Reconciliation of Experiments and Theory on Transport Properties of Iron and the Geodynamo

Youjun Zhang, Mingqiang Hou, Guangtao Liu, Chengwei Zhang, Vitali B. Prakapenka, Eran Greenberg, Yingwei Fei, R. E. Cohen, and Jung-Fu Lin

Phys. Rev. Lett. 125, 078501 – Published 13 August 2020

Abstract

We measure the electrical resistivity of hcp iron up to $\sim 170 GPa$ and $\sim 3000 K$ using a four-probe van der Pauw method coupled with homogeneous flattop laser heating in a DAC, and compute its electrical and thermal conductivity by first-principles molecular dynamics including electron-phonon and electron-electron scattering. We find that the measured resistivity of hcp iron increases almost linearly with temperature, and is consistent with our computations. The results constrain the resistivity and thermal conductivity of hcp iron to $\sim 80 \pm 5 μ Ω cm$ and $\sim 100 \pm 10 W m^{- 1} K^{- 1}$ , respectively, at conditions near the core-mantle boundary. Our results indicate an adiabatic heat flow of $\sim 10 \pm 1 TW$ out of the core, supporting a present-day geodynamo driven by thermal and compositional convection.

Received 6 January 2020
Accepted 13 July 2020

DOI:https://doi.org/10.1103/PhysRevLett.125.078501

Physics Subject Headings (PhySH)

Earth's interior Electrical conductivity Electron thermal conductivity Geomagnetism

Transition metals

Boltzmann theory DFT+DMFT Pressure techniques Resistivity measurements X-ray diffraction

Interdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Youjun Zhang ^1,2,*, Mingqiang Hou ^2,3,*, Guangtao Liu², Chengwei Zhang², Vitali B. Prakapenka ⁴, Eran Greenberg ⁴, Yingwei Fei ⁵, R. E. Cohen ^5,†, and Jung-Fu Lin ^6,‡

¹Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
²Center for High Pressure Science and Technology Advanced Research (HPSTAR), Shanghai 201900, China
³The Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
⁴Center for Advanced Radiation Sources, University of Chicago, Chicago, Illinois 60637, USA
⁵Extreme Materials Initiative, Earth and Planets Laboratory, Carnegie Institution for Science, Washington, DC 20015-1305, USA
⁶Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas 78712, USA

^*These authors contributed equally to this work.
^†rcohen@carnegiescience.edu
^‡afu@jsg.utexas.edu

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Vol. 125, Iss. 7 — 14 August 2020

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Images

Figure 1
Photograph of a shaped iron foil and the sample loaded in a laser-heated diamond anvil cell (DAC). (a) A Greek cross sheet of iron shaped by FIB; and (b) a loaded sample in a DAC with a culet of $300 μ m$ beveled to $75 μ m$ at $\sim 142 GPa$ and 300 K and then laser heated to 2400 K (c). Pt leads had a width of $\sim 20 μ m$ and a thickness of $\sim 4 μ m$ . The laser was focused into a flattop shape on the circular section of the Fe sample with $\sim 6 μ m$ in diameter.
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Figure 2
Measured and calculated resistivity of hcp Fe at $\sim 105 GPa$ with increasing temperature up to $\sim 2980 K$ in a laser-heated DAC. The pressure of 105 GPa was determined at ambient temperature by x-ray diffraction. At $\sim 2980 K$ , the pressure is about 117 GPa according to the thermal equation of state of hcp Fe [41]. Our measurements are compared with the current calculations that considered both $e − ph$ and $e − e$ contributions by $DFPT + DMFT$ (Ref. [20], solid squares) and new calculations by $FPMD + DMFT$ (this study, solid green star), and also compared with previously measured resistivity [14] (open circles) and calculations considered only $e − ph$ contribution (Ref. [20], open red squares). $e − ph$ is the electron-phonon contribution of resistivity calculated by DFPT with inelastic Boltzmann theory, $e − e$ is the electron-electron contribution of resistivity, and “sat” is resistivity saturation effects for the $e − ph$ scattering. The red dashed line through the calculated resistivity is a guide to the eye. The solid blue curve is fitted using the Bloch-Grüneisen formula for the measured resistivity. The black short dashed line represents the saturated resistivity fitted by a shunt resistor model in Ref. [14].
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Figure 3
Electrical resistivity and thermal conductivity of Fe at the relevant $P − T$ conditions of Earth’s core. (a) Our experimental results of hcp Fe at 2000 K (open blue circles) and 3000 K (semiopen blue circles), respectively, and the extrapolated data by Bloch-Grüneisen formula at 3000 K (semiopen blue squares) and 4000 K (solid blue squares), respectively. Our measured resistivities are compared with previous results (Ref. [14]) at 2000 K (open black circles), 3000 K (semiopen black circles), and 4000 K (solid black circles) in laser-heated DACs, respectively, and their extrapolated data by a resistivity saturation model at 3000 K (semiopen black squares) and 4000 K (solid black squares), respectively, and also compared with shock experiments (green open triangle [11] and inverted triangle [10]). (b) Calculated Lorenz number ( $L$ ) as functions of pressure and temperature. The ideal Lorenz number $L_{0} = 2.44 \times 10^{- 8} W Ω K^{- 2}$ . The calculated Lorenz number in liquid Fe (Ref. [7]) is also plotted for comparison (open diamonds). (c) The thermal conductivity of hcp Fe and liquid Fe derived using the Wiedemann-Franz relation at the relevant conditions of the outer core. The thermal conductivities are compared with previous results derived from resistivity experiments (open black circle, Ref. [14]) and from monitoring a heat pulse propagation (solid black diamonds, Ref. [15]), respectively. The star represents the derived thermal conductivity of liquid Fe in this work at near CMB conditions ( $\sim 136 GPa$ and $\sim 4000 K$ ), which is compared with the calculated DFT results for liquid Fe at 4000 K (open black triangle) and 6000 K (solid black triangle) [7], respectively. The lines through the experimental points are guides to the eye, where dashed, short-dashed, and dash-dotted lines represent the electrical resistivity and thermal conductivity as a function of pressure at 2000, 3000, and 4000 K, respectively. The vertical dash-dotted line represents the CMB.
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