Possible chiral topological superconductivity in ${\mathrm{CrO}}_{2}$ bilayers

Possible chiral topological superconductivity in ${CrO}_{2}$ bilayers

Xu Dou, Kangjun Seo, and Bruno Uchoa

Phys. Rev. B 99, 104503 – Published 4 March 2019

Abstract

We address the possible emergence of spin triplet superconductivity in ${CrO}_{2}$ bilayers, which are half-metals with fully spin-polarized conducting bands. Starting from a lattice model, we show that chiral $p + i p$ states compete with nonchiral $p$ -wave ones. At large doping, the $p + i p$ channel has a sequence of topological phase transitions that can be tuned by gating effects and interaction strength. Among several phases, we find chiral topological phases having a single Majorana mode at the edge. We show that different topological superconducting phases could spontaneously emerge in the vicinity of the van Hove singularities of the band.

Received 4 April 2018
Revised 3 November 2018

DOI:https://doi.org/10.1103/PhysRevB.99.104503

Physics Subject Headings (PhySH)

Spin-triplet pairing Superconducting order parameter Superconducting phase transition Topological phase transition

Bilayer films Ultrathin films

Methods in superconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Xu Dou, Kangjun Seo, and Bruno Uchoa^*

Department of Physics and Astronomy, University of Oklahoma, Norman, Oklahoma 73069, USA

^*uchoa@ou.edu

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Vol. 99, Iss. 10 — 1 March 2019

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Figure 1
Top: Lattice of a ${CrO}_{2}$ bilayer, with $d_{x y}$ and $d_{x z}$ ( $d_{y z}$ ) orbitals in sublattice $A$ ( $B$ ). The blue orbitals sit in the top layer ( $A$ sites), and red orbitals in the lower one ( $B$ sites). Hopping energies are indicated by $t_{j}^{α}$ for intraorbital hopping between next-nearest sites, with $α = x y, x z$ for $j = A$ and $α = x y, y z$ for $j = B$ , and $t_{i}$ ( $i = 1, 2, 3, 4$ ) for nearest-neighbor hopping. (c) Energy spectrum of the lattice model along the diagonal (1,1) direction. Energy axes in eV units. Red dots indicate the location of van Hove singularities, where the DOS (d) diverges logarithmically.
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Figure 2
Phase diagram for (a) $p$ -wave and (b) $p + i p$ state for intraorbital ( $g_{1}$ ) and interorbital ( $g_{2}$ ) couplings at $μ = 4 t / 3 = 0.4$ eV. $g_{1}$ and $g_{2}$ in eV units. Green area: Intraorbital $p$ -wave ( $p SC$ I) for $g > {\tilde{g}}_{1 c} (μ)$ . Yellow: Interorbital $p$ -wave ( $p SC$ II). Gray area: Strong-coupling topological phase (TSC) with Chern number $N = 1$ . Light blue: Gapped weak-coupling one ( ${\bar{g}}_{1 c} < g < g_{1 c}$ ) with $N = - 3$ [see Fig. 3]. Light red: Gapless $p + i p$ , which is topologically trivial. Dashed lines: First-order phase transitions. Solid black lines: Second-order transition (blue and green arrows). (c) Scaling of $Δ_{1}$ vs $g_{1}$ . Red and brown circles: $p + i p$ for $μ = 0.4$ , and 0, respectively. Blue square: $p$ -wave for $μ = 0.4$ eV. Green and blue arrows: Critical coupling ${\bar{g}}_{1 c} \approx t / 7 = 0.045$ eV and ${\tilde{g}}_{1 c} \approx t / 12 = 0.025$ eV, respectively, for $μ = 0.4$ eV. Inset: $Δ_{1}^{C}$ vs $log x$ , with $x = (1 - {\bar{g}}_{1 c} / g)$ , showing power-law scaling in the chiral phase near ${\bar{g}}_{1 c} (μ)$ . Green dots: $μ = 0.3 eV$ . Red: $μ = 0.4 eV$ . (d) Brillouin zone. Red line: Anisotropic Fermi surface at $μ = 0.33$ eV.
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Figure 3
Mean-field phase diagram as a function of the chemical potential $μ$ and intraorbital pairing coupling $g_{1}$ , both in eV units. The phase diagram does not depend on $g_{2}$ for $g_{2} ≲ 0.1$ eV (see Fig. 2). (a) Normal phase (N), $p_{x}$ superconducting phase ( $p SC$ I), chiral $p + i p$ state (CSC), and topological $p + i p$ phase (TSC). (b) Possible topological phases in the chiral $p + i p$ channel. The integers indicate the corresponding BdG Chern number $N$ . For fixed $g_{1}$ , the system has a sequence of topological phase transitions near the van Hove singularities of the band, where the topology of the Fermi surface changes. The blue regions correspond to the weak-coupling gapped $p + i p$ phases, which are topological. Gray and maroon regions: Strong-coupling phases. In the mean field, the gapped $p + i p$ state wins over the nonchiral $p$ state in the strong-coupling sector.
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Figure 4
Majorana edge modes in the different topological phases in the gapped $p + i p$ state. Energy units in eV. (a) BdG Chern number $N = - 3$ state, at $μ = 0.44 eV$ . (b) $N = - 5$ at $μ = 0.33 eV$ . (c) $N = - 4$ at $μ = - 0.49$ eV and (d) $N = - 6$ at $μ = - 0.38$ eV in the weak-coupling regime. The lower panels give the corresponding phases in the strong-coupling sector: (e) $N = 1$ at $μ = 0.44 eV$ , (f) $N = - 1$ at $μ = 0.33 eV$ ; (g) $N = 0$ at $μ = - 0.49$ eV, which is topologically trivial and (h) $N = - 2$ at $μ = - 0.38 eV$ . At the crossing from the weak- to strong-coupling phases, when $g = g_{1 c} (μ)$ , all Chern numbers increase by 4.
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Physical Review B

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Possible chiral topological superconductivity in ${CrO}_{2}$ bilayers

Xu Dou, Kangjun Seo, and Bruno Uchoa

Phys. Rev. B 99, 104503 – Published 4 March 2019

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Possible chiral topological superconductivity in CrO2 bilayers

Xu Dou, Kangjun Seo, and Bruno Uchoa

Phys. Rev. B 99, 104503 – Published 4 March 2019

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Possible chiral topological superconductivity in ${CrO}_{2}$ bilayers