Interorbital Cooper pairing at finite energies in Rashba surface states

Multiband effects in hybrid structures provide a rich playground for unconventional superconductivity. We combine two complementary approaches based on density-functional theory (DFT) and effective low-energy model theory in order to investigate the proximity effect in a Rashba surface state in contact with an $s$-wave superconductor. We discuss these synergistic approaches and combine the effective model and DFT analysis at the example of a Au/Al heterostructure. This allows us to predict finite-energy superconducting pairing due to the interplay of the Rashba surface state of Au, and hybridization with the electronic structure of superconducting Al. We investigate the nature of the induced superconducting pairing, and we quantify its mixed singlet-triplet character. Our findings demonstrate general recipes to explore real material systems that exhibit interorbital pairing away from the Fermi energy.

Figure 1

(a) Localization of the electron density (arb. units) around the Fermi energy throughout the Al/Au heterostructure. The background shows a cut through $x=0$. (b) Illustration of three kinds of Cooper pair tunneling and the formation of different singlet/triplet components due to the Rashba surface state. Cooper pairs formed by electrons originating from different orbitals are denoted by different colors.

Figure 2

DFT wave-function localization in the Al/Au hybrid structure consisting of six layers of each element at the two $k$-points (a) $k_x=0.1{\text{Å}}^{-1}$ and (b) $k_x=0.3{\text{Å}}^{-1}$ for the four states labeled in panel (d). (c) Corresponding large-scale DFT band structure. The color bar shows the localization of the states. (d) Enlarged view of the spectrum close to the Fermi energy denoted by the blue rectangle in panel (e). The color bar shows the spin polarization $\langle s_y\rangle$ (arb. units). (e) [(f)] Excitation spectrum of the normal state obtained by the low-energy model Hamiltonian, given in Eq. (1), close to the Fermi energy in the absence (presence) of band hybridization and including third-order Rashba spin-orbit coupling, i.e., $F_0=g=0$ ($F_0=0.2$ and $g=-8.45$). Other model parameters are given in Table 1.

Figure 3

Schematic overview of a KS-BdG simulation that starts from the crystal structure which (in a standard DFT calculation) gives the ground-state density ${\rho }_0$. For the superconducting state, the KS-BdG equations are then solved self-consistently to obtain charge and anomalous densities $\rho ,\chi$ in the superconducting state, which determines the superconducting band structure.

Figure 4

Superconducting band structure of the Al/Au heterostructure obtained by (a) DFT and (b) the low-energy model. The red/green and gray bands indicate the particle and hole character of the BdG bands, respectively. The red/green color of the particle bands indicates the localization of the states. Panels (c) and (d) show an enlarged view of the region marked by the blue box in (a) where six different superconducting avoided crossings emerge (labeled ${\delta }_{\mathrm{Al}}^{\pm }$, ${\delta }_{\mathrm{Au}}^{\pm }$, and ${\delta }_{\mathrm{IOP}}^{\pm }$). The absence of avoided crossings marked by black circles in (c) and (d) is due to pseudo-spin-rotational symmetry. For illustration purposes, we show results for scaled-up values of the superconducting pairing. The model parameters for the analytical model are those given in Table 1 and $\mathrm{\Delta }=0.4F_0$.

Figure 5

Effective superconducting excitation spectra for (a) ${\widehat{H}}_{\mathbf{k},\text{eff}}^{++}$ with $\mathrm{\Delta }=0.4F_0$, (b) ${\widehat{H}}_{\mathbf{k},\text{eff}}^{--}$ with $\mathrm{\Delta }=0.8F_0$ ($\mathrm{\Delta }$ is scaled up for the sake of clarity), and (c) ${\widehat{H}}_{\mathbf{k},\text{eff}}^{+-}$ with $\mathrm{\Delta }=0.4F_0$. (d)–(f) Real and imaginary part of the pseudo-spin-singlet and pseudo-spin-triplet components of the effective pairing matrix associated with the dispersion relation illustrated in the top panels. Black crosses in (d)–(f) highlight $k$-points where particle and hole bands cross in the absence of pairing. The model parameters are the same as those given in Table 1.

Figure 6

(a) Magnitude of the effective superconducting avoided crossings for different pairing potentials. (b) Strength of pseudo-spin-triplet $d_{\mathbf{k},z}^{\nu {\nu }^{\prime }}$ compared to pseudo-spin-singlet ${\phi }_{\mathbf{k},z}^{\nu {\nu }^{\prime }}$ for the effective intraorbital pairing potential, namely ${\widehat{\mathrm{\Delta }}}_{\mathbf{k},\text{eff}}^{++}$ and ${\widehat{\mathrm{\Delta }}}_{\mathbf{k},\text{eff}}^{--}$, as well as the interorbital pairing potential ${\widehat{\mathrm{\Delta }}}_{\mathbf{k},\text{eff}}^{+-}$. The cross marks indicate the momenta where electron and hole bands are crossed in the absence of pairing.

Figure 7

Superconducting band structure of Al/Au obtained by (a) DFT calculations and (b) analytical model in the presence of Zeeman magnetic field of size $B=2\mathrm{mRy}$. Finite-energy Cooper pairings, highlighted by blue circles, emerge due to the interplay between superconductivity and magnetic field. The color bar indicates particle (red/green) and hole (gray) components of the BdG spectra. The model parameters for the analytical model are those given in Table 1 and $\mathrm{\Delta }=0.4F_0$.

Figure 8

Density of states of the Al/Au heterostructure. The faint orange and green lines indicate the contributions of the individual Al and Au layers to the total DOS (blue line).

Figure 9

Superconducting electronic structure of six layers of Au on top of semi-infinite Al. The gray background (i.e., with nonzero spectral density) in most parts of the Brillouin zone reflects the continuum of states in the semi-infinite Al. The spectrum clearly shows the superconducting gap of Al (labeled ${\mathrm{\Delta }}_{\mathrm{Al}}$) and the proximity-induced gap in the Au Rashba surface state (marked by the black box). In contrast to the thin-film limit, the finite-energy pairing disappears in the semi-infinite limit as no discrete Al quantum-well states are present.