Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in ${\mathrm{CeCoIn}}_{5}$ revealed by high-resolution magnetostriction measurements

Letter

Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in ${CeCoIn}_{5}$ revealed by high-resolution magnetostriction measurements

Shunichiro Kittaka, Yohei Kono, Kaito Tsunashima, Daisuke Kimoto, Makoto Yokoyama, Yusei Shimizu, Toshiro Sakakibara, Minoru Yamashita, and Kazushige Machida

Phys. Rev. B 107, L220505 – Published 30 June 2023

Abstract

The Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state is an exotic superconducting phase formed by Cooper pairs with finite center-of-mass momentum $q$ . On theoretical grounds, the superconducting order parameter in the FFLO state is spatially modulated along the $q$ vector, and the emergence of an associated anisotropy is expected at the phase transition from the Abrikosov state to the FFLO state. Here, we report the results of high-resolution magnetostriction measurements for a single crystal of ${CeCoIn}_{5}$ around $B ∥ c$ . We find two anomalies in the magnetostriction along the $c$ axis, parallel to the magnetic-field orientation. In sharp contrast, this $B_{K}$ anomaly disappears in the magnetostriction along the $a$ -axis direction, perpendicular to the magnetic-field orientation. To explain this uniaxial expansion, we suggest a possibility that the FFLO transition occurs slightly below the upper critical field, and the FFLO modulation vector parallel to the applied magnetic field gives rise to the anisotropic response.

Received 13 April 2023
Revised 13 June 2023
Accepted 20 June 2023

DOI:https://doi.org/10.1103/PhysRevB.107.L220505

Physics Subject Headings (PhySH)

FFLO Magnetostriction Superconducting phase transition Superconductivity

Heavy-fermion systems

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Shunichiro Kittaka, Yohei Kono, Kaito Tsunashima, and Daisuke Kimoto

Department of Physics, Chuo University, Bunkyo-ku, Tokyo 112-8551, Japan

Makoto Yokoyama

Faculty of Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan and Institute of Quantum Beam Science, Ibaraki University, Mito, Ibaraki 310-8512, Japan

Yusei Shimizu

Institute for Materials Research (IMR), Tohoku University, Oarai, Ibaraki 311-1313, Japan

Toshiro Sakakibara and Minoru Yamashita

Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan

Kazushige Machida

Department of Physics, Ritsumeikan University, Kusatsu, Shiga 525-8577, Japan

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Vol. 107, Iss. 22 — 1 June 2023

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Figure 1
The $c$ -axis magnetostriction coefficient $λ_{c} (B) = (\partial L_{c} / \partial B) / L_{c}$ at several temperatures for $B ∥ c$ . Each set of data is shifted vertically by $10^{- 4} T^{- 1}$ for clarity. Closed (open) circles represent the data taken in the field-increasing (-decreasing) processes. Inset shows a schematic image of the spatial modulation of the order parameter $Δ (r)$ in the FFLO state along the modulation vector $q$ .
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Figure 2
Thermal expansion $Δ L_{c} / L_{c}$ measured at 5, 4.89, 4.85, 4.8, 4.7, 4.5, 4.3, and 4 T (from bottom to top) for $B ∥ c$ . Closed and open circles represent the data taken in the warming and cooling processes, respectively. Inset shows temperature dependence of the thermal expansion coefficient $α_{c} = (\partial L_{c} / \partial T) / L_{c}$ along the $c$ axis at 0 T.
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Figure 3
(a) Field-temperature phase diagram of ${CeCoIn}_{5}$ for $B ∥ c$ . (b) Field dependence of $d λ_{c} / d B$ at several temperatures. Both increasing and decreasing field data are shown above 0.2 K (only the former at 0.11 K). Each set of data in (b) is shifted vertically by $5 \times 10^{- 3} T^{- 2}$ for clarity. The positions of $B_{c 2}$ [squares in (a)] and $B_{K}$ (triangles) are determined by large and small peaks in $d λ_{c} / d B$ , respectively, as indicated by solid arrows in (b). The symbols shown in red (blue) are determined from the field-increasing (-decreasing) data. Crosses represent the field above which $λ_{c} (B)$ starts to change markedly toward $B_{K}$ with increasing $B$ , possibly corresponding to a boundary between the Abrikosov state and the FFLO state.
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Figure 4
Field angle $ϕ$ dependence of (a) $Δ L_{i} / L_{i}$ at 0.11 K and 5 T, and (b) their field-angle derivatives, $λ_{ϕ, i} = (\partial L_{i} / \partial ϕ) / L_{i}$ ( $i = a$ or $c$ ). Closed (open) symbols represent the data taken in the $ϕ$ -increasing (decreasing) process. The gray solid line in (b) represents the simulation spectrum consisting of two Gaussian functions (dashed and dash-dotted lines). Inset in (a) shows a schematic view of the flux lines and the FFLO modulation. The periodic nodal planes, separated by $L = 2 π / q$ , run perpendicular to the flux lines that form a vortex lattice in the $a b$ plane. Inset in (b) shows a wider view of $λ_{ϕ, i} (ϕ)$ in the $ϕ$ -increasing (-decreasing) process for $ϕ > 0$ ( $ϕ < 0$ ).
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Figure 5
(a) The wave number $q$ normalized by the coherent length and (b) the normalized entropy $S = S_{s} (T) / S_{n} (T_{c})$ as a function of $h = H / H_{c 2}$ , taken from Ref. [36]. Here, $S_{s}$ ( $S_{n}$ ) is the entropy in the superconducting (normal) state. (c) $| λ_{c} |$ as a function of $B$ at $T = 0.06 K$ for $B ∥ c$ . (d) $| α_{c} |$ as a function of $T$ at $B = 4.73 T$ for $B ∥ c$ . (e) $| λ_{ϕ, c} |$ as a function of the angle $ϕ$ at $B = 4.94 T$ and $T = 0.11 K$ . The yellow and the pink colored regions represent the Abrikosov state and the FFLO state, respectively; the boundary in $| λ_{c} |$ and $| λ_{ϕ, c} |$ ( $| α_{c} |$ ) is determined by the onset of the $B_{K}$ anomaly [inferred from the phase diagram in Fig. 3].
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Physical Review B

covering condensed matter and materials physics

Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in ${CeCoIn}_{5}$ revealed by high-resolution magnetostriction measurements

Shunichiro Kittaka, Yohei Kono, Kaito Tsunashima, Daisuke Kimoto, Makoto Yokoyama, Yusei Shimizu, Toshiro Sakakibara, Minoru Yamashita, and Kazushige Machida

Phys. Rev. B 107, L220505 – Published 30 June 2023

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Physical Review B

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Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in CeCoIn5 revealed by high-resolution magnetostriction measurements

Shunichiro Kittaka, Yohei Kono, Kaito Tsunashima, Daisuke Kimoto, Makoto Yokoyama, Yusei Shimizu, Toshiro Sakakibara, Minoru Yamashita, and Kazushige Machida

Phys. Rev. B 107, L220505 – Published 30 June 2023

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Modulation vector of the Fulde-Ferrell-Larkin-Ovchinnikov state in ${CeCoIn}_{5}$ revealed by high-resolution magnetostriction measurements