Spin and charge fluctuation induced pairing in ABCB tetralayer graphene

Letter
Open Access

Spin and charge fluctuation induced pairing in ABCB tetralayer graphene

Ammon Fischer, Lennart Klebl, Jonas B. Profe, Alexander Rothstein, Lutz Waldecker, Bernd Beschoten, Tim O. Wehling, and Dante M. Kennes

Phys. Rev. Research 6, L012003 – Published 8 January 2024

Abstract

Motivated by the recent experimental realization of ABCB stacked tetralayer graphene [Wirth et al., ACS Nano 16, 16617 (2022)], we study correlated phenomena in moiré-less graphene tetralayers for realistic interaction profiles using an orbital resolved random phase approximation approach. We demonstrate that magnetic fluctuations originating from local interactions are crucial close to the van Hove singularities on the electron- and hole-doped side promoting layer selective ferrimagnetic states. Spin fluctuations around these magnetic states enhance unconventional spin-triplet, valley-singlet superconductivity with $f$ -wave symmetry due to intervalley scattering. Charge fluctuations arising from long range Coulomb interactions promote doubly degenerate $p$ -wave superconductivity close to the van Hove singularities. At the conduction band edge of ABCB graphene, we find that both spin and charge fluctuations drive $f$ -wave superconductivity. Our analysis suggests a strong competition between superconducting states emerging from long- and short-ranged Coulomb interactions and thus stresses the importance of microscopically derived interaction profiles to make reliable predictions for the origin of superconductivity in graphene-based heterostructures.

Received 6 June 2023
Revised 18 August 2023
Accepted 25 October 2023

DOI:https://doi.org/10.1103/PhysRevResearch.6.L012003

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Ferrimagnetism Superconductivity

Strongly correlated systems

Random phase approximation

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Ammon Fischer ^1,*, Lennart Klebl^2,*, Jonas B. Profe¹, Alexander Rothstein ^3,4, Lutz Waldecker ³, Bernd Beschoten ³, Tim O. Wehling^2,5, and Dante M. Kennes^1,6,†

¹Institute for Theory of Statistical Physics, RWTH Aachen University, and JARA Fundamentals of Future Information Technology, 52062 Aachen, Germany
²I. Institute for Theoretical Physics, Universität Hamburg, Notkestraße 9-11, 22607 Hamburg, Germany
³2nd Institute of Physics and JARA-FIT, RWTH Aachen University, 52074 Aachen, Germany
⁴Peter Grünberg Institute (PGI-9), Forschungszentrum Jülich, 52425 Jülich, Germany
⁵The Hamburg Centre for Ultrafast Imaging, 22761 Hamburg, Germany
⁶Max Planck Institute for the Structure and Dynamics of Matter, Center for Free Electron Laser Science, 22761 Hamburg, Germany

^*These authors contributed equally to this work.
^†dante.kennes@mpsd.mpg.de

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Vol. 6, Iss. 1 — January - March 2024

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Images

Figure 1
Band structure and density of states (DOS) for different tetralayer graphene structures without an applied electric field. Panels (a)–(c): Low-energy band structure near valley $K$ with the relative orbital polarization $P_{o}$ of the bands indicated by the diverging colormap. The different stacking configurations of ABCB, ABCA, and ABAB tetralayer graphene are shown above while the two sites with dominant spectral weight are indicated. Panel (d): DOS within the range of the valley-flat bands. Intrinsic electrical fields caused by broken inversion symmetry open a band gap of $Δ_{g} \approx 20 meV$ in ABCB stacked graphene and flatten the low-energy bands around the valleys $K, K^{'}$ . Hence, ABCB has the highest DOS near the van Hove singularities (VHS) on the electron- and hole-doped side. The inset highlights different Fermi surfaces around valley $K$ as the chemical potential is tuned through the VHS on the hole-doped side. Starting from the edge of the valence band, the Fermi surface undergoes a Lifshitz transition as the three pockets originating from trigonal warping continuously transform to an annular Fermi surface.
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Figure 2
Spin correlations in the presence of local interactions in ABCB tetralayer graphene. (a) Critical onsite interaction $U_{c}$ required for the onset of magnetic order (purple) and DOS (grey) for $μ$ close to the VHS of the valence band. We expect an increased tendency towards magnetic order at the van Hove filling $μ_{VHS} \approx - 3.9 meV$ . The leading magnetic correlations are displayed in panel (b): Each curve corresponds to the magnetization within one layer. At $μ \approx μ_{VHS}$ , local interactions drive ferrimagnetism in the C layer, while away from the VHS antiferromagnetic fluctuations gain importance in all layers. (c) Momentum structure of the leading eigenvalue of the susceptibility matrix ${\hat{χ}}_{0} (q)$ for $μ \approx μ_{VHS}$ and $μ > μ_{VHS}$ [c.f. markers in (a)]. The transfer momenta supported by the Fermi surface are $q \approx Γ$ and $q \approx K^{(')}$ . The $K$ valley (left subpanel) includes the Fermi contour. (d) Band renormalization of up (blue) and down (red) electrons due to the ferrimagnetic instability at $μ_{VHS}$ .
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Figure 3
Superconducting instabilities mediated by spin and charge fluctuations in ABCB graphene. (a),(b) Leading SC coupling constant $λ_{S C}$ as function of the chemical potential $μ$ for different values of the onsite interaction $U$ . Screening ( $d RPA$ ) favors doubly-degenerate $p_{x, y}$ -wave superconducting order near the VHS (black star) and almost degenerate $f / s^{+}$ -wave SC order for low-hole doping (yellow circle). Spin exchange ( $x RPA$ ) exclusively enhances spin-triplet $f$ -wave superconductivity around the Stoner transition, until ferrimagnetic order eventually displaces SC at the VHS for $U ≳ U_{c}$ (b). Combining the effect of spin and charge fluctuations ( $xd RPA$ ) enhances screening driven spin-triplet SC order near the Stoner instability hinting towards significant screening of local interactions by charge fluctuations. The corresponding SC order parameters in the valleys $K$ and $K^{'}$ are displayed in (c).
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Figure 4
Competition of superconducting order at the VHS of ABCB for realistic values of the onsite interaction $U$ separated into contribution from spin fluctuations and screening. (a) Screening of long-ranged Coulomb interactions ( $d RPA$ ) favors $p_{x, y}$ ( $d_{x^{2} - y^{2}, x y}$ )-wave order, while $f / s^{+}$ order is suppressed for all values of $U$ . (b) Spin-fluctuation exchange ( $x RPA$ ) exclusively promotes spin-triplet $f$ -wave order. (c) Combining screening and spin fluctuations ( $xd RPA$ ) lifts the degeneracy between $p / d$ wave as spin-singlet order is suppressed by ferrimagnetic spin fluctuations. Eventually, $f$ -wave order displaces $p$ -wave order at the Stoner transition.
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