Magnetic exchange interactions in B-, Si-, and As-doped Fe${}_2$P from first-principles theory

Di-iron phosphide (Fe${}_2$P) is a parent system for a set of magnetocaloric materials. Although the magnetic ordering temperature ($T_{\mathrm{C}}$ $=$ 215 K) of the stoichiometric composition is too low for room-temperature magnetic refrigeration, the partial replacement of P with B, Si, or As elements results in a steep increase in the magnetic ordering temperature. Doping leads to different equilibrium volumes and hexagonal axial ratios ($c/a$) within the same crystallographic phase over a wide concentration range. Here, using first principles theory, we decompose the change in the total magnetic exchange interaction upon doping into chemical and structural contributions, the latter including the $c/a$-ratio and volume effects. We demonstrate that for the investigated alloys the structural effect can be ascribed mainly to the decrease in the $c/a$ ratio that strengthens the magnetic exchange interactions between the two Fe sublattices.

Figure 1

Experimental Curie temperature ($T_{\mathrm{C}}$) of Fe${}_2$P${}_{1-x}$T${}_x$ ($\mathrm{T}=\mathrm{B}$, Si, As) as a function of the atomic percentage of dopant element:[15, 16, 17] circles represent B; squares, Si; and triangles, As. Dashed lines indicate a crystallographic phase transition for Fe${}_2$P${}_{1-x}$Si${}_x$ and phase separation for Fe${}_2$P${}_{1-x}$B${}_x$.

Figure 2

Calculated Fe-Fe exchange interactions for Fe${}_2$P as a function of distance (in Å). The interactions ${\mathbf{J}}_{\mathbf{ij}}$ are sorted into symmetry-related groups as explained in the text.

Figure 3

Accumulated exchange interactions (see text) including the long-range effects for the six groups of Fe-Fe pairs of Fe${}_2$P as a function of volume at the experimental $c/a$ ratio ($c/a=0.589$).

Figure 4

Accumulated exchange interactions (see text) including the long-range effects for the six groups of Fe-Fe pairs of Fe${}_2$P as a function of $c/a$ ratio at the experimental volume ($V=103.10$ Å${}^3$).

Figure 5

${\mathbf{J}}_{\mathrm{tot}}$ of Fe${}_2$P as a function of volume at different $c/a$ ratios. The $c/a$ ratios correspond to values found experimentally for Fe${}_2$P (solid line), Fe${}_2$P${}_{0.92}$B${}_{0.08}$ (dashed line), Fe${}_2$P${}_{0.9}$Si${}_{0.1}$ (dashed-dotted line), and Fe${}_2$P${}_{0.9}$As${}_{0.1}$ (dotted line).

Figure 6

Total density of states (tDOS) around the Fermi level ($E_{\mathrm{F}}$) of Fe${}_2$P at different $c/a$ ratios, which correspond to the experimental $c/a$-ratio value of Fe${}_2$P (solid line), Fe${}_2$P${}_{0.92}$B${}_{0.08}$ (dashed line), and Fe${}_2$P${}_{0.9}$Si${}_{0.1}$ (dashed-dotted line). The solid vertical line corresponds to the $E_{\mathrm{F}}$ of Fe${}_2$P. The dashed and dashed-dotted vertical lines correspond to the estimated Fermi level of Fe${}_2$P${}_{0.92}$B${}_{0.08}$ and Fe${}_2$P${}_{0.9}$Si${}_{0.1}$, respectively, using the rigid band model. Inset: tDOS around the $E_{\mathrm{F}}$ of Fe${}_2$P (solid line), Fe${}_2$P${}_{0.92}$B${}_{0.08}$ (dashed line), and Fe${}_2$P${}_{0.9}$Si${}_{0.1}$ (dashed-dotted line), respectively, calculated at a fixed (0.589) $c/a$ ratio.