Electronically Driven 1D Cooperative Diffusion in a Simple Cubic Crystal

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Electronically Driven 1D Cooperative Diffusion in a Simple Cubic Crystal

Yong Wang, Junjie Wang, Andreas Hermann, Cong Liu, Hao Gao, Erio Tosatti, Hui-Tian Wang, Dingyu Xing, and Jian Sun

Phys. Rev. X 11, 011006 – Published 11 January 2021

Abstract

Atomic diffusion is a spontaneous process and significantly influences properties of materials, such as fracture toughness, creep-fatigue properties, thermal conductivity, thermoelectric properties, etc. Here, using extensive molecular dynamics simulations based on both ab initio and machine-learning potentials, we demonstrate that an atomic one dimensional cooperative diffusion exists in the simple cubic high-pressure finite-temperature phase of calcium in the premelting regime, where some atoms diffuse cooperatively as chains or even rings, while others remain in the solid state. This intermediate regime is triggered by anharmonicity of the system at high temperature and is stabilized by the competition between the internal energy minimization and the entropy maximization, and has close connections with the unique electronic structures of simple cubic Ca as an electride with a pseudogap. This cooperative diffusion regime explains the abnormal enhancement of the melting line of Ca under high pressure and suggests that the cooperative chain melting is a much more common high-temperature feature among metals under extreme conditions than hitherto thought. The microscopic electronic investigations of these systems combining ab initio and machine-learning data point out the direction for further understanding of other metallic systems such as the glass transition, liquid metals, etc.

Received 1 July 2020
Revised 12 October 2020
Accepted 30 November 2020

DOI:https://doi.org/10.1103/PhysRevX.11.011006

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Crystal melting Electronic structure Liquid-solid phase transition Pressure effects

First-principles calculations Machine learning Molecular dynamics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yong Wang^1,*, Junjie Wang ^1,*, Andreas Hermann ², Cong Liu¹, Hao Gao¹, Erio Tosatti ^3,4, Hui-Tian Wang¹, Dingyu Xing¹, and Jian Sun ^1,†

¹National Laboratory of Solid State Microstructures, School of Physics and Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing, 210093, China
²Centre for Science at Extreme Conditions and SUPA, School of Physics and Astronomy, The University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
³International School for Advanced Studies (SISSA) and CNR-IOM Democritos, Via Bonomea 265, 34136 Trieste, Italy
⁴The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, 34151 Trieste, Italy

^*These authors contributed equally to this work.
^†Corresponding author. jiansun@nju.edu.cn

Popular Summary

Melting is not necessarily an all-or-nothing deal. In certain circumstances, a melted state can coexist with a solid state. While the two states are normally separated, sometimes they are interwoven. One such example in high-pressure scenarios is “chain melting,” in which 1D chains of atoms in a crystalline lattice “melt” while the surrounding atoms remain fixed. Seen in certain elements, it remains unclear how general this phenomenon is. Using simulations based on quantum mechanics and machine learning, we show that even the simplest possible crystal structure can support chain melting.

For this study, we explore the simple cubic phase of calcium—a state in which the calcium atoms create a lattice of simple joined cubes. Through first-principles molecular dynamics simulations, we study the behaviors of individual atoms in supercells at high pressure and a range of temperatures. Starting at about $1000 ° C$ and pressures around half a million atmospheres, the simulations show concerted motion of atoms along lines or rings within the lattice—the onset of chain melting. The number of melted chains and atom exchanges between them increase with temperature before eventually giving way to widespread melting.

The appearance of this cooperative motion is connected to the unique electronic structure of simple cubic calcium, where electrons localize in step with the atomic lattice. This is the first time that researchers have found such an intriguing phenomenon in a simple cubic system. Our study also reveals connections between partial melting, electronic structure, and lattice anharmonicity, providing a new perspective to understand melting mechanisms in general under extreme conditions.

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Vol. 11, Iss. 1 — January - March 2021

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Figure 1
Behavior of sc Ca at finite temperatures from AIMD simulations in a 125 atom supercell at 50 GPa. (a),(b) The averaged mean-squared displacements (MSD) and phonon DOS of sc Ca at 900, 1300, 1600, 2100, 3000 K (dash line, solid line). (c) The MSD at 1300 K and its various directions. (d)–(f) Representations of two diffusive states in sc Ca, and the difference between their MSDs and DOS along chains (cMSD). (g) Snapshots of chains’ melting along the $⟨ 100 ⟩$ in sequence at 50 GPa and 1500 K with two different directions on a section of a layer.
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Figure 2
(a),(b) The decomposition of the normalized density of states for 50 GPa sc calcium at 1500 and 2700 K corresponding to chain melting and liquid, respectively. The 2PT MF solution for gaslike (red solid line) and solidlike (blue solid line) components was generated using the even moments $M_{2}$ . (c) The proportion of atoms that contribute to the diffusive chains inside 50 GPa sc Ca at different temperatures. (d) Temperature dependence of Ca self-diffusion due to collective atomic motion in the form of chain diffusion.
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Figure 3
Pair distribution function $g (r)$ of the AIMD simulations at 40 and 50 GPa and different temperatures. Red, blue, and green lines represent $g (r)$ of simulations at 900, 1500, and 2700 K, which corresponds to the solid, cooperative diffusion and the liquid state, respectively.
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Figure 4
Behavior of sc Ca at finite temperatures from machine-learning MD simulations at 40 GPa. (a), (b) The atomic flux density of planes perpendicular to the $⟨ 100 ⟩$ at different directions in the picosecond scale from ML MD simulations at 40 GPa and 1300 K. Panel (a) is $F_{x} (y, z)$ in a $20 \times 20 \times 20$ supercell (8000 atoms) and (b) is $F_{z} (x, y)$ in a $25 \times 25 \times 25$ supercell (15 625 atoms). The chain-melting behavior along the $⟨ 100 ⟩$ is visible. The positive (red) and negative (blue) values of density represent the diffusion directions of atoms, the green area represents the atoms in solid state. (c) Snapshots of 1D cooperative diffusion state with chains along the $⟨ 100 ⟩$ and rings (the data of 1200 and 1300K were overlapped to show different diffusive behaviors). Chains and rings coexist with solid lattice.
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Figure 5
Intrinsic properties associated with chain-melting regime from DFT calculations. (a) Migration energy barriers along the edge $⟨ 100 ⟩$ , face diagonal $⟨ 110 ⟩$ , body diagonal $⟨ 111 ⟩$ in sc Ca calculated using nudged-elastic-band (NEB) methods. (b) Harmonic phonon spectra and renormalized anharmonic phonon spectra at 600 and 1200 K of sc Ca at 50 GPa. The resoftening of phonon modes near chain melting 1200 K can be found. (c) ELF plot for the transition state of cooperative $⟨ 100 ⟩$ migration in a sc Ca supercell; ELF values are shown from 0 (blue) to 1 (red). Yellow and magenta circles correspond to atoms with solidlike and diffusive atoms. (d) Averaged electronic DOS for solid (blue solid line), chain melting (green solid line), liquid (red solid line) of sc Ca at 50 GPa with the Fermi energy (black dot line). Note the pseudogap, providing an electronic stabilization to sc Ca, which persists even in the chain melt, and eventually disappears in the liquid. (e),(f) Entropy $S$ and relative Gibbs free energy $Δ G$ of the Ca system from 600 to 2700 K at 50 GPa. Free-energy values of the chain-melting states are used as the baseline. Square (blue), triangle (green), and circle (red) represent solid, chain melting, and liquid, respectively, and solid lines are drawn as guides to the eye.
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Figure 6
Phase diagram of Ca based on AIMD simulations up to 3000 K. Orange squares, green stars, and red circles represent solid sc Ca, cooperative diffusion state, and liquid, respectively. Cyan diamonds and blue triangles represent fcc and bcc solid calcium. Black crosses correspond to results of phase boundaries calculated from Gibbs free-energy calculations. The experimental melting lines and phase boundaries (black dashed and dash-dotted lines) were taken from Refs. [42, 43].
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