Modeling the thickness dependence of the magnetic phase transition temperature in thin FeRh films

FeRh and its first-order phase transition can open new routes for magnetic hybrid materials and devices under the assumption that it can be exploited in ultra-thin-film structures. Motivated by experimental measurements showing an unexpected increase in the phase transition temperature with decreasing thickness of FeRh on top of MgO, we develop a computational model to investigate strain effects of FeRh in such magnetic structures. Our theoretical results show that the presence of the MgO interface results in a strain that changes the magnetic configuration which drives the anomalous behavior.

Figure 1

Experimentally determined magnetization as a function of temperature during heating (red triangular points) and cooling (blue circular points). The black curves are fits to the data in the region of the transition, and the data has been offset for clarity. The left inset shows the extracted transition temperature as a function of thickness using the procedure described in the text. The schematic inset on the right shows diagrammaticality the FeRh/MgO structure.

Figure 2

(a) Schematic of the system configuration used in the numerical simulations. The top (dark red) layer is FeRh of varying thickness ($x$) and the bottom (dark blue) is the region of strained FeRh of thickness 1.5 nm and can be either antiferromagnetic ($J_{\mathrm{m}}<0$) as depicted here or ferromagnetic ($J_{\mathrm{m}}>0$), not shown. Depending on the exchange within the strained layer ($J_{\mathrm{m}}$) and the exchange at the interface ($J_{\mathrm{int}}$) with the unstrained FeRh there are four possible configurations of the spins as depicted in (b) AFM and (c) FM. Depending on the combination of $J_{\mathrm{m}}$ and $J_{\mathrm{int}}$, either above or below the phase transition, there is always some frustration (green rectangles) where the exchange cannot be satisfied exactly. Note that the four-spin exchange is not included in the interface layer.

Figure 3

Calculated thermal hysteresis curves for a range of FeRh thicknesses from 2 nm (lower) to 8 mn (upper) with the red lines representing the heating curve and the blue lines representing the cooling curve. The inset shows the transition temperature as a function of film thickness (taken as the midpoint between the cooling and the heating curves and found by fitting the data as discussed in the text).

Figure 4

Schematic of the difference between the bulk (left-hand side) and the surface (right-hand side) exchange interactions for the bilinear (upper row) and four-spin (lower row) terms. For the bilinear terms there are fewer dangling exchange bonds as the surface (in-plane) interactions are still included (for the nearest neighbors six interactions in bulk and five at the surface). However, for the four-spin terms there are half the number of complete quartets as the central atom (denoted with a white cross) must always interact with three spins from the other (antiferromagnetic) sublattice in the formation of one of the basic four quartets. Therefore, a fully coordinated atom in the bulk has 32 quartets (not all shown) and at the surface has just 16.

Figure 5

Numerically determined transition temperatures as a function of FeRh thickness for a 1.5-nm strained FeRh layer. The red curves show the case for an antiferromagnetic interface material ($J_{\mathrm{m}}<0$), which both show an increase in the transition temperature, although the effect is reduced for a ferromagnetic interface coupling ($J_{\mathrm{int}}$). For a ferromagnetic interface material ($J_{\mathrm{m}}>0$), the transition temperature is reduced, although the change is small for a ferromagnetic interface coupling ($J_{\mathrm{int}}>0$).

Figure 6

Simulated thermal hysteresis curves (blue curve for cooling and red curve for heating) for a 6-nm FeRh layer and both (a) ferromagnetic and (b) antiferromagnetic strained layers for a range of exchange values of $|J_{\mathrm{m}}|$ and $|J_{\mathrm{int}}|$. The ferromagnetic interface shows a decreasing transition temperature with increasing exchange, whereas the antiferromagnetic is increasing.

Figure 7

Calculated transition temperature for a range of FeRh thicknesses for an antiferromagnetic strained FeRh layer ($J_{\mathrm{m}}<0$ and $J_{\mathrm{int}}<0$) for a range of exchange $J_{\mathrm{m}}$ and $J_{\mathrm{int}}\left(J_{\mathrm{m}}=J_{\mathrm{int}}\right)$. For low exchange in the strained layer (e.g., the red curve) there is an increasing transition temperature with thickness. The trend is reversed as the exchange is increased in the interface layer (the blue curve).