Tunable reentrant Kondo effect in quantum dots coupled to metal-superconducting hybrid reservoirs

Peter Zalom and Tomáš Novotný

Phys. Rev. B 104, 035437 – Published 30 July 2021

Abstract

We elaborate on the recently introduced concept of the reentrant Kondo effect in quantum impurities/dots coupled to hybrid metal-semiconductor contacts [G. Diniz et al., Phys. Rev. B 101, 125115 (2020)]. By noticing the equivalence of the originally suggested semiconducting arrangement to an analogous three-terminal quantum dot setup with a normal and two phase-biased superconducting leads introduced in our recent work [P. Zalom et al., Phys. Rev. B 103, 035419 (2021)], we put this effect into a new physical context, which gives us a fresh look on the problem. First, we identify the superconducting counterpart of the reentrant Kondo effect and, consequently, reveal its fragility with respect to an underlying doublet-singlet quantum phase transition induced by the particle-hole asymmetry of the reservoir densities of states. This is pertinent (even if previously unnoticed) also in the original as well as extended semiconducting setups, where it puts stringent conditions on the symmetry of the contact density of states. Furthermore, we analyze the experimental feasibility of observing the reentrant Kondo effect in its superconducting realization concluding that even present-day experiments might see the onset of the reentrant behavior, but fully developed Kondo features cannot be reached due to the required vast separation of energy/temperature scales.

Received 20 May 2021
Accepted 9 July 2021

DOI:https://doi.org/10.1103/PhysRevB.104.035437

Physics Subject Headings (PhySH)

Quantum phase transitions

Quantum dots

Renormalization group Scanning tunneling microscopy

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Peter Zalom ^1,* and Tomáš Novotný ^2,†

¹Institute of Physics, Czech Academy of Sciences, Na Slovance 2, CZ-18221 Praha 8, Czech Republic
²Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, CZ-12116 Praha 2, Czech Republic

^*zalomp@fzu.cz
^†tno@karlov.mff.cuni.cz

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Vol. 104, Iss. 3 — 15 July 2021

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Figure 1
Compliance of the above-gap peak ( $| ω | \geq Δ$ ) of the normal spectral function $A_{n}^{d} (ω)$ with the Friedel sum rule in the parameter space of $U / Δ$ and $Γ_{S} / Δ$ on the scale between 0 (above-gap peak not present) and 1 (peak fulfilling the Friedel sum rule) for $φ = π$ and $Γ_{N} = Δ / 2$ . NRG solution is given according to Sec. 2b.
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Figure 2
(a)–(c) NRG flow of rescaled energy eigenvalues $E_{n} Λ^{n / 2}$ at the $n th$ NRG iteration for the TDOS (9) calculated for selected values of $Γ_{S}$ . Since several lines may overlap, we indicate their multiplicity at the end of the NRG flow by the attached numbers. The gray vertical line at $n_{Δ} = 34$ denotes the NRG iteration with the energy resolution $Δ \approx B Λ^{- (n_{Δ} - 1) / 2}$ . It divides the NRG flow into high-temperature ( $n < 34$ ) and low-temperature parts ( $n > 34$ ). In panel (a) with $U / Γ_{S} = 50 / 41 \approx 1.22$ the high-temperature NRG flow ( $n > 34$ ) just barely develops the strong-coupling (SC) fixed point. The low-temperature part clearly flows into the SC fixed point for $n \approx 50$ . In panel (b) $U / Γ_{S} = 2.5$ , which causes the high-temperature NRG flow to stop in the crossover region before the SC fixed point. The low-temperature NRG flow develops SC fixed point for $n > 54$ . In panel (c) $U / Γ_{S} = 5$ so that the high-temperature NRG flow misses the SC and local-moment fixed points entirely. The low-temperature part flows evidently toward the SC point of the ordinary SIAM for $n > 60$ which is however much smaller than $n \approx 90$ resulting from the NRG flow of the ordinary SIAM at the corresponding model parameters. (d)–(f) Normal spectral functions (11) for the parameters corresponding to the top panels. All panels have been calculated using NRG Ljubljana in the wide-band limit with $Λ = 2$ . In the SC fixed point, $E_{n}$ depend only on $Λ$ making the corresponding effective Hamiltonian identical for all panels.
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Figure 3
(a)–(c) Parametric $Γ_{S}$ plot of $ln (T_{K 2} / Δ)$ vs $Δ / Γ_{N}$ for the hybrid superconducting system corresponding to the TDOS (9) and various values of $U$ at $φ = π$ . Points represent the actual NRG data while the lines are just for visual guidance. Notice that the dependencies appear linear indicating a good compliance with the Haldane expression (12) and allowing a reliable fitting of the proportionality factor $U_{eff}$ . (d)–(f) Values of $U_{eff}$ for the parameters corresponding to the top panels show nontrivial dependence on $U$ and $Γ_{S}$ . The comparison to the energy of the ABS states $E_{ABS}$ when $Γ_{N} = 0$ (SCIAM) shows that it correlates with $U_{eff}$ but the two quantities do not coincide.
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Figure 4
(a)–(c) NRG flow of $E_{n}$ for the TDOS functions (8) at $U = 100 Δ$ , $Γ_{N} = Δ$ , $U / Γ_{S} = 5$ , and selected values of $φ$ . Notation is the same as in Fig. 2 including the gray vertical line at $n_{Δ} = 34$ denoting the NRG iteration with the energy resolution roughly $Δ$ . It divides the NRG flow into the high-temperature ( $n < 34$ ) and the low-temperature parts ( $n > 34$ ). In panel (a), $φ = 0.8 π$ and the superconducting hybrid system is in the 0-like phase. Increasing $φ$ we reach the transition region at $φ \approx 0.90 π$ , see panel (b), and then the $π$ -like phase for $φ = 0.95 π$ in panel (c). The high-temperature NRG flow is largely independent of $φ$ . On the contrary, the low-temperature NRG flow shows various regimes typical of the particle-hole asymmetric SIAM. (d)–(f) Spectral functions of the particle-hole asymmetric semiconducting TDOS. Notation is the same as in Fig. 2. Parameter values correspond to the presented NRG flows from the top panels (a)–(c). All panels have been calculated using NRG Ljubljana in the wide-band limit with $Λ = 2$ . Although the $z$ -discretization scheme has been employed, noticeable numeric artifacts remain as unphysical oscillatory features of the low-temperature plateau.
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Figure 5
(a) $φ$ evolution of the subgap part of the normal spectral function $A^{w} (ω)$ in the scalar $w$ basis for TDOS (8) and parameters $U = 100 Δ$ , $Γ_{S} = 20 Δ$ , and $Γ_{N} = Δ$ . Here, $φ$ merely parametrizes the particle-hole asymmetry of the TDOS and has no direct physical interpretation. We observe one asymmetric central peak and two asymmetrically placed smaller peaks. All features show strong $φ$ dependence. (b) $φ$ evolution of the subgap part of the normal spectral function $A_{n}^{d} (ω)$ in the natural basis $d$ with parameter values as in panel (a). Spectral functions in panels (a) and (b) correspond to each other via symmetrization (10). Correspondingly, in the superconducting scenario we observe one symmetric central peak and two pairs of smaller ABS peaks which merge together only at $φ = π$ . (c) Phase dependence of the Kondo temperature $T_{K 2}$ for three selected values of $U$ at $Γ_{N} = Δ$ . $T_{K 2}$ is determined as the HWHM of the central Kondo-like peak observed in the $π$ -like phase of the superconducting three-terminal setup. $T_{K 2}^{π}$ denotes $T_{K 2}$ at $φ = π$ . Points represent the NRG data while lines are fits in the corresponding $π$ -like phase regions that terminate in the crossover region to the 0-like region. Since $Γ_{N} = Δ$ , we observe not only a good agreement of the data with $ln T_{K 2} \propto {cos}^{2} (φ / 2)$ but also enhancement of the $π$ -like phase region as $φ^{*} \to 0$ with increasing $U$ .
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Figure 6
(a) $φ$ evolution of the subgap normal spectral function $A^{w} (ω)$ in the scalar $w$ basis for $U = 10^{4} Δ$ , $U / Γ_{S} = 10$ , and $Γ_{N} = 2.5 Δ$ for system with TDOS (8). (b) $φ$ evolution of the subgap normal spectral function $A_{n}^{d} (ω)$ in the natural basis $d$ with parameter values as in panel (a). Spectral functions in panels (a) and (b) correspond to each other via symmetrization (10). Correspondingly, in the superconducting scenario we observe one symmetric central peak and two pairs of smaller ABS peaks which merge together only at $φ = π$ . (c) Phase dependence of the Kondo temperature $T_{K 2}$ for $Γ_{N} ≳ Δ$ shows significant differences when compared to the results of Fig. 5. HWHM is determined as previously; $T_{K 2}^{π}$ denotes $T_{K 2}$ at $φ = π$ . Points represent the NRG data while lines are fits in the corresponding $π$ -like phase regions that terminate in the crossover region to the 0-like region. In all three cases, $ln T_{K 2} \propto {cos}^{2} (φ / 2)$ but $φ^{*} \to π$ with increasing $U$ which is in stark contrast to the opposite shift in the $Γ_{N} ≲ Δ$ case.
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