Quasiparticle bands and structural phase transition of iron from Gutzwiller density-functional theory

We use the Gutzwiller density-functional theory to calculate ground-state properties and band structures of iron in its body-centered-cubic (bcc) and hexagonal-close-packed (hcp) phases. For a Hubbard interaction $U=9\mathrm{eV}$ and Hund's-rule coupling $J=0.54\mathrm{eV}$, we reproduce the lattice parameter, magnetic moment, and bulk modulus of bcc iron. For these parameters, bcc is the ground-state lattice structure at ambient pressure up to a pressure of $p_{\mathrm{c}}=41\mathrm{GPa}$ where a transition to the nonmagnetic hcp structure is predicted, in qualitative agreement with experiment $(p_{\mathrm{c}}^{\exp }=10,...,15\mathrm{GPa})$. The calculated band structure for bcc iron is in good agreement with ARPES measurements. The agreement improves when we perturbatively include the spin-orbit coupling.

Figure 1

Cubic lattice parameter $a(U,J)$ for bcc iron in units of the Bohr radius $a_{\mathrm{B}}=0.529\text{Å}$ as a function of the Hubbard interaction $U$ for $J/U=0.05,0.06,\text{and}0.075$. The horizontal dashed line indicates the experimental value $a^{\exp }=5.42a_{\mathrm{B}}=2.87\text{Å}$.

Figure 2

Magnetization $m(U,J)$ in units of Bohr magneton ${\mu }_{\mathrm{B}}$ as a function of the Hubbard interaction $U$ for $J/U=0.05,0.06,\text{and}0.075$. The horizontal dashed line indicates the experimental value $m=2.22{\mu }_{\mathrm{B}}$.

Figure 3

Energy per atom $e\left(v\right)$ in units of eV as a function of the unit-cell volume $v$ in units of $a_{\mathrm{B}}^3$ for nonmagnetic and ferromagnetic bcc iron and nonmagnetic hcp iron at $U=9\mathrm{eV}$ and $J=0.54\mathrm{eV}$ and ambient pressure. The energies are shifted by the same constant amount.

Figure 4

Enthalpy difference per atom $h_{\mathrm{hcp}}-h_{\mathrm{bcc}}$ in units of eV between nonmagnetic hcp iron and ferromagnetic bcc iron as a function of applied pressure $p$ for $U=8.0,9.0,10\mathrm{eV}$ and $J/U=0.06$. Inset: magnetization as a function of pressure.

Figure 5

Local charge distribution of iron atoms $n_d\left(n\right)$ for $U=0$ (LDA limit) and $U=9\mathrm{eV}$ and $J/U=0.05,0.06,\text{and}0.075$ (Gutzwiller-DFT) for optimal lattice parameters.

Figure 6

Local spin distribution of iron atoms $n_s\left(s\right)$ for $U=0$ (LDA limit) and $U=9\mathrm{eV}$ and $J/U=0.05,0.06,\text{and}0.075$ (Gutzwiller-DFT) for optimal lattice parameters.

Figure 7

Comparison between DFT(LDA) bands (blue, dashed lines) for $a^{\mathrm{LDA}}=5.21a_{\mathrm{B}}$ and bands from Gutzwiller-DFT (red, full lines) for the optimal atomic parameters $U=9.0\mathrm{eV}$, $J=0.54\mathrm{eV}$, and $a=5.39a_{\mathrm{B}}$ for ferromagnetic bcc iron. For clarity, we do not discriminate between majority and minority spin bands. The Fermi energy is at $E_{\mathrm{F}}=0$ (dashed black horizontal line).

Figure 8

Comparison between DFT(GGA) bands (black, dashed lines) for $a_{\mathrm{GGA}}=5.39a_{\mathrm{B}}$ and bands from Gutzwiller-DFT (red, full lines) for the optimal atomic parameters $U=9.0\mathrm{eV}$, $J=0.54\mathrm{eV}$, and $a=5.39a_{\mathrm{B}}$ for ferromagnetic bcc iron. For clarity, we do not discriminate between majority and minority spin bands. The Fermi energy is at $E_{\mathrm{F}}=0$ (dashed horizontal line).

Figure 9

Comparison between DFT(LDA) bands (blue, dashed lines) and bands from Gutzwiller-DFT (red, full lines) for the atomic parameters $U=9.0\mathrm{eV}$ and $J=0.54\mathrm{eV}$ for paramagnetic hcp iron at $a=4.60a_{\mathrm{B}}$ and $c/a=1.60$. The Fermi energy is at $E_{\mathrm{F}}=0$ (dashed black horizontal line).

Figure 10

Overlay of quasiparticle bands from LDA+Gutzwiller for $U=9.0\mathrm{eV},J=0.54\mathrm{eV}$, and lattice parameter $a=5.39a_{\mathrm{B}}$ with ARPES data along the $\mathrm{\Gamma }\text{−}P$ direction [35].

Figure 11

Overlay of quasiparticle bands from LDA+Gutzwiller for $U=9.0\mathrm{eV},J=0.54\mathrm{eV}$, and lattice parameter $a=5.39a_{\mathrm{B}}$ with ARPES data along the $\mathrm{\Gamma }\text{−}H$ direction [35].

Figure 12

Overlay of quasiparticle bands from LDA+Gutzwiller for $U=9.0\mathrm{eV},J=0.54\mathrm{eV}$, and lattice parameter $a=5.39a_{\mathrm{B}}$ with ARPES data along the $N\text{−}\mathrm{\Gamma }$ direction [35].

Figure 13

Overlay of quasiparticle bands from LDA+Gutzwiller for $U=9.0\mathrm{eV},J=0.54\mathrm{eV}$, and lattice parameter $a=5.39a_{\mathrm{B}}$ with ARPES data along the $H\text{−}P$ direction [35].