Electron-phonon coupling and superconductivity-induced distortion of the phonon lineshape in ${\mathrm{V}}_{3}\mathrm{Si}$

Electron-phonon coupling and superconductivity-induced distortion of the phonon lineshape in $V_{3} Si$

A. Sauer, D. A. Zocco, A. H. Said, R. Heid, A. Böhmer, and F. Weber

Phys. Rev. B 99, 134511 – Published 15 April 2019

Abstract

Phonon measurements in the $A 15$ -type superconductors were complicated in the past because of the unavailability of large single crystals for inelastic neutron scattering, e.g., in the case of ${Nb}_{3} Sn$ , or unfavorable neutron scattering properties in the case of $V_{3} Si$ . Hence, only few studies of the lattice dynamical properties with momentum resolved methods were published, in particular below the superconducting transition temperature $T_{c}$ . Here, we overcome these problems by employing inelastic x-ray scattering and report a combined experimental and theoretical investigation of lattice dynamics in $V_{3} Si$ with the focus on the temperature-dependent properties of low-energy acoustic phonon modes in several high-symmetry directions. We paid particular attention to the evolution of the soft phonon mode of the structural phase transition observed in our sample at $T_{s} = 18.9 K$ , i.e., just above the measured superconducting phase transition at $T_{c} = 16.8 K$ . Theoretically, we predict lattice dynamics including electron-phonon coupling based on density-functional-perturbation theory and discuss the relevance of the soft phonon mode with regard to the value of $T_{c}$ . Furthermore, we explain superconductivity-induced anomalies in the lineshape of several acoustic phonon modes using a model proposed by Allen et al, [Phys. Rev. B 56, 5552 (1997)].

Received 21 January 2019

DOI:https://doi.org/10.1103/PhysRevB.99.134511

Physics Subject Headings (PhySH)

Acoustic phonons Electron-phonon coupling Superconducting gap

Density functional theory X-ray scattering

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

A. Sauer¹, D. A. Zocco^1,2, A. H. Said³, R. Heid¹, A. Böhmer¹, and F. Weber¹

¹Institute for Solid State Physics, Karlsruhe Institute of Technology, 76021 Karlsruhe, Germany
²Institute of Solid State Physics, Vienna University of Technology, 1040 Vienna, Austria
³Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA

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Vol. 99, Iss. 13 — 1 April 2019

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Figure 1
First-principles lattice dynamical calculations of $V_{3} Si$ compared to the experimental results obtained at room temperature. (a)–(d) Calculated phonon dispersion lines (lines) compared to the observed phonon energies at room temperatures (squares). Vertical lines represent the calculated electronic contributions to the phonon linewidth of the respective mode. (e) Calculated total (solid line) and partial (dashed/dotted lines) phonon density of states. The inset shows a comparison to the results from inelastic neutron scattering for $T = 25 K$ (blue dash-dotted line) and $300 K$ (red dashed line) published in Ref. [21]. (f) Calculated Eliashberg function $α^{2} F (ω)$ (black solid line) and electron-phonon coupling constant $λ$ (red dashed line).
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Figure 2
(a) Temperature dependence of the thermal expansion coefficient $α = \frac{1}{L} \cdot \frac{Δ L}{Δ T}$ derived from dilatometry measurements of the same single crystal used for the inelastic x-ray scattering experiments. The sharp peak and the lower-temperature kink indicate the structural and superconducting phase transition in our sample at $T_{s} = 18.9 K$ and $T_{c} = 16.8 K$ , respectively. (b)–(j) Evolution of IXS raw data (normalized to monitor) on cooling from room temperature (black dots, right-hand scale) to low temperatures $T = 20 - 3 0 K$ (blue circles, left-hand scale). We observe an LA phonon dispersing along the [100] direction (b)–(d) and a TA phonon dispersing along the [111] direction (h)–(j). The focus along the [110] direction (e)–(g) was on the corresponding TA phonon but the LA mode has also a finite structure factor [see (e)]. Lines are fits to the data using a DHO function for phonon peaks convoluted with the experimental resolution and a pseudo-Voigt function for the elastic line. (k)–(m) Phonon energies and (n)–(p) linewidths deduced from the above presented raw data (b)–(j) (same color/symbol code) compared to the corresponding ab initio calculations (lines).
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Figure 3
(a) Raw data of IXS measurements taken at $T = 300 K$ (black dots, right-hand scale), 30 K (blue circles, left-hand scale) and 4 K (blue squares, left-hand scale) for the wave vector $Q = (3.85, 0.15, 0)$ . Lines are fits to the data using a DHO function for phonon peaks convoluted with the experimental resolution and a pseudo-Voigt function for the elastic line. (b) Temperature dependence of the phonon energy (blue dots) and linewidth (red circles) of the soft phonon mode at $Q = (3.85, 0.15, 0)$ when analyzed with a DHO function at all temperatures (see text). The inset shows a zoom of the low-temperature range $T \leq 30 K$ .
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Figure 4
Analysis of the superconductivity-induced changes of the phonon lineshape for the structural soft phonon mode observed at $Q = (3.85, 0.15, 0)$ . (a) IXS raw data taken at $Q = (3.85, 0.15, 0)$ at $T = 20 K$ (green dots) and 4 K (blue circles), i.e., just above $T_{s} (= 18.9 K)$ and $T_{c} (= 16.8 K)$ and well below $T_{c}$ . The (blue) solid line denotes the fit of the data taken in the normal state consisting of two DHO functions for the TA and TO modes (blue dashed lines) along with a pseudo-Voigt function for the elastic line (black short-dashed line) and a constant experimental background (black dotted line). The green solid line represents a calculation of the phonon lineshape of the TA mode based on the theory of Allen et al. [15] to which the DHO function corresponding to the TO mode, the elastic line, and the experimental background were added (see text). (b) Upper and lower blue dashed lines represent the DHO fit of the TA mode at $T = 20 K$ and the same DHO function except that the temperature factor was set to the equivalent of $T = 4 K$ , respectively. The black dash-dotted and red dashed lines are calculations based on the theory of Allen et al. [15] and on the fitting parameters obtained at $T = 20 K$ for the TA mode [see (a)] for two different values of the superconducting energy gap 2Δ. The inset shows the calculation for $2 Δ = 5.2 meV$ with (black dash-dotted line) and without (red dashed line) a convolution with the experimental resolution. (c) Difference between the raw IXS data shown in panel (a) (green half-dots) compared to the differences obtained by subtracting the calculated lines in panel (b) from the DHO fit for $T = 20 K$ for the respective values of 2Δ (dotted/dash-dotted lines). The blue solid line represents the difference between the DHO fit at $T = 20 K$ and the curve obtained by setting the temperature factor of the DHO fit to the equivalent of $T = 4 K$ .
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Figure 5
(a)–(d) IXS raw data taken at four different wave vectors for $T = 20 K$ (green dots) and 4 K (blue circles), i.e., just above $T_{s} (= 18.9 K)$ and $T_{c} (= 16.8 K)$ and well-below $T_{c}$ . Lines are fits to the normal state data including DHO functions for phonons (convoluted with the experimental resolution) and pseudo-Voigt function for the respective elastic line. The inset in (b) shows data for $Q = (4, 0.15, 0.15)$ as an example with no superconductivity-induced changes of the phonon lineshape. (e)–(h) Comparison of the IXS data taken at $T = 4 K$ (green dots) with the calculated phonon lineshape in the superconducting phase employing $2 Δ = 5.6 meV$ (orange dashed lines) and 4.2 meV (red dash-dotted lines). (i)–(l) Differences between raw IXS data on cooling from $T = 20$ to 4 K (half-green dots) compared to the respective theoretical predictions for $2 Δ = 4.2 meV$ (red dash-dotted lines) and $5.6 meV$ (orange dashed lines) [see text and Fig. 4] with the corresponding raw data shown in the top row of panels. Dashed vertical lines denote the color-coded value of 2Δ, for which calculations show the best agreement with experimental results. Blue arrows mark the observed phonon energies at the respective wave vectors at $T = 20 K$ .
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Figure 6
Analysis of INS data taken for ${Nb}_{3} Sn$ [12]. (a) Fit (red solid line) of a DHO function (blue dashed line) and an estimated experimental background (black dotted line) to INS data (black dots) taken above the superconducting transition temperature. (b) INS data taken in the superconducting phase (black dots) along with the calculated phonon lineshape (green solid line) using the model theory [15] on top of the experimental background (black dotted line). The green solid line denotes the calculated result scaled by a factor of 0.73 on top of the background.
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Physical Review B

covering condensed matter and materials physics

Electron-phonon coupling and superconductivity-induced distortion of the phonon lineshape in $V_{3} Si$

A. Sauer, D. A. Zocco, A. H. Said, R. Heid, A. Böhmer, and F. Weber

Phys. Rev. B 99, 134511 – Published 15 April 2019

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Physical Review B

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Electron-phonon coupling and superconductivity-induced distortion of the phonon lineshape in V3Si

A. Sauer, D. A. Zocco, A. H. Said, R. Heid, A. Böhmer, and F. Weber

Phys. Rev. B 99, 134511 – Published 15 April 2019

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Electron-phonon coupling and superconductivity-induced distortion of the phonon lineshape in $V_{3} Si$