Multigap superconductivity at an unconventional Lifshitz transition in a three-dimensional Rashba heterostructure at the atomic limit

It is well known that the critical temperature of multigap superconducting three-dimensional (3D) heterostructures at atomic limit (HAL) made of a superlattice of atomic layers with an electron spectrum made of several quantum subbands can be amplified by a shape resonance driven by the contact exchange interaction between different gaps. The $T_C$ amplification is achieved tuning the Fermi level near the singular nodal point at a Lifshitz transition for opening a neck. Recently high interest has been addressed to the breaking of inversion symmetry which leads to a linear-in-momentum spin-orbit induced spin splitting, universally referred to as Rashba spin-orbit coupling (RSOC) also in 3D layered metals. However the physics of multigap superconductivity near unconventional Lifshitz transitions in 3D HAL with RSOC, being in a non-BCS regime, is not known. The key result of this work getting the superconducting gaps by Bogoliubov theory and the 3D electron wave functions by solution of the Dirac equation is the feasibility of tuning multigap superconductivity by suitably matching the spin-orbit length with the 3D superlattice period. It is found that the presence of the RSOC amplifies both the $k$ dependent anisotropic gap function and the critical temperature when the Fermi energy is tuned near the circular nodal line. Our results suggest a method to effectively vary the effect of RSOC on macroscopic superconductor condensates via the tuning of the superlattice modulation parameter in a way potentially relevant for spintronics functionalities in several existing experimental platforms and tunable materials needed for quantum devices for quantum computing.

Figure 1

The dispersion along $k_x$ is shown together with the dispersion along $k_z$, in arbitrary units, both for $n$ odd (bottom panel) and for $n$ even (top panel). In this figure $\mathrm{\Delta }{E}_{zn}$ is the dispersion in the $z$ direction equal to ${\varepsilon }_z$ for both $n$ even and odd (in the numerical model we will assume that $\mathrm{\Delta }{E}_{z2}$ is equal to the cut-off energy ${\omega }_0$). $\mathrm{\Delta }{E}_{\text{RSOC}}=-k_0^2$ is, instead, the shift of the Dirac point (defined as the point at which the in-plane dispersions with opposite helicity meet) as a consequence of the dispersion along $k_z$. The $\mathrm{\Gamma }$ and $Z$ points are the center and the edge of the first Brillouin zone (IBZ). We then indicate the three different regimes in which to study the system: (I) ${\varepsilon }_z<\mu$ (light-yellow box); (II) $0<\mu <{\varepsilon }_z$ (light-green box); and (III) $-k_0^2<\mu <0$ (light-blue box).

Figure 2

Top panel: Contour plots in the $(k_{\parallel },k_z)$ plane for $\lambda =-1$ (left) and $\lambda =+1$ (right) for a single-harmonic tight-binding model for the first subband of Eq. (11). Parameters are $d=1, t=1$ so that ${\varepsilon }_z=2$. The RSOC momentum $k_0=1.5$. On top of some of the isoenergetic curves are shown the corresponding Fermi surfaces. Bottom panel: Contour plots as above for the second subband of Eq. (12). The orange numbers are the values of the Lifshitz parameter, defined in Eq. (8) of the different level curves, for the choice of the parameters made in this simplified model, while the dashed green curves on the 3D Fermi surfaces of (a) and (c) represent the nodal line of singular points at an unusual van Hove singularity.

Figure 3

Total and partial DOS as a function of ${\eta }_R$ [Eq. (9)]. Top panel: The partial DOS $N_{-}$ [Eq. (19)] and $N_{+}$ [Eq. (20)] as a function of the rescaled Lifshitz parameter ${\eta }_R$. The black curve is $k_0=0$. The other curves have increasing values of $k_0=0.1,0.2,0.3,0.41,0.5,0.6,0.7,0.8$. Here $t=1$. Bottom panel: The total DOS $N_{-}+N_{+}$ given in Eqs. (19) and (20) as a function of rescaled Lifshitz parameter ${\eta }_R$. The black curve is $k_0=0$. The other curves have increasing values of $k_0=0.1,0.2,0.3,0.41,0.5,0.6,0.7,0.8$. Here $t=1$.

Figure 4

Isoenergetic curves, Fermi surfaces for the first and second subbands at three different values of Lifshitz parameter. The top panels show the case of $\lambda =1$ (left) and $\lambda =-1$ (right) for the first subband and for $\mathrm{\Delta }{E}_{\text{RSOC}}=\mathrm{\Delta }{E}_{2z}$ (${\alpha }_{\text{SO}}=0.41$). The bottom panels show the same analysis carried out for the second subband. The DOS maximum is observed at ${\eta }_L$ where the system develops a van Hove singularity. In this case the Fermi surface develops a nodal line highlighted in (a) and (c) with a dashed green line. For $\lambda =1$, both for the first and for the second subband, ${\eta }_L$ is independent of the value of ${\alpha }_{\text{SO}}$ and it is equal to $-3.9$ and 0.92 meV, respectively. While for $\lambda =-1, {\eta }_L$ changes with ${\alpha }_{\text{SO}}$ (as underlined in Fig. 5), for the first subband ${\eta }_L=-4.6\text{meV}$, for the second ${\eta }_L=-0.11\text{meV}$. In (c) and (d), for $\eta ={\eta }_L$, we highlight the phase factor with the colors of the rainbow and the nodal line (white dashed curve) of the singular points.

Figure 5

Partial density of the states vs $\eta$ and projection of the FS in the plane $(k_x,k_y)$ at $k_z=Z$. In the left panel the continuous curves represent the partial DOS for $\lambda =-1$, the dots those for $\lambda =1$, the shades of blue refer to the first subband for three different values of ${\alpha }_{\text{SO}}=0.20,0.41,0.50$, while the shades of red to the second subband. The figure shows that the value of ${\eta }_L$, for which the FS have a nodal line (Fig. 4), decreases as ${\alpha }_{\text{SO}}$ increases by an amount equal to $E_0/{\omega }_0$, while the peak of the partial DOS $\lambda =+1$ increases. In the right panel the light blue curve represents the projection of the FS relative to $\lambda =-1$ in the plane for the first subband, the light blue dashed curve is relative to $\lambda =1$. Similarly, the orange curves refer to the second subband. This panel is built for ${\alpha }_{\text{SO}}=0.41$ and $\eta =3$.

Figure 6

Normalized band-edge energy as a function of the RSOC constant. The orange empty circles represent the band edge energy vs ${\alpha }_{\text{SO}}$ for the second subband and $\lambda =1$, while the blue empty squares are relative to the first subband at the same helicity value. The red dots, on the other hand, represent the band edge energy for the second subband at $\lambda =-1$, the light blue dots refer to the first subband for the same helicity value. As noted earlier, a Rashba shift can only be observed for negative helicity bands.

Figure 7

The DOS for the first and the second subband as the ${\alpha }_{\text{SO}}$ changes. In particular, we have chosen values for this parameter between 0.2 and 0.8 in order to reproduce the three cases previously discussed, $\mathrm{\Delta }{E}_z⋛\mathrm{\Delta }{E}_{\text{RSOC}}$, and compare them with the case of no RSOC, ${\alpha }_{\text{SO}}=0$. What is observed is that as the RSOC increases, the DOS peak becomes more pronounced and shifts to gradually smaller energies, since the peak occurs at $E_0/{\omega }_0$.

Figure 8

Histogram of the matrix elements defining the superposition integral of Eq. (38). The green and blue bars refer, respectively, to the intraband pairings $I_{1,1}(k_z,q_z)$ and $I_{2,2}(k_z,q_z)$, while the red and yellow bars refer to, respectively, to the interband couplings $I_{1,2}(k_z,q_z)$ and $I_{2,1}(k_z,q_z)$. The histogram shows a marked anisotropy.

Figure 9

Terms of the matrix of the exchange integral defined in Eq. (38) as a function of the wave vectors in the direction of the confinement potential. The green plan corresponds to $I_{1,1}(k_z,q_z)$, the blue plan corresponds to $I_{2,2}(k_z,q_z)$, the red plan corresponds to $I_{1,2}(k_z,q_z)$, and the yellow plan corresponds to $I_{2,1}(k_z,q_z)$.

Figure 10

Properties of the normal phase and the superconductive phase vs the rescaled Lifshitz parameter without and with RSOC for different ${\alpha }_{\text{SO}}$ values such that $\mathrm{\Delta }{E}_{\text{RSOC}}⪌\mathrm{\Delta }{E}_{2z}$. Panel (a) left side Starting from the bottom, the first panel shows the DOS for the first subband (blue curve) and the second subband (red curve) and the trend of the gap relative to the first subband (blue curve) and to the second subband (red curve) as a function of the rescaled Lifshitz parameter. The other panels in (a) are related to the four values of ${\alpha }_{\text{SO}}$ previously discussed, ${\alpha }_{\text{SO}}=0.30,0.41,0.50,0.70$, and show the partial DOS for the first subband with positive helicity (light blue curves) and with negative helicity (blue curves) and the partial DOS for the second subband with positive helicity (orange curve) and with negative helicity (red curves). We also report the trend of the gap relative to the first subband (blue curve) and to the second subband (red curve) as a function of the rescaled Lifshitz parameter. An anomalous behavior and an amplification of the parameters of the superconductive phase are observed in a range of rescaled Lifshitz parameter $0<{\eta }_R<1$. That is, in the proximity to the unusual van Hove singularity. Panel (b) right side The values of the gaps for the second subbands at ${\alpha }_{\text{SO}}=0$ and ${\alpha }_{\text{SO}}=0.41$ vs the rescaled Lifshitz parameter for different values of $k_z=0,\pi /2d,\pi /d$ show a variation in a neighborhood of van Hove unusual singularity. We have highlighted the variation of the gap values on the Fermi surface at different $k_z$ for three values of the Lifshitz parameter in ${\eta }_R={\eta }_L$ and ${\eta }_R={\eta }_L\pm 0.5$ by choosing the black color for the low gap values at $k_z=\pi /d$, the red color for $k_z=\pi /2d$, and the orange color for the high gap values at $k_z=0$.

Figure 11

(a) The critical temperature $T_C$ versus Lifshitz parameter for different values of Rashba coupling ${\alpha }_{\text{SO}}$ on a semilogarithmic scale. The critical temperature appears as an asymmetric function of the Lifshitz parameter and if ${\alpha }_{\text{SO}}<0.41$ the maximum of $T_C$ grows slowly, while if ${\alpha }_{\text{SO}}>0.41$ it grows faster and faster. (b) In this Uemura plot the critical temperature $T_C$ is plotted on a log-log scale versus the Fermi temperature for different values of Rashba coupling ${\alpha }_{\text{SO}}$. The white box refers to the Fano resonance appearing near the BEC-BCS crossover indicated by the dashed line.

Figure 12

The maximum value of the critical temperature (red curve) and critical temperature predicted by the BCS theory ratio for different values of ${\alpha }_{\text{SO}}$ parameter. What is observed is a marked amplification of the maximum of the $T_C$ when a Rashba coupling is introduced into the system. In particular, the maximum of the critical temperature increases slowly for ${\alpha }_{\text{SO}}<0.41$ and always becomes faster for ${\alpha }_{\text{SO}}>0.41$.

Figure 13

Properties of the superconductive phase vs the Lifshitz parameter without and with RSOC for different ${\alpha }_{\text{SO}}$ values such that $\mathrm{\Delta }{E}_{\text{RSOC}}⪌\mathrm{\Delta }{E}_{2z}$. This figure shows both the trend of the gap ratio for the first subband (blue curves) and the second subband (red curves) compared to the constant value predicted by the BCS theory (black curves).

Figure 14

The trend of the critical temperature vs the ${\mathrm{\Delta }}_2/{\mathrm{\Delta }}_1$ ratio for different values of the parameter ${\alpha }_{\text{SO}}$. The $T_C$ trend shows a strong asymmetry which becomes maximum when the ${\mathrm{\Delta }}_2/{\mathrm{\Delta }}_1$ ratio is maximum.

Figure 15

Properties of the superconductive phase vs the Lifshitz parameter without and with RSOC for different ${\alpha }_{\text{SO}}$ values such that $\mathrm{\Delta }{E}_{\text{RSOC}}⪌\mathrm{\Delta }{E}_{2z}$. This figure shows the variation of the critical temperature with the cut-off energy via the isotope coefficient. The constant value predicted by the BCS theory is the black line.

Figure 16

The isotope coefficient as a function of the critical temperature for different values of ${\alpha }_{\text{SO}}$.