Thermal Conductivity and Electrical Resistivity of Solid Iron at Earth’s Core Conditions from First Principles

We compute the thermal conductivity and electrical resistivity of solid hcp Fe to pressures and temperatures of Earth’s core. We find significant contributions from electron-electron scattering, usually neglected at high temperatures in transition metals. Our calculations show a quasilinear relation between the electrical resistivity and temperature for hcp Fe at extreme high pressures. We obtain thermal and electrical conductivities that are consistent with experiments considering reasonable error. The predicted thermal conductivity is reduced from previous estimates that neglect electron-electron scattering. Our estimated thermal conductivity for the outer core is $77\pm 10\mathrm{W}{\mathrm{m}}^{-1}{\mathrm{K}}^{-1}$ and is consistent with a geodynamo driven by thermal convection.

Figure 1

Calculated resistivity of hcp Fe at atomic volume 57.9 and $53.3{\text{bohr}}^3$, corresponding to about 110 and 160 GPa at 2500 K [56, 57]. ${\rho }_{e\text{−}\text{ph}}$ is the electron-phonon contribution of resistivity. DFPT represents calculating ${\rho }_{e\text{−}\text{ph}}$ using the $\mathrm{DFPT}+\text{inelastic}$ Boltzmann theory. “$+$ sat” includes resistivity saturation effects for the $e\text{−}\text{ph}$ scattering using Eq. (1). “KKR-CPA” represents using the KKR-CPA method with the Kubo-Greenwood formula. FPMD is first-principles molecular dynamics with the Kubo-Greenwood formula. Both the KKR-CPA and the FPMD have naturally included resistivity saturation effects.

Figure 2

Resistivity along the Hugoniot from shock data. $\times$ are from Ref. [9], and $+$ are from Ref. [55]. ${\rho }_{\text{tot}}={\rho }_{e\text{−}\text{ph}}+{\rho }_{e\text{−}e}$, where ${\rho }_{e\text{−}\text{ph}}$ considers resistivity saturation effects using Eq. (1), and ${\rho }_{e\text{−}e}$ is electrical resistivity due to the electron-electron scattering. ${\rho }_{\text{tot}}^{(\text{no sat})}={\rho }_B+{\rho }_{e\text{−}e}$, where ${\rho }_B$ is from Boltzmann theory and does not consider resistivity saturation. The purple line is the linear fit of the shock compression data. The blue lines are the 95% mean confidence interval.

Figure 3

Calculated resistivity at fixed atomic volumes—57.9, 54.9, 53.3, and $51.6{\text{bohr}}^3$, corresponding to pressures of about 110, 140, 160, and 190 GPa at 2500 K [56, 57]—compared with shock data and experimental data by Ohta et al. [11]. ${\rho }_{\text{tot}}$ and ${\rho }_{\text{tot}}^{(\text{no sat})}$ are total resistivity considering and not considering resistivity saturation effects, respectively. ${\rho }_{\text{tot}}^{(2)}$ is the total resistivity to which the $e\text{−}\text{ph}$ contribution is calculated using the KKR-CPA and the Kubo-Greenwood formula. The dotted lines are their fits to the diamond anvil cell (DAC) data.

Figure 4

Resistivity versus temperature of hcp Fe at Earth’s inner core density. ${\rho }_{\text{tot}}$ and ${\rho }_{\text{tot}}^{(\text{no sat})}$ are our calculated total resistivity considering and not considering resistivity saturation effects, respectively. The crosses are results using the KKR-CPA, which naturally includes saturation effects. The open triangle is FPMD results [58], where there is no $e\text{−}e$ contribution, at a little higher density. The navy squares are extrapolations to this density using the systematics of Stacey and Anderson [59] based on the melting curve (which has no fundamental justification). The dark diamond is an interpolation to this density and an extrapolation to the 6658 K temperature of previous shock compression results [9, 55]. The green line is from Ref. [11] and is the extrapolation of their experimental data.

Figure 5

Our calculated thermal conductivity of hcp Fe at fixed atomic volumes—57.9, 54.9, 53.3, and $51.6{\text{bohr}}^3$, corresponding to pressures about 110, 140, 160, and 190 GPa at 2500 K [56, 57]—compared with experimental data [13]. ${\kappa }_{\text{tot}}$ is the total thermal conductivity and is equal to ${\kappa }_{\text{tot}}=[{\kappa }_{e\text{−}\text{ph}}^{-1}+{\kappa }_{e\text{−}e}^{-1}{]}^{-1}$.

Figure 6

Calculated thermal conductivity of solid hcp Fe at Earth’s core density, $13.04\mathrm{g}{\mathrm{cm}}^{-3}$, or atomic volume $47.8{\text{bohr}}^3$. At 6000 K, the corresponding pressure is about 305 GPa. ${\kappa }_{\text{tot}}$ is our calculated total thermal conductivity using the DFPT and DMFT. The black line is the extrapolation model at 330 GPa based on the experimental data at 112 GPa [13]. A uncertainty envelope of the model (gray) is given according to our estimate that their experimental data at 2000 K and the extrapolation to pressure 330 GPa may both have errors of 10%–30% and the thermal conductivity may decrease slower than $T^{-0.5}$, as assumed in their model. The green open triangles are the theoretical values of thermal conductivity at 329 GPa due to only $e\text{−}\text{ph}$ scattering using FPMD [60], which includes saturation effects consistent with system size and $k$-point sampling. The dark cyan open triangle and blue open stars are the theoretical thermal conductivity of liquid Fe due to only $e\text{−}\text{ph}$ scattering using FPMD [5, 6].