Temperature-dependent electron-phonon spectral function and the intrinsic resistivity of a metal: A case study of monolayer ${\mathrm{Ti}}_{2}\mathrm{N}$

Temperature-dependent electron-phonon spectral function and the intrinsic resistivity of a metal: A case study of monolayer ${Ti}_{2} N$

Binyuan Zhang, Mingfeng Zhu, Zhe Liu, Fei Guo, and Yisong Zheng

Phys. Rev. B 102, 165402 – Published 5 October 2020

Abstract

The transport spectral function of electron-phonon ( $e$ -ph) interaction in the double $δ$ -function approximation (DDFA) is extensively employed to calculate the intrinsic resistivity (arising from $e$ -ph scattering) of metallic materials in recent works of first-principles calculations. In contrast, a more fundamental transport spectral function with the Fermi smearing effect due to finite temperature ( $T$ ) and nonzero phonon frequency is less involved. In this work, we perform first-principles calculations of the intrinsic resistivity of ${Ti}_{2} N$ monolayer, a potential MXene material, by employing the two kinds of spectral function. We find that the spectral function with the DDFA fails to describe correctly the temperature dependence of the intrinsic resistivity of ${Ti}_{2} N$ monolayer at $T > 250$ K, much lower than the Debye temperature. The underlying reason is that ${Ti}_{2} N$ monolayer has a multisheet Fermi surface formed by several bands, and some band edges are very close to the Fermi surface. Our results suggest that the transport spectral function with the Fermi smearing effect, instead of the one with the DDFA, is always adequate for studying the intrinsic resistivity of realistic materials on the level of first-principles calculations. In addition, we give a brief remark on the intrinsic resistivity of ${Ti}_{2} N$ monolayer, in contrast with other typical two-dimensional materials, which is significant from the viewpoint of application of such an MXene material.

Received 19 August 2020
Revised 18 September 2020
Accepted 18 September 2020

DOI:https://doi.org/10.1103/PhysRevB.102.165402

Physics Subject Headings (PhySH)

Electrical conductivity Electron-phonon coupling

2-dimensional systems Monolayer films

Boltzmann theory Density functional calculations First-principles calculations

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Binyuan Zhang, Mingfeng Zhu, Zhe Liu , Fei Guo, and Yisong Zheng ^*

Key Laboratory of Physics and Technology for Advanced Batteries (Ministry of Education), Department of Physics, Jilin University, Changchun 130012, China

^*Author to whom all correspondence should be addressed: zhengys@jlu.edu.cn

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Vol. 102, Iss. 16 — 15 October 2020

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Figure 1
(a) Side view and (b) top view of the ${Ti}_{2} N$ monolayer. In (b) $a_{1}$ and $a_{2}$ are the lattice vectors of ${Ti}_{2} N$ monolayer, and the unit cell of ${Ti}_{2} N$ lattice is shown by a dashed line. (c) Brillouin zone of ${Ti}_{2} N$ with the Fermi surface embedded in it, high-symmetry points labeled; $b_{1}$ and $b_{2}$ are the inverse lattice vectors.
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Figure 2
(a) Electronic and (b) phononic dispersions of ${Ti}_{2} N$ monolayer along the high-symmetry path $Γ - M - K - Γ$ . The electronic band structures of ${Ti}_{2} N$ monolayer obtained by DFT (blue solid line) and Wannier interpolation (black dashed line) along the high-symmetry path are exhibited in (a). Fermi energy is shifted to be $0 eV$ . The difference between the band edge of the small electron (hole) pockets and the Fermi level is $83 meV$ ( $87 meV$ ). The phonon modes ordered by frequency are identified by magnitude from 1 (the lowest) to 9 (the highest). The three phonon modes with zero frequency at the high-symmetry point $Γ$ are acoustic branches ( $ν = 1$ –3), and the remaining six are optical branches ( $ν = 4$ –6).
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Figure 3
Intrinsic resistivity of ${Ti}_{2} N$ monolayer with varied Gaussian broadening parameter $σ$ and $n_{s} \times n_{s} \times 1 k$ - and $q$ -mesh calculated by (a) $α^{2} F_{T}$ and (b) $α^{2} {\tilde{F}}_{T}$ . All results are calculated at 300 K.
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Figure 4
The variation of $ρ_{θ} / ρ_{x}$ with angle $θ$ . $ρ_{θ}$ is the intrinsic resistivity driven by the electric field in the direction with the angle $θ$ with respect to the $x$ -axis. This result is calculated by $α^{2} F_{T}$ at 300 K.
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Figure 5
(a) Resistivity of ${Ti}_{2} N$ monolayer calculated by $α^{2} {\tilde{F}}_{T}$ (black solid line) and $α^{2} F_{T}$ (red dashed line) vs temperature $T$ within 20–45 K, with the $T^{4}$ fitting curve plotted for comparison. Resistivity of ${Ti}_{2} N$ calculated by (b) $α^{2} F_{T}$ and (c) $α^{2} {\tilde{F}}_{T} (Ω)$ vs temperature $T$ within 50–2000 K, with the linear fitting curve plotted for comparison. (d) Resistivity of ${Ti}_{2} N$ monolayer calculated by $α^{2} {\tilde{F}}_{T}$ (black solid line) and $α^{2} F_{T}$ (red dashed line) vs temperature $T$ within 50–2000 K.
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Figure 6
(a) The spectral function $α^{2} {\tilde{F}}_{T} (Ω)$ and $α^{2} F_{T} (Ω)$ at temperatures of 50–250 K. (b) The plot of the function $F (T / T_{0})$ . When the argument $x < 0.22$ , the function $F (T / T_{0})$ begins to deviate from a linear function. (c) The plot of the function $N (T / T_{0})$ . When the argument $x < 0.5$ , the function $F (T / T_{0})$ begins to deviate from a linear function. (d) The percentages of the intrinsic resistivity contributed by AP (black line) and OP (red line) at different temperatures. The solid line represents the result calculated by $α^{2} {\tilde{F}}_{T} (Ω)$ , the dashed line represents the result calculated by $α^{2} F_{T} (Ω)$ . (e) The contributions of AP (black line) and OP (red line) to $α^{2} F_{T} (Ω)$ at $T = 100$ K. (f) The spectral function $α^{2} F_{T} (Ω)$ at temperatures of 800–2000 K. (g) The spectral function $α^{2} {\tilde{F}}_{T} (Ω), α^{2} F_{T} (Ω)$ , and $α^{2} {\tilde{F}}_{T} (Ω) + \frac{π^{2}}{6} {(k_{B} T)}^{2} {\frac{\partial^{2} I (ɛ, Ω)}{\partial^{2} ɛ}|}_{ɛ = E_{f}}$ at 300 K.
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Physical Review B

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Temperature-dependent electron-phonon spectral function and the intrinsic resistivity of a metal: A case study of monolayer ${Ti}_{2} N$

Binyuan Zhang, Mingfeng Zhu, Zhe Liu, Fei Guo, and Yisong Zheng

Phys. Rev. B 102, 165402 – Published 5 October 2020

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Physical Review B

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Temperature-dependent electron-phonon spectral function and the intrinsic resistivity of a metal: A case study of monolayer Ti2N

Binyuan Zhang, Mingfeng Zhu, Zhe Liu, Fei Guo, and Yisong Zheng

Phys. Rev. B 102, 165402 – Published 5 October 2020

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Temperature-dependent electron-phonon spectral function and the intrinsic resistivity of a metal: A case study of monolayer ${Ti}_{2} N$