Anharmonic theory of superconductivity in the high-pressure materials

Chandan Setty, Matteo Baggioli, and Alessio Zaccone

Phys. Rev. B 103, 094519 – Published 29 March 2021; Erratum Phys. Rev. B 106, 139902 (2022)

Abstract

Electron-phonon superconductors at high pressures have displayed the highest values of critical superconducting temperature $T_{c}$ on record, now rapidly approaching room temperature. Despite the importance of high- $P$ superconductivity in the quest for room-temperature superconductors, a mechanistic understanding of the effect of pressure and its complex interplay with phonon anharmonicity and superconductivity is missing, as numerical simulations can bring only system-specific details, clouding out key players controlling the physics. Here we develop a minimal model of electron-phonon superconductivity under an applied pressure which takes into account the anharmonic decoherence of the optical phonons. We find that $T_{c}$ behaves nonmonotonically as a function of the ratio $Γ / ω_{0}$ , where $Γ$ is the optical phonon damping and $ω_{0}$ is the optical phonon energy at zero pressure and momentum. Optimal pairing occurs for a critical ratio $Γ / ω_{0}$ when the phonons are on the verge of decoherence (“diffusonlike” limit). Our framework gives insights into recent experimental observations of $T_{c}$ as a function of pressure in the complex BCS material ${TlInTe}_{2}$ .

Received 24 October 2020
Revised 27 February 2021
Accepted 1 March 2021

DOI:https://doi.org/10.1103/PhysRevB.103.094519

Physics Subject Headings (PhySH)

Lattice dynamics Pressure effects Superconductivity

Superconductors

BCS theory

Condensed Matter, Materials & Applied Physics

Erratum

Erratum: Anharmonic theory of superconductivity in the high-pressure materials [Phys. Rev. B 103, 094519 (2021)]

Chandan Setty, Matteo Baggioli, and Alessio Zaccone

Phys. Rev. B 106, 139902 (2022)

Authors & Affiliations

Chandan Setty^*

Department of Physics, University of Florida, Gainesville, Florida, USA

Matteo Baggioli^†

Instituto de Fisica Teorica UAM/CSIC, c/Nicolas Cabrera 13-15, Universidad Autonoma de Madrid, Cantoblanco, 28049 Madrid, Spain

Alessio Zaccone ^‡

Department of Physics “A. Pontremoli,” University of Milan, via Celoria 16, 20133 Milan, Italy and Cavendish Laboratory, University of Cambridge, JJ Thomson Avenue, Cambridge CB30HE, United Kingdom

^*csetty@ufl.edu
^†matteo.baggioli@uam.es
^‡alessio.zaccone@unimi.it

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Vol. 103, Iss. 9 — 1 March 2021

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Images

Figure 1
Mechanism of $T_{c}$ enhancement through anharmonicity with two phonon modes. Left: In the absence of anharmonic decoherence ( $D = 0$ ), the Stokes and anti-Stokes processes are insensitive to their phase and are thus indistinguishable. This scenario leads to ordinary Cooper pairing. Middle: Weak anharmonic decoherence ( $D \sim D^{*}$ ) sensitizes the phase of the Stokes and anti-Stokes processes and enables them to act coherently and enhance the effective coupling of electrons and phonons, leading to strong Cooper pairs. Right: For very strong anharmonicity ( $D ≫ D^{*}$ ), the Stokes and anti-Stokes processes are only weakly sensitive to their phases, making them effectively indistinguishable while acting to reduce the effective coupling of electrons and phonons, leading to weak pairing.
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Figure 2
An illustration of the two regimes present in our model: the “coherent” regime, where the critical temperature grows with $D \equiv Γ / ω_{0}$ , and the “incoherent” regime, where the functional dependence is inverted. In the coherent regime, the optical phonons behave like independent quasiparticles with frequencies renormalized by anharmonicity, whereas in the incoherent regime the quasiparticle coherence breaks down due to the large anharmonic damping.
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Figure 3
The normalized critical temperature with $D = α ω_{0}^{4}$ and $ω_{0}$ given by Eq. (2). $α$ decreases from purple to red, $α = {0.5, 0.48, 0.45, 0.4, 0.35} \times 10^{- 7}$ . The $y$ axis is dimensionless in our units. $ω_{0} (P)$ is taken from the experimental fit shown in Fig. 4.
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Figure 4
(a) The normalized Raman shift (proportional to $ω^{'}$ ) of the $A_{g}$ phonon mode in ${TlInTe}_{2}$ as a function of pressure and its fit to an empirical function. The value at zero pressure is $\approx 128$ ${cm}^{- 1}$ . Data are taken from [36]. (b) Comparison between the best empirical fit of [36] [shown in (a)] and Eq. (2) in terms of $ω_{0} \approx ω^{'}$ ( $Γ ≪ ω_{0}$ ). The parameters are set to the values shown in Eq. (13).
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Figure 5
(a) The normalized linewidth of the $A_{g}$ phonon mode in ${TlInTe}_{2}$ and three different sets of fits. The zero pressure value is taken to be 3.2 ${cm}^{- 1}$ . Data are taken from [36]. (b) The corresponding theoretical calculations for the critical temperature (solid lines) are compared with the experimental data (symbols). The colors of the theoretical curves for $T_{c}$ match the respective models for the linewidth in (a). The parameters used in the model correspond to $\hat{a} = 1, α = (6, 5.4, 5.4, 4.2) \times 10^{- 8} {eV}^{- 4}, μ^{'} = \frac{μ}{\sqrt{λ}} = (37.3, 32.5, 35.6, 24)$ for the orange, blue, green, and purple curves, respectively. We also choose $λ = 1, ω_{0} \sim 15$ meV $= \frac{1}{2} v$ .
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Figure 6
Plot of the dimensionless critical temperature $T_{c}$ as a function of $D$ for various $v^{'}$ (left) and $M^{'}$ (right). We have chosen $M^{'} = 2$ (left) and $v^{'} = 0.5$ (right). The Klemens factor has a negligible effect in both cases.
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