Spin-dependent boundary conditions for isotropic superconducting Green’s functions

The quasiclassical theory of superconductivity provides the most successful description of diffusive heterostructures comprising superconducting elements, namely, the Usadel equations for isotropic Green’s functions. Since the quasiclassical and isotropic approximations break down close to interfaces, the Usadel equations have to be supplemented with boundary conditions for isotropic Green’s functions (BCIGF), which are not derivable within the quasiclassical description. For a long time, the BCIGF were available only for spin-degenerate tunnel contacts, which posed a serious limitation on the applicability of the Usadel description to modern structures containing ferromagnetic elements. In this paper, we close this gap and derive spin-dependent BCIGF for a contact encompassing superconducting and ferromagnetic correlations. This finally justifies several simplified versions of the spin-dependent BCIGF, which have been used in the literature so far. In the general case, our BCIGF are valid as soon as the quasiclassical isotropic approximation can be performed. However, their use requires the knowledge of the full scattering matrix of the contact, an information usually not available for realistic interfaces. In the case of a weakly polarized tunnel interface, the BCIGF can be expressed in terms of a few parameters, i.e., the tunnel conductance of the interface and five conductancelike parameters accounting for the spin dependence of the interface scattering amplitudes. In the case of a contact with a ferromagnetic insulator, it is possible to find explicit BCIGF also for stronger polarizations. The BCIGF derived in this paper are sufficiently general to describe a variety of physical situations and may serve as a basis for modeling realistic nanostructures.

Figure 1

(Color online) Model of a planar contact between two diffusive conductors $L$ and $R$. The inner part of the model consists of two ballistic zones (light gray areas), connected together by a scattering barrier $V_b\left(z\right)$ located at $z∊\left[-b_L,b_R\right]$ (dark gray area, green online). We set the end of the left/right ballistic zone at $z=\mp c_{L\left(R\right)}$ with $c_{L\left(R\right)}-b_{L\left(R\right)}$ a distance smaller or equal to the elastic mean-free path ${\ell }_{\text{e}}^{L\left(R\right)}$ of $L\left(R\right)$. At this place, the ballistic Green’s function $\tilde{g}\left(z,z,\epsilon \right)$ equals ${\tilde{g}}_{L\left(R\right)}$. The effect of $V_b\left(z\right)$ can be described with electronic-transmission amplitudes $t_{L\left(R\right)}$ and reflection amplitudes $r_{L\left(R\right)}$. There exists a simple relation between ${\tilde{g}}_R$ and ${\tilde{g}}_L$, based on the scattering parameters $t_{L\left(R\right)}$ and $r_{L\left(R\right)}$ [see Eq. (22)]. The two ballistic zones are connected to two isotropization zones which have a size $d_{L\left(R\right)}$ of the order of a few ${\ell }_{\text{e}}^{L\left(R\right)}$ (dotted areas). In the isotropization zones, the electron dynamics is dominated by elastic-impurity scattering. Hence, $\tilde{g}\left(z,z,\epsilon \right)$ becomes isotropic away from the contact region, and one can define, beyond the isotropization zones, two diffusive zones (gray areas, purple online) described by the isotropic Green’s function $\check{G}\left(z,\epsilon \right)$. For $z\to \mp d_{L\left(R\right)}$, $\tilde{g}\left(z,z,\epsilon \right)$ tends to $\check{G}\left(z,\epsilon \right)={\check{G}}_{L\left(R\right)}$. Note that since the transition between the ballistic, isotropization, and diffusive zones is smooth, the choice of the coordinates $c_{L\left(R\right)}$ and $d_{L\left(R\right)}$ in Fig. 1 is somewhat arbitrary, i.e., defined only up to an uncertainty of the order of a fraction of ${\ell }_{\text{e}}^{L\left(R\right)}$ or ${\ell }_{\text{e}}^{L\left(R\right)}$, respectively. However, this choice does not affect the BCIGF as we show in the text.

Figure 2

(Color online) Density of states $N\left(\epsilon ,x=d_S\right)$ in a $S$ layer contacted to a $FI$. The black dashed lines correspond to $\Gamma =0.1$ and ${\gamma }_{\varphi ,2}=0$, and the blue full lines correspond to $\Gamma =0.01$ and ${\gamma }_{\varphi ,2}=0.03$. The left panel corresponds to $d_S/{\xi }_S=0.5$ and the right panel corresponds to $d_S/{\xi }_S=3$. In all cases, we have used ${\gamma }_{\varphi ,1}=0.15$, ${\gamma }_{\varphi ,3}={\gamma }_{\varphi ,4}=0$, and $k_{B}T=0.1{\Delta }_{\text{BCS}}$.

Figure 3

(Color online) Top panel: density of states $N\left(\epsilon ,x=d_S\right)$ in a $S$ layer contacted to a $FI$. The red dashed line corresponds to $\Gamma =0$ and ${\gamma }_{\varphi ,2}=0$ in both panels. The black full lines in the top panel correspond to $\Gamma \ne 0$ and ${\gamma }_{\varphi ,2}=0$, and the blue full lines in the bottom panel correspond to $\Gamma =0$ and ${\gamma }_{\varphi ,2}\ne 0$. In all cases, we have used $d_S/{\xi }_S=0.5$, ${\gamma }_{\varphi ,1}={\gamma }_{\varphi ,3}={\gamma }_{\varphi ,4}=0$, and $k_{B}T=0.1{\Delta }_{\text{BCS}}$.

Figure 4

(Color online) Density of states in the $S$ layer contacted to a $FI$, for $x=0$ (left panel) and $x=d_S$ (right panel), and different values of $d_S/{\xi }_S$. In all cases, we have used ${\gamma }_{\varphi ,2}=0.1$, ${\gamma }_{\varphi ,1}={\gamma }_{\varphi ,3}={\gamma }_{\varphi ,4}=0$, $\Gamma =0.025$, and $k_{B}T=0.1{\Delta }_{\text{BCS}}$.

Figure 5

(Color online) Scheme representing two particular types of Andreev reflection processes which can occur on a $S/F$ interface modeled like in Fig. 1. The ballistic, isotropization, and diffusive zones of $S$ and $F$ are represented by gray, dotted, and purple areas, respectively. Full (dashed) lines represent trajectories of electrons (holes) from the $\uparrow$ $(\downarrow )$ spin band. The superconducting gap $\Delta$ is taken into account in the diffusive part of $S$ only so that one can consider that Andreev reflections occur at the interface between the diffusive and isotropization zones of $S$. The upper part of the scheme represents and electron incident from the $F$ side, which is transmitted by the barrier $V_b$ as an electron, Andreev reflected on the diffusive part of $S$ as a hole, and transmitted again by $V_b$ as a hole, before joining the diffusive part of $F$ again. The probability associated to this process is proportional to $T_n^2\left(1-P_n^2\right)$. The lower part of the scheme represents a more complicated trajectory which also involves two reflections on $V_b$. The joint probability of these reflections is ${\left(1-T_n\right)}^2-T_n^2{P}_n^2$.