Phonon softening in paramagnetic bcc Fe and its relationship to the pressure-induced phase transition

The structural stability of paramagnetic (PM) body-centered-cubic (bcc) Fe under pressure is investigated based on first-principles phonon calculations. Spin configurations of the PM phase are approximated using a binary special quasirandom structure with a supercell approach. The behavior of phonon modes can be associated with pressure-induced phase transitions to the face-centered-cubic (fcc) and hexagonal-close-packed (hcp) structures as follows: For the PM phase, it is found that the low-frequency transverse mode at the $N$ point ($N_4^{-}$ mode), which corresponds to a bcc-hcp phase transition pathway, exhibits strong softening under isotropic volume compression. The frequency of this mode becomes zero by a 2% volume decrease within the harmonic approximation. This result is not consistent with the experimental fact that the phase transition from the PM bcc to hcp phases does not occur under volume compression. The seeming contradiction can be explained only when the anharmonic behavior of the $N_4^{-}$ mode is taken into consideration; A potential energy curve along the $N_4^{-}$ mode becomes closer to a double-well shape for the PM phase under volume compression. On the other hand, softening of the longitudinal mode at the 2/3[111] point under volume compression is also found for the PM phase, which indicates the pressure-induced bcc-fcc phase transition along this mode. Such behaviors are not seen in ferromagnetic bcc Fe, implying that the magnetic structure plays an essential role on the phase transition mechanism.

Figure 1

Experimental phase diagram of Fe [1, 2, 3, 4]. Solid and dashed bold lines represent structural and magnetic phase boundaries, respectively. For the magnetic structure of the hcp phase, both nonmagnetic (NM) and antiferromagnetic phases have been proposed.

Figure 2

Two-dimensional schematic illustration of the procedure to obtain the force constants of the PM phase. Red and blue circles represent atoms with spin-up and spin-down magnetic moments, respectively. Green small arrows on the circles represent the forces acting on the atoms. (a) The completely disordered spin configuration is mimicked by the binary SQS. (b) Symmetrically inequivalent displacements ${\mathbf{U}}_i^{\prime }$ are applied to the SQS. (c) The displaced atoms are moved to one of the crystallographically equivalent sites of the supercell without magnetic moments. By this operation, ${\mathbf{U}}_i^{\prime }$ is changed to ${\mathbf{U}}_i$. (d) The forces acting on the atoms ${\mathbf{F}}_i$ are calculated. (e) From all the configurations with the displacements, the forces are collected, and the force constants of the PM phase are obtained via Eq. (6).

Figure 3

Two kinds of binary SQSs for the bcc structure investigated in this study. These SQSs are constructed from a supercell obtained by a $2\times 2\times 2$ isotropic expansion of the conventional bcc unit cell and hence include 16 atoms. The completely disordered collinear spin configuration is mimicked by these SQSs. Up (red) and down (blue) arrows represent atoms with spin-up and spin-down magnetic moments, respectively. Visualization is performed using the vesta code [22].

Figure 4

Calculated phonon dispersion relations at the volume of (a) $V_0$, (b) $0.96V_0$, (c) $0.92V_0$, and (d) $0.80V_0$ for bcc Fe, where $V_0$ is the equilibrium volume of each magnetic phase shown in Table 2. Blue solid and red dashed curves are for the PM and FM phases, respectively. Note that a negative value of a phonon frequency corresponds to an imaginary mode.

Figure 5

Potential energy curves along the $N_4^{-}$ mode with respect to volumes for the (a) PM and (b) FM phases. Square, circle, and triangle symbols are for the results at $V_0,0.96V_0$, and $0.92V_0$, respectively, where $V_0$ denotes the equilibrium volume. Lines connecting the symbols are guides for the eyes.