Giant superconducting proximity effect on spintronic anisotropy

We investigate theoretically the interplay of proximity effects due to the presence of a superconductor and normal ferromagnetic leads on the formation of the spintronic quadrupolar exchange field in a large-spin molecule. We show that the spintronic anisotropy can be enhanced by a few orders of magnitude and tuned by changing the strength of coupling to the superconductor. Especially large anisotropy is generated in the vicinity of the charge parity changing transition of the molecule. We also provide predictions of measurable spectral properties being the hallmarks of these phenomena.

Figure 1

(a) Example of an energy spectrum for a model system with the ground spin-multiplet $\mathcal{S}=3/2$ and uniaxial spintronic anisotropy $D<0$. (b) Schematic of a SC-proximized large-spin molecule [with a molecular level (ML) occupied by a single electron] connected to two metallic ferromagnetic leads. For details see the main text.

Figure 2

(a) Spin-quadrupole moment $\langle {\mathcal{Q}}_{zz}\rangle$ for a molecule of spin $\mathcal{S}=3/2$ plotted as a function of tunnel coupling $\mathrm{\Gamma }$ for ${\mathrm{\Gamma }}_{\text{S}}=0$ and several spin polarizations $p$ of the leads. (b) $\langle {\mathcal{Q}}_{zz}\rangle$ shown for different values of ${\mathrm{\Gamma }}_{\text{S}}$ indicated in (d) and $p=0.5$. Panels (c) and (d) present the linear-response conductance $G$ (normalized to $G_0\equiv 2e^2/h$) and the spin polarization of transported electrons $\gamma$, respectively, which correspond to curves shown in panel (b). Other parameters: $U/W=0.5$, $J/U={10}^{-3}$ and $T/U={10}^{-10}$.

Figure 3

Dependence of (a) normalized conductance $G/G_0$ and (b) spin-quadrupole moment $\langle {\mathcal{Q}}_{zz}\rangle$ on the coupling ${\mathrm{\Gamma }}_{\text{S}}$, representing the SC-proximity effect on the molecule, for selected values of the tunnel coupling $\mathrm{\Gamma }$. Other parameters as described in the caption of Fig. 2.

Figure 4

(a) Spectral function $\mathcal{A}\left(\omega \right)$ [scaled by ${\mathcal{A}}_0\equiv \mathcal{A}(\omega =0)$] shown as a function of energy $\omega$ for selected values of ${\mathrm{\Gamma }}_{\text{S}}$ [given in panel (b)] and $\mathrm{\Gamma }/U={10}^{-3}$. All spectral functions are symmetric with respect to $\omega =0$. (b) Evolution of the excitation gap $\mathrm{\Delta }\mathcal{E}$ with $\mathrm{\Gamma }$ (note a log-log scale). Points represent $\mathrm{\Delta }\mathcal{E}$ estimated from the inflection point of the step in $\mathcal{A}\left(\omega \right)$ [as, for example, indicated in (a)], whereas lines correspond to $\mathrm{\Delta }\mathcal{E}$ obtained using a complementary theoretical method based on the analysis of $\langle {\mathcal{Q}}_{zz}\rangle$; see Appendix pp3. The resolution limit for $\mathrm{\Delta }\mathcal{E}$ is set here by thermal energy $T$. (c) The height $\mathrm{\Delta }\mathcal{A}/{\mathcal{A}}_0$ of the step in $\mathcal{A}\left(\omega \right)/{\mathcal{A}}_0$ used for estimating $\mathrm{\Delta }\mathcal{E}$ in (b) plotted as a function of $\mathrm{\Gamma }$. All other parameters as described in the caption of Fig. 2.

Figure 5

Extension of Figs. 2–2 to different values of the magnetic-core spin $S_{\text{c}}$. (a) Spin-quadrupole moment $\langle {\mathcal{Q}}_{zz}\rangle$ plotted as a function of $\mathrm{\Gamma }$ for different values of ${\mathrm{\Gamma }}_{\text{S}}$ (different colors) and $S_{\text{c}}$ (different styles of dashing). Panels (b) and (c) present the linear-response conductance $G$ (as previously normalized to $G_0\equiv 2e^2/h$) and the spin polarization of transported electrons $\gamma$, respectively, which correspond to curves shown in (a). Other parameters as described in the caption of Fig. 2.

Figure 6

Extension of Fig. 4 to different values of the magnetic-core spin of the molecule, $S_{\text{c}}$. (a) Spectral function $\mathcal{A}\left(\omega \right)$ [scaled by ${\mathcal{A}}_0\equiv \mathcal{A}(\omega =0)]$ shown as a function of energy $\omega$ for selected values of ${\mathrm{\Gamma }}_{\text{S}}$ and $\mathrm{\Gamma }/U={10}^{-3}$. All spectral functions are symmetric with respect to $\omega =0$. (b) Evolution of the excitation gap $\mathrm{\Delta }\mathcal{E}$ with $\mathrm{\Gamma }$ (note a log-log scale). (c) The height $\mathrm{\Delta }\mathcal{A}/{\mathcal{A}}_0$ of the step in $\mathcal{A}\left(\omega \right)/{\mathcal{A}}_0$ used for estimating $\mathrm{\Delta }\mathcal{E}$ in (b) plotted as a function of $\mathrm{\Gamma }$. All other parameters as described in the caption of Fig. 4.

Figure 7

Generalization of Figs. 2 and 2 for ${\mathrm{\Gamma }}_{\text{S}}=0$ and $p=0.5$ to different values of intrinsic magnetic anisotropy of the molecule, $D_0$. (a) Spin-quadrupole moment $\langle {\mathcal{Q}}_{zz}\rangle$ plotted as a function of $\mathrm{\Gamma }$ for different values of $D_0$. Panel (b) presents the spin polarization of transported electrons $\gamma$, corresponding to curves shown in (a). Inset in panel (b) shows the linear-response conductance $G$ (normalized to $G_0\equiv 2e^2/h$) as a function of $\mathrm{\Gamma }$. Other parameters as described in the caption of Fig. 2.

Figure 8

Generalization of Fig. 4 for ${\mathrm{\Gamma }}_{\text{S}}=0$ to different values of intrinsic magnetic anisotropy of the molecule, $D_0$. All other parameters as described in the caption of Fig. 4.

Figure 9

Analogous to Fig. 4, but now plotted also for selected values of the intrinsic magnetic anisotropy constant $D_0$. Note that fine-dashed lines serve as reference lines, since they correspond to the results for $D_0=0$ shown in Fig. 4. All other parameters as described in the caption of Fig. 4.

Figure 10

The energies of the Andreev bound states $\mathrm{\Delta }{E}_{\text{a}\left(\text{b}\right)}^{\text{ABS}}$ [see Eq. (A11)] presented as a function of the coupling strength ${\mathrm{\Gamma }}_{\text{S}}$ of the molecule to the superconductor in the case of the ferromagnetic exchange coupling between spins of the ML and the magnetic core of the molecule. Parameters used: $S_{\text{c}}=1$, $\delta /U=0.1$, $J/U={10}^{-3}$ and $D_0=0$.

Figure 11

Normalized spectral function of the ML, $\mathcal{A}\left(\omega \right)$, corresponding to Fig. 4 but now plotted without scaling to ${\mathcal{A}}_0\equiv \mathcal{A}(\omega =0)$ and in the full range of energies $\omega$. Arrows indicate the expected positions of the step related to $\mathrm{\Delta }\mathcal{E}$ obtained from the compensation method explained in Appendix pp3. Note that, unlike in Fig. 4, here $\mathcal{A}\left(\omega \right)$ is shown on the logarithmic scale. The vertical thin dashed line indicates the maximal range of energies $\omega$ presented in Fig. 4.

Figure 12

Illustration of the compensation method for calculating the spintronic component of magnetic anisotropy for two different examples of the spin magnitude: $\mathcal{S}=1$ (dashed lines) and $\mathcal{S}=3/2$ (solid lines). Moreover, blue lines correspond to the case of nonmagnetic ($p=0$) leads, whereas the red ones represent the case of ferromagnetic leads with $p=0.5$. Parameters used in calculations: $U/W=0.5$, $T/U={10}^{-10}$, $J/U={10}^{-3}$, and $\mathrm{\Gamma }/U=0.01$. See Appendix pp3 for further explanation.

Figure 13

Comparison of the spectral functions of ML, $\mathcal{A}\left(\omega \right)$, for a finite spintronic component $D_{\text{spin}}$ to magnetic anisotropy (shown as solid lines with the same color code as in Fig. 11) with those for $p=0$ (and thus, for $D_{\text{spin}}=0$, but with $D_0=-\tilde{D}$; dashed lines). The arrows indicate $\omega =D_0$ for the corresponding curve. Dotted lines present the results for the spin-anisotropic case ($p=0$ and $D_0=0$). Other parameters as described in the caption of Fig. 11.