Unconventional proximity-induced superconductivity in bilayer systems

We study the proximity-induced superconducting state in a general bilayer–conventional $s$-wave superconductor hybrid structure. For the bilayer we include a general parabolic dispersion, Rashba spin-orbit coupling, and finite-layer tunneling as well as the possibility to apply a bias potential and a magnetic Zeeman field, in order to address experimentally relevant bilayer systems, ranging from topological insulator thin films to generic double quantum well systems. By extracting the proximity-induced anomalous Green's function in the bilayer, we show a very rich structure for the superconducting pairing, including different spin states and odd-frequency pairing. Equal-spin spin-triplet $\left(p_x\pm i{p}_y\right)$-wave pairing is induced in both layers in the presence of a finite spin-orbit coupling and opposite-spin spin-triplet $s$-wave pairing with odd-frequency dependence appears for an applied magnetic Zeeman field. Finite interlayer pairing is also generally present in the bilayer. The interlayer pairing can be either even or odd in the layer index, with a complete reciprocity between parity in frequency and in layer index. We also find that a bilayer offers the possibility of sign reversal of the superconducting order parameters, both between the two layers and between multiple Fermi surfaces.

Figure 1

(a) Bilayer system (pink) in proximity to a spin-singlet $s$-wave superconductor (yellow). The superconductor is connected by a single-particle tunneling amplitude $t_0$ to the bottom layer (1) of the bilayer system, whereas the two bilayers are coupled by an interlayer tunneling $\delta$. A finite magnetization $M$ in the $z$ direction and a bias voltage $2V$ can also be applied to the bilayer. (b) Energy dispersion (eV) of a TI thin film as function of the momentum scaled by the spin-orbit interaction $\alpha$ (eV) for $\delta =0.03,M=0.01$, and $V=0.05$ eV.

Figure 2

(a), (b) Energy dispersion (meV) as a function of $\alpha k$ (meV) and (c), (d) LDOS $\left(\right[\frac{L}{2\pi \alpha }{]}^2$ states/meV) for each layer of a TI thin film in proximity to an external $s$-wave superconductor. The TI film is described by $K=\mu =M=0$ and $V=5$ meV to model an undoped TI thin film under low bias. The tunneling between the two TI surfaces is $\delta =2$ meV (a), (c) and $\delta =4$ meV (b), (d). A generic superconductor is assumed by setting $m_{sc}=m_e$ and ${\mu }_{sc}=1$ eV, which remove the dispersion of the superconductor from the low-energy regime.

Figure 3

Momentum-dependent order parameters ${\Delta }^{11}$ (intralayer pairing in layer 1, solid black), ${\Delta }^{22}$ (intralayer pairing in layer 2, dashed red), and ${\Delta }^{12}$ (interlayer pairing, dotted blue) as a function of momentum scaled with the spin-orbit coupling constant $\alpha$ (eV) for spin-singlet (left) and spin-up spin-triplet (right) pairing. Additional parameters are $V=0.05,\delta =0.05$, and $\mu =0.07$ eV. Arrows indicate the position of the two different Fermi momenta, green thin line indicates the zero level.

Figure 4

Contour plots showing the parameter region in $V-\mu$ (eV) space where the spin-singlet order parameter changes sign between the two Fermi surfaces ${\Delta }_s^{ii}\left(k_{F1}\right){\Delta }_s^{ii}\left(k_{F2}\right)<0$ (blue) for a TI thin film, demonstrating $s_{\pm }$-wave pairing in layer (surface) 1 (left figure) and 2 (right figure). The film thickness is represented by $\delta =0.05$ eV.

Figure 5

Momentum-integrated order parameters ${\Delta }^{11}$ (intralayer pairing in layer 1, solid black), ${\Delta }^{22}$ (intralayer pairing in layer 2, dashed red), and ${\Delta }^{12}$ (interlayer pairing, dotted blue) as a function of applied bias field $V$ (eV). Additional parameters are $\delta =0.05$ eV, $\mu =0.1$ eV. The insets show the even- and odd-interlayer pairing terms.