Electrically Tunable Superconductivity Through Surface Orbital Polarization

We investigate the physical mechanisms for achieving an electrical control of conventional spin-singlet superconductivity in thin films by focusing on the role of surface orbital polarization. Assuming a multiorbital description of the metallic state, due to screening effects the electric field acts by modifying the strength of the surface potential and, in turn, yields nontrivial orbital Rashba couplings. The resulting orbital polarization at the surface and in its close proximity is shown to have a dramatic impact on superconductivity. We demonstrate that, by varying the strength of the electric field, the superconducting phase can be either suppressed, i.e., turned into normal metal, or undergo a 0-$\pi$ transition with the $\pi$ phase being marked by nontrivial sign change of the superconducting order parameter between different bands. These findings unveil a rich scenario to design heterostructures with superconducting orbitronics effects.

Figure 1

(a) Schematic view of multilayered spin-singlet SC with $n_z$ layers labeled by the index $i_z$. The electric field penetrates only at the surface layers (blue) by inducing processes with intra- and interlayer orbital mixing. In the remaining layers (orange), the electric field is absent. (b) Sketch of the surface electronic hybridization due to orbital Rashba coupling, ${\alpha }_{\mathrm{OR}}$, between $xy$ and $(xz,yz)$ orbitals along the symmetry allowed directions. The standard nearest-neighbor hopping between $d$ orbitals is mainly relevant for homolog orbitals along $x,y,z$ axes, e.g., in (c) $xz$ orbitals hybridize along $x$ and $z$ directions, with $t_{\parallel }$ and $t_{\perp }$, respectively. Panel (d) depicts the effect of $\lambda$ with orbital mixing involving $xy$ and $xz,yz$ states. The term $\lambda$ is active only between the first two surface layers (blue). The orbital Rashba coupling is considered to be nonvanishing only at the surface layers.

Figure 2

(a) Phase diagram in the (${\alpha }_{\mathrm{OR}},\lambda$) plane with conventional superconducting (0 SC), unconventional ($\pi$ SC), and normal state. The parameters are $n_z=6,\mu =-0.4t,t_{\perp }=1.5t,\eta =0.1$. (b),(c) Behavior of the order parameter ${\overline{\mathrm{\Delta }}}_{\alpha }(\alpha =a,b,c)$ in the central layer at $i_z=n_z/2$, as a function of $\lambda$ in the regimes of weak (b) and strong (c) orbital Rashba interaction, namely ${\alpha }_{\mathrm{OR}}=0.2t$ and ${\alpha }_{\mathrm{OR}}=3.0t$, respectively. In (b) we observe a sharp transition to the $\pi$ SC, with a sign change in ${\mathrm{\Delta }}_c$ and hence a relative $\pi$ phase between $c$ and $a,b$ OPs. In (c) we demonstrate that all the OPs go to zero. Insets: ${\mathrm{\Delta }}_c$ along the $\hat{z}$ direction is shown for different values of $\lambda$. In (d) and (e) we show the analogous transitions of (b) and (c), but for $n_z=12$. In (f) and (g) we present the profile of ${\mathrm{\Delta }}_c$ for $n_z=30$, for weak and strong ${\alpha }_{\mathrm{OR}}$, respectively. In (f) we show the sign change (for $\lambda =0.3t$ and $0.4t$), while in (g) the OP is suppressed by increasing $\lambda$. In (b)–(g) ${\mathrm{\Delta }}_0$ is the superconducting OP for a monolayer without the OR and sets its scale (Appendix pp2).

Figure 3

(a)–(e) Plots of the superconducting free energy as a function of the inversion-asymmetric interlayer parameter $\lambda$ for the two most relevant configurations [i.e., conventional (0 SC) blue lines and unconventional $(\pi$ SC) purple lines] by considering different values of the pairing strength $g$. The orbital Rashba parameter is fixed to ${\alpha }_{\mathrm{OR}}=0.2t$ for (a)–(d), while it is ${\alpha }_{\mathrm{OR}}=0.1t$ in (e). A system of $n_z=12$ layers is considered and the other parameters are $t_{\perp }=1.5t,\mu =-0.4t,\eta =0.1$. The crossing of the lines in (a)–(c) and (e) gives the transition value of the inversion-asymmetric interlayer parameter $\lambda$. This critical value is almost unaffected by the change of $g$ when the same value of ${\alpha }_{\mathrm{OR}}$ is considered. Namely we have ${\lambda }_c(g=1.2t)=0.0904t\simeq 0.09t$, ${\lambda }_c(g=1.4t)=0.0987t\simeq 0.10t$ and ${\lambda }_c(g=2.0t)=0.1014t\simeq 0.10t$. For $g=1.0t$ and ${\alpha }_{\mathrm{OR}}=0.2t$ the intermediate 0-$\pi$ transition is not observed any more and the system undergoes a direct changeover into a normal state. Taking a smaller value of ${\alpha }_{\mathrm{OR}}$ [see (e)], we recover again the intermediate 0-$\pi$ transition also for pairing strength $g=1.0t$.

Figure 4

We report the spatial profile of the superconducting order parameter ${\mathrm{\Delta }}_{\alpha }(\alpha =a,b,c)$ along the $z$ direction for a slab with $n_z=12$ layers, pairing strength $g=1.2t$, and orbital Rashba coupling $\alpha =0.2t$. The OP is calculated self-consistently described in the main text. In (a) for $\lambda =0.05t$ we find that all the three components of the OP have the same sign (hence the system is in the conventional $0$ SC phase), in (b) and (c) for $\lambda =0.15t$ and $\lambda =0.22t$, ${\mathrm{\Delta }}_c$ has opposite sign with respect to ${\mathrm{\Delta }}_{a,b}$ thus realizing the unconventional $\pi$ SC phase.

Figure 5

We compare the results obtained in Figs. 2 and 2 (which are here reproduced as gray lines) with those obtained adding orbital mixing hopping terms. We see that the 0 SC to $\pi$ SC transition and the 0 SC to the normal state one are unaffected by the inclusion of orbital mixing terms. Specifically, we show the behavior of the order parameter in the inner side of the system $\overline{\mathrm{\Delta }}$ as a function of the surface interlayer coupling $\lambda$ in the regimes of weak (a) and strong (b) orbital Rashba interaction, namely ${\alpha }_{\mathrm{OR}}=0.2t$ and ${\alpha }_{\mathrm{OR}}=3.0t$, respectively. In both cases the critical value of $\lambda$ is not changed with the inclusion of an orbital mixing hopping term $t_m=t_{m\perp }=0.2t$, the other parameters are as in Figs. 2 and 2 of the main paper, i.e., $n_z=6,t_{\perp }=1.5t,\mu =-0.4t,\eta =0.1$, and $\beta =0.1$.

Figure 6

Behavior of the superconducting order parameter components ${\mathrm{\Delta }}_{\alpha \beta }=|{\mathrm{\Delta }}_{\alpha \beta }|\exp (i{\varphi }_{\alpha \beta })$ (with $\alpha ,\beta =a,b,c$) [(a) amplitude and (b) phase] in the central layer ($i_z=n_z/2$) as a function of the interlayer asymmetric interaction $\lambda$ for a fixed value of the orbital Rashba coupling ${\alpha }_{\mathrm{OR}}=0.2t$. The other parameters used are the same as in Fig. 2 of the main paper (i.e., $n_z=6,t_{\perp }=1.5t,\mu =-0.4t,\eta =0.1$), with the additional inclusion of the interorbital pairing interaction $g_{od}$. The amplitude of the interorbital OP is multiplied by a factor of 10 to be visible in the same plot. ${\mathrm{\Delta }}_{aa},{\mathrm{\Delta }}_{bb}$, and ${\mathrm{\Delta }}_{ab}$ are real and positive, thus their phases are zero for any $\lambda$. We stress that the values of the diagonal OP, ${\mathrm{\Delta }}_{\alpha \alpha }\equiv {\mathrm{\Delta }}_{\alpha }$ (with $\alpha =a,b,c$), do not change appreciably, if compared with Fig. 2 of the main text, in the self-consistent evaluation upon the inclusion of the interorbital pairing, hence the onset of the $\pi$ phase remain unchanged. We point out that here the interorbital hopping amplitude is zero.

Figure 7

Behavior of the superconducting order parameter components ${\mathrm{\Delta }}_{\alpha \beta }=|{\mathrm{\Delta }}_{\alpha \beta }|\exp (i{\varphi }_{\alpha \beta })$ (with $\alpha ,\beta =a,b,c$) [(a) amplitude and (b,c) phase of the OP] in the central layer ($i_z=n_z/2$) as a function of the interlayer asymmetric interaction $\lambda$ for fixed value of the orbital Rashba coupling ${\alpha }_{\mathrm{OR}}=0.2t$. In this analysis orbital mixing hoppings and interorbital pairing are included. The value of the interorbital hopping parameter here used is $t_m=t_{m\perp }=0.2t$, while other parameters are the same as in Fig. 2 of the main paper (i.e., $n_z=6,t_{\perp }=1.5t,\mu =-0.4t,\eta =0.1$). It is evident that, even if the amplitudes of the several components have changed from the case with the absence of interorbital hoppings and pairing, a $\pi$ phase is still present and its onset occurs at the same value of the parameter $\lambda$.

Figure 8

Spatial profile of the superconducting order parameter ${\mathrm{\Delta }}_{\alpha }(\alpha =a,b,c)$ along the $z$ direction at $n_z=6$ for five different values of $t_{\perp }$, in the case of weak [(a) and (b)] and strong [(c) and (d)] orbital Rashba coupling. The surface interlayer interaction $\lambda$ is set to zero.

Figure 9

(a)–(c) Behavior of the order parameter ${\overline{\mathrm{\Delta }}}_{\alpha }$ inside the system (i.e., in the central layer $i_z=3$) for $t_{\perp }=0.9t$. In the insets a schematic phase diagram in the $({\alpha }_{\mathrm{OR}},\lambda )$ plane is shown, where the blue line marks the path followed in the figure and the red dot denotes the transition point. Specifically, in (a) we present the results for ${\alpha }_{\mathrm{OR}}=0.5t$, in (b) for ${\alpha }_{\mathrm{OR}}=3.0t$, while in (c) we follow a horizontal path in the (${\alpha }_{\mathrm{OR}},\lambda$) plane, with $\lambda =0.6t$ varying ${\alpha }_{\mathrm{OR}}$. Finally, in (d) we report the spatial profile of ${\mathrm{\Delta }}_c$ along the $z$ direction corresponding to the analysis performed in (c). Here we see that while ${\overline{\mathrm{\Delta }}}_c$ is always negative, ${\mathrm{\Delta }}_c$ becomes positive in the outer layers for ${\alpha }_{\mathrm{OR}}\gtrsim 1.3t$.

Figure 10

Schematic figure describing the atomic positions for the determination of the electrostatic energy associated to the intra- and interlayer electronic processes. (a) Sketch of the nearest-neighbor atomic positions along the $(x,y,z)$ symmetry directions. (b) and (c) describe schematically the in-plane displacements that are related with the interlayer orbital Rashba coupling.

Figure 11

Plot of the self-consistent superconducting order parameter ${\mathrm{\Delta }}_{\alpha }$ ($\alpha =a,b,c$) as a function of ${\alpha }_{\mathrm{OR}}/t$ for $g=2.0t$ in a monolayer system. ${\mathrm{\Delta }}_0$ is the pairing amplitude in the absence of OR effect (i.e., for ${\alpha }_{\mathrm{OR}}=0$) for the orbitally isotropic case $({\mathrm{\Delta }}_a={\mathrm{\Delta }}_b={\mathrm{\Delta }}_c)$.

Figure 12

Behavior of the superconducting OP $\mathrm{\Delta }$ in a monolayer as a function of the orbital Rashba interaction parameter (${\alpha }_{\mathrm{OR}}$), for different values of the pairing strenght $g$. The OP is here assumed orbitally uniform, i.e., with ${\mathrm{\Delta }}_a={\mathrm{\Delta }}_b={\mathrm{\Delta }}_c$. We observe that the smaller the value of $g$ the stronger the effect of suppression and reduction of the OP due to the orbital Rashba effect. Inset: behavior of the SC OP in a monolayer in the absence of OR effects (${\mathrm{\Delta }}_0$) as a function of $g$. This value is used as a scale for $\mathrm{\Delta }$.

Figure 13

(a),(b) Plot of the value of $\mathrm{\Delta }$ that minimizes the free energy varying $\lambda$ (with $|{\mathrm{\Delta }}_a|=|{\mathrm{\Delta }}_b|=|{\mathrm{\Delta }}_c|=\mathrm{\Delta }$), for weak (a) and strong (b) orbital Rashba strength. The blue line corresponds to the analysis in which ${\mathrm{\Delta }}_{\alpha }$ ($\alpha =a,b,c$) have all the same sign, the magenta line instead refers to a configuration with ${\mathrm{\Delta }}_c$ having an opposite sign with respect to ${\mathrm{\Delta }}_{a,b}$. We observe that in (a) we have a transition (around $\lambda \simeq 0.16t$ to the $\pi$ SC state). (c)–(f) Plots of the free energy as a function of $\mathrm{\Delta }$ for several values of $\lambda$ in different cases, namely, as follows: (c) for weak ${\alpha }_{\mathrm{OR}}$ and ${\mathrm{\Delta }}_{\alpha }>0$ with $\lambda /t\in [0,0.4]$; (d) for strong ${\alpha }_{\mathrm{OR}}$ and ${\mathrm{\Delta }}_{\alpha }>0$ and $\lambda /t\in [0,0.24]$; (e) for weak ${\alpha }_{\mathrm{OR}}$ and ${\mathrm{\Delta }}_c<0$ and $\lambda /t\in [0,0.4]$; and finally in (f) for strong ${\alpha }_{\mathrm{OR}}$ and ${\mathrm{\Delta }}_c<0$ and $\lambda /t\in [0,0.12]$. For the four free-energy plots the different lines refer to inequivalent values of $\lambda$, for the reported ranges, starting from $\lambda =0$ (red bottom line), up to the highest value of $\lambda$ analyzed, following the rainbow colors with step $0.02t$.

Figure 14

Plot of the energy difference [$E(\mathrm{\Delta })-E(0)$] as a function of the order parameter $\mathrm{\Delta }$ (assuming that $|{\mathrm{\Delta }}_a|=|{\mathrm{\Delta }}_b|=|{\mathrm{\Delta }}_c|=\mathrm{\Delta }$) for several values of $\lambda$ close to the critical point for strong OR coupling (${\alpha }_{\mathrm{OR}}=3.0t$). The system has $n_z=6$ layers and $t_{\perp }=1.5t$. The analysis is made by assuming that the superconducting order parameter is zero in the outer layers and uniform in the inner ones. In the inset (a) there is an enlargement for four values of $\lambda$, underlying the presence of multiple minima and hence the occurrence of a first-order phase transition. (b) Behavior of the order parameter as a function of the orbital mixing term $\lambda$. After the small discontinuity the second minimum goes smoothly to zero.

Figure 15

Vector plots of $\mathbf{L}(k_x,k_y)$ per layer in the Brillouin zone, for a superconducting system with ten layers, in the case of weak (${\alpha }_{\mathrm{OR}}=0.5t$) and strong (${\alpha }_{\mathrm{OR}}=3.0t$) orbital Rashba effect. $L_z=0$ everywhere. In the first and third column we show the vector plots for $\lambda =0$, while in the second and fourth we show the difference $\mathrm{\Delta }\mathbf{L}$ between the vectors $\mathbf{L}$ for $\lambda =0.3t$ and $\lambda =0$. In each row we present the behavior in each layer labeled by $i_z$. Since the system is symmetric, layers from six to ten are not shown. The arrows are colored according to the magnitude of the vector field (see the legend at the bottom), with intensities that are scaled to unity. In each panel, the magnitude of $\mathbf{L}$ is also represented by the dimension of the arrows.