Inverse proximity effect in $s$-wave and $d$-wave superconductors coupled to topological insulators

Inverse proximity effect in $s$ -wave and $d$ -wave superconductors coupled to topological insulators

Henning G. Hugdal, Morten Amundsen, Jacob Linder, and Asle Sudbø

Phys. Rev. B 99, 094505 – Published 7 March 2019

Abstract

We study the inverse proximity effect in a bilayer consisting of a thin $s$ - or $d$ -wave superconductor (S) and a topological insulator (TI). Integrating out the topological fermions of the TI, we find that spin-orbit coupling is induced in the S, which leads to spin-triplet $p$ -wave $(f$ -wave) correlations in the anomalous Green's function for an $s$ -wave $(d$ -wave) superconductor. Solving the self-consistency equation for the superconducting order parameter, we find that the inverse proximity effect can be strong for parameters for which the Fermi momenta of the S and TI coincide. The suppression of the gap is approximately proportional to $e^{- 1 / λ}$ , where $λ$ is the dimensionless superconducting coupling constant. This is consistent with the fact that a higher $λ$ gives a more robust superconducting state. For an $s$ -wave S, the interval of TI chemical potentials for which the suppression of the gap is strong is centered at $μ_{TI} = \pm \sqrt{2 m v_{F}^{2} μ}$ , and increases quadratically with the hopping parameter $t$ . Since the S chemical potential $μ$ typically is high for conventional superconductors, the inverse proximity effect is negligible except for $t$ above a critical value. For sufficiently low $t$ , however, the inverse proximity effect is negligible, in agreement with what has thus far been assumed in most works studying the proximity effect in S-TI structures. In superconductors with low Fermi energies, such as high- $T_{c}$ cuprates with $d$ -wave symmetry, we again find a suppression of the order parameter. However, since $μ$ is much smaller in this case, a strong inverse proximity effect can occur at $μ_{TI} = 0$ for much lower values of $t$ . Moreover, the onset of a strong inverse proximity effect is preceded by an increase in the order parameter, allowing the gap to be tuned by several orders of magnitude by small variations in $μ_{TI}$ .

Received 9 August 2018

DOI:https://doi.org/10.1103/PhysRevB.99.094505

Physics Subject Headings (PhySH)

Superconductors Topological insulators Topological materials Topological phases of matter

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Henning G. Hugdal, Morten Amundsen, Jacob Linder, and Asle Sudbø^*

Center for Quantum Spintronics, Department of Physics, NTNU, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

^*Corresponding author: asle.sudbo@ntnu.no

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Vol. 99, Iss. 9 — 1 March 2019

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Figure 1
(a) Plot of the superconducting gap at $T = 0$ for an $s$ -wave superconductor as a function of $μ_{TI}$ and $t$ and with an upper cutoff $ω_{+} = 0.0025$ eV, normalized to the bulk value $| Δ_{0} |$ for parameter values relevant for Nb-HgTe bilayers. The $k_{F}$ values for the TI appear vertical on this plot as a function of $μ_{TI}$ due to the small value of the cutoff $ω_{+}$ . The numerical results show that the zero-temperature gap essentially is unaffected by the proximity to the TI for small values of $t$ , where the suppression is severe only for values of $μ_{TI}$ close to $\sqrt{2 m v_{F}^{2} μ}$ , a value far too large to be experimentally achievable. However, for increasing $t$ , the region where superconductivity is suppressed increases quadratically with $t$ , eventually leading to a suppression also for $μ_{TI} = 0$ . The inset shows the normalized gap at $t = 0.1$ , 0.2, and $0.3 eV$ , indicating that the gap is not suppressed entirely in most cases, but rather to a reduced value of $Δ_{0} e^{- 1 / λ}$ (dashed line), consistent with there being only one band contributing to superconductivity in this region. The exception is close to $μ_{TI} = 0$ for $t = 0.3 eV$ , where there are no bands with Fermi wave vector between $k_{-}$ and $k_{+}$ , resulting in $Δ = 0$ . This is the case in the area restricted by the dotted line in the main figure. (b) The upper left panel is a plot of the integrand in the gap equation, Eq. (32), evaluated at $Δ_{0}$ for wave vectors $k_{-} < | k | < k_{+}$ and $t = 0.1 eV$ , where light colors correspond to high values of the integrand. The three remaining panels show the magnitude of the Fermi wave vectors $k_{F}$ of the bands defined in Eq. (24) (left axis) in the same interval at $t = 0.1$ , 0.2 and $0.3 eV$ , and the normalized gap (right axis). Notice that the plots are close to symmetric around $μ_{TI} = 0$ since $ω_{D} ≪ μ$ . The dash-dotted lines are $k_{F}^{S}$ and $k_{F}^{TI} (μ_{TI})$ , the Fermi wave vectors of the S and TI for $t = 0$ , respectively. Comparing the two left panels it is clear that the main contribution to the integral in the gap equation comes from wave vectors close to the Fermi wave vectors of the bands in the relevant $| k |$ interval. $μ_{TI}^{α, \pm} (t)$ are plotted as dashed $(α = 1)$ and dotted $(α = - 1)$ lines in all plots, indicating the onset of the region in parameter space where superconductivity is greatly suppressed.
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Figure 2
(a) Plot of the superconducting gap at $T = 0$ for a $d$ -wave superconductor as a function of $μ_{TI}$ and $t$ with upper cutoff $ω_{+} = 0.15 eV$ , normalized to the bulk value $| Δ_{0} |$ for parameter values relevant for bilayers consisting of HgTe and high- $T_{c}$ superconductors. The gap is strongly suppressed for $μ_{TI}^{-} < μ_{TI} < μ_{TI}^{+}$ , where the approximate (exact numerical) functions $μ_{TI}^{\pm} (t)$ in Eq. (36) are plotted as dotted (dashed) lines. The approximate solution is only valid for $k_{F} \approx k_{F}^{S}$ , corresponding to small $t$ . For $μ_{TI} \approx μ_{TI}^{\pm} (t)$ the gap increases beyond $Δ_{0}$ . (b) Plot of the magnitude of the Fermi wave vectors of the bands in Eq. (24) in the interval $k_{-} < k_{F} < k_{+}$ (left axis), together with the normalized gap (right axis) for $ω_{+} = 0.15$ and $0.04 eV$ . The upper limit $k_{+}$ in the left axis corresponds to $ω_{+} = 0.04 eV$ . The black dash-dotted lines show the S and TI Fermi wave vectors for $t = 0$ . As for the $s$ -wave case, the strong suppression of the gap is due to only one band having a Fermi wave vector in the integration interval. Note how the values of $k_{F} (μ_{TI})$ of the hybridized bands (originating with the left $t = 0$ crossing of the $k_{F}$ 's of the TI and the S) bend back in a pronounced way as a function of $μ_{TI} (k_{F}$ is a multivalued function of $μ_{TI}$ since there are four bands). This leads to an enhanced density of states for these values of $μ_{TI}$ . This in turn gives an enhancement of the gap in the immediate vicinity of the region of $μ_{TI}$ where the gap is suppressed by the disappearance of bands crossing the TI Fermi surface. This effect is not seen in the $s$ -wave case, where the pronounced back bending of $k_{F} (μ_{TI})$ does not occur inside the integration interval with the much lower values of $ω_{\pm}$ [see Fig. 3].
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Figure 3
Plots of the bands $ε_{α}^{γ} (k)$ in Eq. (24) for (a) $s$ -wave and (b) $d$ -wave parameter values and different values of $μ_{TI}$ . The linewidths are proportional to the spectral weights $w_{α}^{γ} (k)$ of the bands [see Eq. (25)]. In (a) the values of $μ_{TI}$ correspond to a barely suppressed $(μ_{TI} = 0.0 eV)$ and strongly suppressed $(μ_{TI} = - 3.3 eV \approx - \sqrt{2 m v_{F}^{2} μ})$ gap for coupling $t = 0.2 eV$ . The inset shows that there is no hybridization of bands close to the Fermi level (dashed line) for the lowest $μ_{TI}$ , while the strong hybridization for $μ_{TI} = 3.3 eV$ leads to only one band crossing the Fermi level in the interval $[k_{-}, k_{+}]$ (dotted lines). In (b) we see that only one band crosses the Fermi level for $μ_{TI} = - 0.25 eV$ , explaining the strong suppression in this case. At $μ_{TI} = μ_{TI}^{\pm}$ we have an increase in $| Δ |$ , which can be explained by the bands having minima/maxima at the Fermi level in these cases, leading to high densities of states.
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Figure 4
The figure shows how the dimensionless coupling constant $λ$ affects the suppression of the superconducting gap for $s$ -wave S with $t = 0.2 eV$ (top) and $d$ -wave S with $t = 0.05 eV$ and $ω_{+} = 0.15 eV$ (bottom). Increasing $λ$ makes the superconducting state more robust, reducing both the suppression of $Δ$ , and also the increase in $Δ$ at $μ_{TI}^{\pm}$ in the $d$ -wave case.
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