Kohn anomaly and elastic softening in body-centered cubic molybdenum at high pressure

Chao Yang, Youjun Zhang, Nilesh P. Salke, Yan Bi, Ahmet Alatas, Ayman H. Said, Jiawang Hong, and Jung-Fu Lin

Phys. Rev. B 105, 094105 – Published 9 March 2022

Abstract

Transition metals in body-centered cubic (bcc) structures under compression can display several novel physical properties because of their complex electronic structures and electron-phonon interactions. Here, we used inelastic x-ray scattering experiments in a diamond-anvil cell up to ∼45 GPa and density-functional theory calculations up to 210 GPa to investigate the phonon dispersions, and electronic and elastic properties of single-crystal molybdenum (Mo). Our results show a pressure-induced Kohn anomaly at $q \sim 0.5$ along the [ξ00] direction in the longitudinal acoustic mode at ∼45 GPa; this anomaly is triggered by the pressure-enhanced Fermi-surface nesting effect. Theoretical calculations show that electron redistributions in the $s$ -to- $d$ orbitals of bcc-Mo contribute to the shear modulus anomaly at ∼50 GPa. In contrast, the Young's modulus anomaly in bcc-Mo at ∼210 GPa results from a Lifshitz-type electronic topological transition. Our results shed light on the complex electronic behaviors that are associated with macroscopic elastic properties in typical bcc $d$ -block transition metals under compression.

Received 6 October 2021
Revised 13 February 2022
Accepted 22 February 2022

DOI:https://doi.org/10.1103/PhysRevB.105.094105

Physics Subject Headings (PhySH)

Elastic modulus Electron-phonon coupling Fermi surface Mechanical & acoustical properties Nesting

3-dimensional systems Transition metals

Band structure methods Density functional theory Resonant inelastic x-ray scattering Wannier function methods

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Chao Yang ^1,2, Youjun Zhang ^3,*, Nilesh P. Salke ⁴, Yan Bi⁵, Ahmet Alatas ⁶, Ayman H. Said⁶, Jiawang Hong^2,†, and Jung-Fu Lin^7,‡

¹Department of Physics, Jishou University, Hunan 416000, China
²School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China
³Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, China
⁴Department of Physics, University of Illinois Chicago, Chicago, Illinois 60607, USA
⁵Center for High Pressure Science and Technology Advanced Research (HPSTAR), Shanghai 201900, China
⁶Argonne National Laboratory, Lemont, Illinois 60439, USA
⁷Department of Geological Sciences, Jackson School of Geosciences, The University of Texas at Austin, Austin, Texas 78712, USA

^*zhangyoujun@scu.edu.cn
^†hongjw@bit.edu.cn
^‡afu@jsg.utexas.edu

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Vol. 105, Iss. 9 — 1 March 2022

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Images

Figure 1
High-energy resolution inelastic x-ray scattering experiments in single-crystal Mo at high pressure. (a) Microphotograph of a Mo crystal loaded with He in a DAC at 45.0(0.5) GPa. The sample was ∼40 × 50 μm long and ∼15 μm thick. The culet size of the diamond anvils was 300 μm. (b) Representative longitudinal acoustic (LA) spectra of Mo at 45.0(0.5) GPa measured in the (200) Brillouin zone along the [ξ00] direction from $q = 0.1$ to $q = 1.0$ . Error bars (vertical ticks) were estimated from counting statistics. Solid lines were fit to experimental data (open circles) using a Gaussian function. An additional Gaussian function was used for the elastic line centered around 0 meV.
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Figure 2
Phonon dispersions in bcc single-crystal Mo at high pressure along the (a) Γ- $H$ and (b) Γ- $N$ directions. Blue and red solid circles represent our experimental data at 22 and 45 GPa, respectively, while black and green solid circles represent results at ambient condition and at 17 GPa from previous inelastic neutron and x-ray scattering measurements [19, 56]. Errors for data are smaller than the circle size and are not plotted for clarity. Black, blue, and red lines represent DFT calculations of phonon dispersions at 0, 22, and 45 GPa, respectively. Longitudinal dispersions along the Γ- $N$ direction in experiments at 45 GPa were not completed because of an anvil failure. Dark-green and black arrows indicate the Kohn anomalies at $q \sim 0.5$ and near the $H$ point, respectively.
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Figure 3
DFT results of phonon dispersions, electron-phonon coupling, Fermi surface nesting, and 3D Fermi surfaces in bcc Mo under pressure. (a) Calculated phonon dispersions at pressures of 0–210 GPa. The black arrow indicates the Kohn anomalies of bcc Mo at $q \sim 0.5$ along the Γ- $H$ direction. Electron-phonon coupling coefficients ( $λ_{q, v}$ ) (b) and Fermi-surface nesting functions $[χ (q)]$ (c) in bcc Mo along the Γ- $H$ direction at high pressures of 0–100 GPa. Three-dimensional Fermi surfaces of bcc Mo are shown at (d) 0 GPa and (e) 210 GPa. Red arrows indicate the Fermi-surface nesting vectors along the Γ- $H$ direction. The 3D Fermi surface structures simulated with vasp and wannier90 were visualized using xcrysden [50].
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Figure 4
Elastic moduli of solid Mo at high pressures. Shear modulus (a) and Young's modulus (b) of Mo when under high pressure. Solid red circles represent our experimental results, and the solid red line is for our DFT calculations. The hollow circles represent the results reported from Ref. [64] at high pressures and room temperatures. Diamonds (Ref. [63]) and squares (Ref. [28]) represent results from shock-wave experiments.
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Figure 5
Number of electrons and 3D Fermi surfaces of Mo calculated using vasp at high pressure. Number of electrons of $s$ orbital (a) and $d$ orbital (b), (c) as a function of pressure. Three-dimensional Fermi surface of Mo at 0 (d), 110 (e), and 210 (f) GPa. Fermi surfaces in panel (f) show an occurrence of an electronic topological transition at ∼210 GPa (blue areas). Three-dimensional Fermi surfaces were visualized using the xcrysden package [50].
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Figure 6
$d$ -Orbital projected 3D Fermi surfaces of bcc Mo at 210 GPa calculated using quantum espresso. Corresponding $d$ -orbital projected 3D Fermi surfaces of seventh band (a)–(e) and eighth band (f)–(j). The s- and $p$ orbitals are ignored because they have small contributions near the Fermi level. $d$ -Orbital projected 3D Fermi surfaces were displayed using fermisurfer [54].
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