Phonon Softening due to Melting of the Ferromagnetic Order in Elemental Iron

We study the fundamental question of the lattice dynamics of a metallic ferromagnet in the regime where the static long-range magnetic order is replaced by the fluctuating local moments embedded in a metallic host. We use the ab initio density functional theory $+$ embedded dynamical mean-field theory functional approach to address the dynamic stability of iron polymorphs and the phonon softening with an increased temperature. We show that the nonharmonic and inhomogeneous phonon softening measured in iron is a result of the melting of the long-range ferromagnetic order and is unrelated to the first-order structural transition from the bcc to the fcc phase, as is usually assumed. We predict that the bcc structure is dynamically stable at all temperatures at normal pressure and is thermodynamically unstable only between the bcc-$\alpha$ and the bcc-$\delta$ phases of iron.

Figure 1

(a) The electronic free energy per atom versus $V$ of a cubic unit cell. (b) The temperature dependence of the ordered ferromagnetic moment of bcc iron using both the density-density (“Ising”) and the rotationally invariant (“full”) Coulomb interaction form. (c) The single-particle spectral function of the bcc-$\alpha$ phase at 300 K (the majority and minority spectra are plotted in blue and red, respectively).

Figure 2

Phonon spectrum at a low temperature ($T=300\mathrm{K}$), in the paramagnetic bcc-$\alpha$ phase ($T=1.125T_c$) and in the paramagnetic bcc-$\delta$ phase ($T=1800\mathrm{K}$) evaluated at experimental lattice constants. The dots correspond to the experimental data from Refs. [11, 66, 67] at 300 K, $1.125T_c$, and $T=1743\mathrm{K}$, respectively.

Figure 3

(a) The calculated phonon dispersions in the bcc-$\alpha$ phase below and above the Curie temperature at an experimental equilibrium volume. (b) Phonons in the metastable paramagnetic bcc phase, but at a constant volume (experimental volume at $T=T_c$).

Figure 4

The change of the phonon frequencies for representative modes with the temperature calculated by $\mathrm{DFT}+\mathrm{DMFT}$ (red dots) compared to experimental data (green triangle). The dashed blue lines denote the change predicted by the quasiharmonic model: ${\omega }^{\mathrm{qh}}(T)={\omega }^{300\mathrm{K}}(1-{\gamma }_{\mathrm{th}})(V_T-V_{300\mathrm{K}}/V_{300\mathrm{K}})$, where ${\omega }^{300\mathrm{K}}$ is the calculated value of the phonon frequency at 300 K, $V_T$ is the experimental volume of the unit cell at temperature $T$, and ${\gamma }_{\mathrm{th}}$ is the thermal Grüneisen parameter, approximated by a constant value of 1.81, as suggested in Refs. [12, 68].

Figure 5

(a) The total energy computed along the Bain crystallographic transition from the bcc to the fcc phase in the FM state. Note that at $T=1547\mathrm{K}$ the FM and PM phases are indistinguishable. (b) The ordered ferromagnetic moment along the same path.