Critical temperature and tunneling spectroscopy of superconductor-ferromagnet hybrids with intrinsic Rashba-Dresselhaus spin-orbit coupling

We investigate theoretically how the proximity effect in superconductor/ferromagnet hybrid structures with intrinsic spin-orbit coupling manifests in two measurable quantities, namely, the density of states and critical temperature. To describe a general scenario, we allow for both Rashba- and Dresselhaus-type spin-orbit coupling. Our results are obtained via the quasiclassical theory of superconductivity, extended to include spin-orbit coupling in the Usadel equation and in the Kupriyanov-Lukichev boundary conditions. Unlike previous works, we have derived a Riccati parametrization of the Usadel equation with spin-orbit coupling which allows us to address the full proximity regime and not only the linearized weak proximity regime. First, we consider the density of states in both SF bilayers and SFS trilayers, where the spectroscopic features in the latter case are sensitive to the phase difference between the two superconductors. We find that the presence of spin-orbit coupling leaves clear spectroscopic fingerprints in the density of states due to its role in creating spin-triplet Cooper pairs. Unlike SF and SFS structures without spin-orbit coupling, the density of states in the present case depends strongly on the direction of magnetization. Moreover, we show that the spin-orbit coupling can stabilize spin-singlet superconductivity even in the presence of a strong exchange field $h\gg \mathrm{\Delta }$. This leads to the possibility of a magnetically tunable minigap: changing the direction of the exchange field opens and closes the minigap. We also determine how the critical temperature $T_c$ of an SF bilayer is affected by spin-orbit coupling and, interestingly, demonstrate that one can achieve a spin-valve effect with a single ferromagnet. We find that $T_c$ displays highly nonmonotonic behavior both as a function of the magnetization direction as well as the type and direction of the spin-orbit coupling, offering a new way to exert control over the superconductivity of proximity structures.

Figure 1

(a) The SF bilayer in Secs. 3a, 3b, and 3d. (b) The SFS trilayer in Sec. 3c. We take the thin-film layering direction along the $z$ axis, and assume an $xy$ plane Rashba-Dresselhaus coupling in the ferromagnetic layer.

Figure 2

Geometric interpretation of the SO field (21) in polar coordinates: the Hamiltonian couples the momentum component $k_x$ to the spin component $({\sigma }_{x}cos\chi +{\sigma }_{y}sin\chi )$ with a coefficient $+a/m$, and the momentum component $k_y$ to the spin component (${\sigma }_{x}sin\chi +{\sigma }_{y}cos\chi$) with a coefficient $-a/m$. Thus $a$ determines the magnitude of the coupling, and $\chi$ the angle between the coupled momentum and spin components.

Figure 3

Density of states $D\left(\varepsilon \right)$ for the SF bilayer with energies normalized to the superconducting gap $\mathrm{\Delta }$ and SO coupling normalised to the inverse ferromagnet length $1/L_F$. The table shows the spectroscopic effect of increasing SO coupling with $\alpha =\beta$ when the magnetization $\underline{h}=3\mathrm{\Delta }\widehat{z}$, i.e., with the field perpendicular to the interface, and the effect of increasing difference between the Rashba and Dresselhaus coefficients for both $\underline{h}=3\mathrm{\Delta }\widehat{z}$ and $\underline{h}=3\mathrm{\Delta }\widehat{y}$. Although the conditions for LRT generation are fulfilled in the latter case, it is clear that no spectroscopic signature of this is present.

Figure 4

Zero-energy density of states $D\left(0\right)$ as a function of the spin-orbit angle $\chi$ and magnetization angle $\theta$. We have used a ferromagnet of length $L_F/{\xi }_S=0.5$ with an exchange field $h/\mathrm{\Delta }=3$ and a spin-orbit magnitude $a{\xi }_S=2$.

Figure 5

Density of states $D\left(\varepsilon \right)$ for the SF bilayer with energies normalized to the superconducting gap $\mathrm{\Delta }$ and SO coupling normalized to the inverse ferromagnet length $1/L_F$. The table shows the spectroscopic effect of equal Rashba and Dresselhaus coefficients when the magnetization is oriented entirely in the $y$ direction, and also the correlation between the SO-induced zero-energy peak with the long-range triplet component $|\mathrm{Re}\left(d_x\right)|\equiv \mathrm{Re}\left(d_{\perp }\right)$. It is clear that the predominant effect of the LRT component, which appears only when the SO coupling is included, is to increase the peak at zero energies. Increasing the field strength rapidly suppresses the density of states towards that of the normal metal.

Figure 6

The dependence of the density of states of the SF bilayer at zero energy on the angle $\theta$ between the $x$ and $y$ components of the magnetization exchange field $\underline{h}/\mathrm{\Delta }=6(cos\left(\theta \right),sin\left(\theta \right),0)$ for increasing SO coupling. As the strength of the SO coupling increases, we see increasingly sharp variations in the density of states from an optimal peak at around $\theta \approx 7\pi /32$ and $\theta \approx 9\pi /32$ to a gap around $\theta =\pi /4$.

Figure 7

Density of states $D\left(\varepsilon \right)$ in the SF bilayer for energies normalized to the superconducting gap $\mathrm{\Delta }$ and SO coupling normalized to the inverse ferromagnet length $1/L_F$. The table shows the spectroscopic features of the SF bilayer with rotated exchange field in the $xy$ plane. Again we see a peak in the density of states at zero energy due to the LRT component, i.e., the component of $\underline{d}$ perpendicular to $\underline{h}, d_{\perp }$. The height of this zero-energy peak is strongly dependent on the angle of the field vector in the plane, as shown in Fig. 6. For near-optimal field orientations, increasing the SO coupling leads to a dramatic increase in the peak of the density of states at zero energy.

Figure 8

Density of states $D\left(\varepsilon \right)$ in the SF bilayer for energies normalized to the superconducting gap $\mathrm{\Delta }$ and SO coupling normalized to the inverse ferromagnet length $1/L_F$. The table shows the spectroscopic features of the SF bilayer with a rotated exchange field in the $xz\equiv yz$ plane. Note that when the field component along the junction is twice the component in the $y$ direction, here $\underline{h}=(0,3\mathrm{\Delta },6\mathrm{\Delta })$, the density of states is equivalent to the case $\underline{h}=(6\mathrm{\Delta },3\mathrm{\Delta },0)$ illustrated in Fig. 7, as predicted in the limit of weak proximity effect.

Figure 9

(a) Plot of the zero-energy density of states $D\left(0\right)$ in an S/F structure with pure Rashba spin-orbit coupling. We have set $h/\mathrm{\Delta }=4$ and $L/{\xi }_S=0.5$. (Inset) Stronger SO coupling $\alpha =1.5$, demonstrating that the angular variation of $D\left(0\right)$ remains, although the enhancement due to triplets is absent. (b) Plot of the magnitude of the LRT anomalous Green function $|d_{\perp }|$ at $\varepsilon =0$. As seen, its enhancement correlates with an accompanying increase in the density of states for the same angle $\phi$, and beyond an optimal SO coupling value there is anticorrelation between the density of states peak and $|d_{\perp }|$. The only angle of importance is the angle $\phi$ between the out-of-plane and in-plane components of the exchange field, shown in the inset.

Figure 10

The table shows the density of states $D\left(\varepsilon \right)$ in the SFS junction with increasing SO coupling and exchange field in a single direction, with $D\left(\varepsilon \right)$ normalised to the superconducting gap $\mathrm{\Delta }$ and SO coupling normalised to the inverse ferromagnet length $1/L_F$. With no SO coupling and very weak exchange field we see a phase-dictated gap-to-peak qualitative change in the density of states at zero energy. When the field is strong enough to destroy this gap, i.e., above the resonant condition, increasing the phase difference simply lowers the density of states towards that of the normal metal, which is achieved at a phase difference of $\varphi =\pi$. With the addition of SO coupling we see a clear difference in the density of states due to the long range triplet component, which is present when the field is oriented in $y$ but not in $z$. When LRTs are present with weak exchange fields, a phase-dictated gap-to-peak feature is retained and increased as the strength of SO coupling increases the gap, with the peak shown here at a phase difference of $0.75\pi$. For stronger exchange fields, increasing the SO coupling produces the minigap when there is no LRT component, whereas the existence of an LRT component again introduces an increasing peak at zero energy when no minigap is present.

Figure 11

Density of states $D\left(\varepsilon \right)$ in the SFS junction for energies normalized to the superconducting gap $\mathrm{\Delta }$ and SO coupling normalized to the inverse ferromagnet length $1/L_F$. The table shows the spectroscopic effects of increasing SO coupling in SFS with rotated exchange field. In the absence of SO coupling, the density of states is flat and featureless at low energies. Increasing the SO coupling again leads to a strong increase in the peak of the density of states at zero energy, while increasing the phase difference reduces the peak and shifts the density of states weight toward the gap edge for higher SO coupling strengths. With a component of the field in the junction direction a qualitative change in the density of states from strongly suppressed to enhanced at zero energy can be achieved by altering the phase difference between the superconductors. This change can occur in the presence of stronger exchange fields when SO coupling is included. Increasing the exchange field destroys the ability to maintain a gap in the density of states and the LRT component of the SO coupling increases the zero-energy peak as it did in the bilayer case.

Figure 12

Spatial distribution of the density of states $D\left(\varepsilon \right)$ throughout the ferromagnet of an SFS junction with phase difference $\varphi =0$, spin-orbit coupling $\alpha =\beta =1$, and magnetization $\underline{h}=(1.5\mathrm{\Delta },3.5\mathrm{\Delta },0)$.

Figure 13

Zero-energy density of states $D\left(0\right)$ as a function of the phase-difference $\varphi$ and magnetization angle $\theta$, both tunable parameters experimentally. The other parameters used are $L_F/{\xi }_S=0.5, h/{\mathrm{\Delta }}_0=3, a{\xi }_S=2$. In the top panel, we have dominant Dresselhaus coupling ($\chi =0.15\pi$) while in the bottom panel we have equal magnitude of the Rashba and Dresselhaus coefficients ($\chi =\pi /4$).

Figure 14

Plot of the critical temperature $T_c/T_{cs}$ as a function of the length $L_S/{\xi }_S$ of the superconductor for $a{\xi }_S=0$. Below a critical length $L_S$, superconductivity can no longer be sustained and $T_c$ becomes zero. For larger thicknesses of the superconducting layer, $T_c$ reverts back to its bulk value.

Figure 15

Plot of the critical temperature $T_c/T_{cs}$ as a function of the ferromagnet length $L_F/{\xi }_S$ for $a{\xi }_S=0$. Increasing the thickness of the ferromagnet gradually suppresses the $T_c$ of the superconductor, causing a stronger inverse proximity effect.

Figure 16

Plot of the critical temperature $T_c/T_{cs}$ as a function of the SO angle $\chi$, when $L_S/{\xi }_S=1.00, L_F/{\xi }_S=0.2$, and $\underline{h}\parallel \underline{\widehat{z}}$. Increasing the SO coupling causes $T_c$ to move closer to its bulk value, since the triplet proximity effect channel becomes suppressed.

Figure 17

Plot of the critical temperature $T_c/T_{cs}$ as a function of the SO angle $\chi$, when $L_S/{\xi }_S=0.55, L_F/{\xi }_S=0.2$, and $\underline{h}\parallel \underline{\widehat{z}}$.

Figure 18

Plot of the critical temperature $T_c/T_{cs}$ as a function of the SO angle $\chi$, when $L_S/{\xi }_S=1.00, L_F/{\xi }_S=0.2$, and $\underline{h}\parallel \underline{\widehat{x}}$. The critical temperature depends on the relative magnitudes of the Rashba and Dresselhaus coefficients.

Figure 19

Plot of the critical temperature $T_c/T_{cs}$ as a function of the SO angle $\chi$, when $L_S/{\xi }_S=0.55, L_F/{\xi }_S=0.2$, and $\underline{h}\parallel \underline{\widehat{x}}$.

Figure 20

Plot of critical temperature $T_c/T_{cs}$ as a function of the exchange-field angle $\theta$, when $L_S/{\xi }_S=1.00, L_F/{\xi }_S=0.2$, and $a{\xi }_S=2$. In contrast to ferromagnets without SO coupling, $T_c$ now depends strongly on the magnetization direction. This gives rise to a spin-valve like functionality with a single ferromagnet featuring SO coupling.

Figure 21

Plot of critical temperature $T_c/T_{cs}$ as a function of the exchange-field angle $\theta$, when $L_S/{\xi }_S=0.55, L_F/{\xi }_S=0.2$, and $a{\xi }_S=2$.