf}}=1\mathrm{T}$. The arrows illustrate the direction of the magnetic moments of Mn and Fe atoms. The green arrows in the top five atomic layers correspond to the spin components of Fe atoms. The bottom-layer green arrows and the yellow arrows directly below represent the spin moments of the Mn atoms near the interface, while red and cyan arrows correspond to the two Mn sublattices of the MnN.

Figure 8

Calculated in-plane hysteresis loops as a function of the AF thickness, $t_{\mathrm{AF}}$ (in ML), for $t_{\mathrm{FM}}=5$ ML. Here, $M_x^{\mathrm{tot}}$ and $M_x^{\mathrm{FM}}$ denote the $x$ magnetization component of the total system and the ferromagnetic side, respectively.

Figure 9

Calculated coercive, $H_C$, and exchange-bias fields, $H_{\text{EB}}$, as a function of the AF thickness, $t_{\text{AF}}$.

Figure 10

Calculated in-plane hysteresis loops as a function of the FM thickness, $t_{\mathrm{FM}}$ (in ML), in the case of $t_{\mathrm{AF}}=25\mathrm{ML}$. Here, $M_x^{\mathrm{tot}}$ and $M_x^{\mathrm{FM}}$ denote the $x$ magnetization component of the total system and the ferromagnetic side, respectively.

Figure 11

Calculated relative remanence magnetization, coercive, $H_C$, and exchange-bias fields, $H_{\text{EB}}$, as a function of the FM thickness, $t_{\mathrm{FM}}$, in the case of $t_{\mathrm{AF}}=25\mathrm{ML}$.