Theory of superconducting and magnetic proximity effect in S/F structures with inhomogeneous magnetization textures and spin-active interfaces

We present a study of the proximity effect and the inverse proximity effect in a $\text{superconductor}\mid \text{ferromagnet}$ bilayer, taking into account several important factors which mostly have been ignored in the literature so far. These include spin-dependent interfacial phase shifts (spin-DIPS) and inhomogeneous textures of the magnetization in the ferromagnetic layer, both of which are expected to be present in real experimental samples. Our approach is numerical, allowing us to access the full proximity effect regime. In Sec. II of this work, we study the superconducting proximity effect and the resulting local density of states in an inhomogeneous ferromagnet with a nontrivial magnetic texture. Our two main results in Sec. II are a study of how Bloch and Néel domain walls affect the proximity-induced superconducting correlations and a study of the superconducting proximity effect in a conical ferromagnet. The latter topic should be relevant for the ferromagnet Ho, which was recently used in an experiment to demonstrate the possibility to generate and sustain long-range triplet superconducting correlations. In Sec. III of this work, we investigate the inverse proximity effect with emphasis on the induced magnetization in the superconducting region as a result of the “leakage” from the ferromagnetic region. It is shown that the presence of spin-DIPS modifies conclusions obtained previously in the literature with regard to the induced magnetization in the superconducting region. In particular, we find that the spin-DIPS can trigger an antiscreening effect of the magnetization, leading to an induced magnetization in the superconducting region with the same sign as in the proximity ferromagnet.

Figure 1

(Color online) The three types of inhomogeneous ferromagnets we will consider in this work: Bloch walls, Néel walls, and conical ferromagnets (such as Ho).

Figure 2

(Color online) Plot of the energy-resolved LDOS evaluated at $x/d_F=0.5$ in the case of a thin domain wall $d_W/d_F=0.2$. We consider three values of the exchange field $h$ and also investigate how the LDOS changes with the phase shift $G_{\varphi }$ at the interface.

Figure 3

(Color online) Plot of the energy-resolved LDOS evaluated at $x/d_F=0.5$ in the case of a thick domain wall, $d_W/d_F=0.8$. We consider three values of the exchange field $h$ and also investigate how the LDOS changes with the phase shift $G_{\varphi }$ at the interface.

Figure 4

(Color online) Plot of the zero-energy LDOS induced in the ferromagnet for $h/{\Delta }_0=5$ and $d_F/{\xi }_S=0.5$. The lines correspond to $d_W/d_F$ in the range [0.1,0.9] in steps of 0.1 along the arrow. Here, $G_{\varphi }$ is set to zero.

Figure 5

(Color online) Plot of the LDOS at $x=d_F$ for a conical ferromagnet with $h/{\Delta }_0=5$ for several values of the ferromagnetic layer thickness $d_F$. In each case, we investigate the role of the spin-DIPS $\left(G_{\varphi }\right)$ at the $\text{S}\mid \text{F}$ interface.

Figure 6

(Color online) Plot of the zero-energy LDOS at $x=d_F$ for a conical ferromagnet with $h/\Delta =5$ as a function of the normalized spin-DIPS parameter $G_{\varphi }/G_T$. Below a critical value for $d_F$, a qualitatively different behavior is observed for the LDOS.

Figure 7

(Color online) Plot of the zero-energy LDOS at $x=d_F$ as a function of $d_F/{\xi }_S$ for several values of $G_{\varphi }$. As seen, there is a critical thickness at which the zero-energy LDOS becomes nonzero. We have used $h/{\Delta }_0=5$.

Figure 8

(Color online) Plot of the LDOS at $x=d_F$ for a conical ferromagnet with $h/{\Delta }_0=5$ and $d_F/{\xi }_S=0.1$ for several values of the spin-DIPS $\left(G_{\varphi }\right)$ at the $\text{S}\mid \text{F}$ interface. In (a), $G_{\varphi }/G_T$ is below the critical value, while in (b) $G_{\varphi }/G_T$ is larger than the critical value.

Figure 9

(Color online) Plot of the zero-energy LDOS at $x=d_F$ as a function of $\theta$ for several values of $\alpha$. We have fixed $h/{\Delta }_0=5$, $d_F/{\xi }_S=0.5$, and $G_{\varphi }=0$ to focus on the effect of the magnetic structure. The dotted lines give the result for $h_y=h_z=0$, which corresponds to the saturating behavior when $\theta$ increases since the average value of $h_y$ and $h_z$ vanishes in this limit. The inset shows the case $\alpha =\pi /2$ corresponding to $h_x=0$. For increasing $\theta$, the ferromagnetic layer effectively acts as a normal metal, thus causing a complete suppression of the zero-energy LDOS.

Figure 10

(Color online) Self-consistent solution for the spatial profile of the superconducting order parameter using the approach described in Sec. 3 of this paper. The choice for parameter values are specified in the main text. To add stability to the numerical calculations, we added a small imaginary number $i\eta$ to the quasiparticle energies $\epsilon$, with $\eta =0.05{\Delta }_0$, effectively modeling inelastic scattering.

Figure 11

(Color online) Plot of the zero-energy LDOS at $x=d_F$ in the case of a Bloch domain wall with $h/{\Delta }_0=5$. In (a) and (c), we plot the LDOS as a function of the spin-DIPS $G_{\varphi }$, while in (b) and (d) we plot it as a function of the ferromagnetic layer thickness $d_F$. For thin layers $d_F/{\xi }_S⪡1$, one observes an abrupt transition from a fully suppressed DOS to a nonzero DOS at a critical value for either $G_{\varphi }$ or $d_F$.

Figure 12

(Color online) The experimental setup proposed in this paper: a $\text{superconductor}\mid \text{ferromagnet}$ bilayer.

Figure 13

(Color online) (a) Plot of the proximity-induced magnetization for ${\gamma }_{\varphi ,F}=0$ upon varying the spin-DIPS ${\gamma }_{\varphi ,S}$ on the superconducting side. (b) Plot of the proximity-induced magnetization for ${\gamma }_{\varphi ,S}=0$ upon varying the spin-DIPS ${\gamma }_{\varphi ,F}$ on the ferromagnetic side. We have used $d_S/{\xi }_S=0.2$, and the other parameter values are discussed and provided in the main text. Note that the lines with a (blue) circle marker are equal in (a) and (b), corresponding to ${\gamma }_{\varphi ,S}={\gamma }_{\varphi ,F}=0$. As seen, ${\gamma }_{\varphi ,S}$ affects $\delta M_S/M_0$ much more strongly than ${\gamma }_{\varphi ,F}$. In (b), the magnetization switches sign upon increasing ${\gamma }_{\varphi ,F}$ further (for the present parameters, the sign switch occurs around ${\gamma }_{\varphi ,F}\simeq 1.3$).

Figure 14

(Color online) Plot of the proximity-induced magnetization for ${\gamma }_{\varphi ,F}=0$ upon varying the spin-DIPS ${\gamma }_{\varphi ,S}$ on the superconducting side using $d_S/{\xi }_S=1.0$.

Figure 15

(Color online) Plot of the spatial dependence of the total magnetization at zero temperature for $d_S/{\xi }_S=0.2$ and ${\gamma }_{\varphi ,F}={\gamma }_{\varphi ,S}=0.0$.

Figure 16

(Color online) Plot of the spatial dependence of the total magnetization at zero temperature for $d_S/{\xi }_S=0.2$, ${\gamma }_{\varphi ,F}=0.0$, and ${\gamma }_{\varphi ,S}=1.0$.

Figure 17

(Color online) Plot of the proximity-induced magnetization at $T=0$ as a function of ${\gamma }_{\varphi ,S}$ for two different values of $d_S/{\xi }_S$.

Figure 18

(Color online) Junction consisting of two regions 1 and 2 with an interface perpendicular to the $\mathit{x}$ axis.