Scattering problem in nonequilibrium quasiclassical theory of metals and superconductors: General boundary conditions and applications

I derive a general set of boundary conditions for quasiclassical transport theory of metals and superconductors that is valid for equilibrium and nonequilibrium situations and includes multiband systems, weakly and strongly spin-polarized systems, and disordered systems. The formulation is in terms of the normal state scattering matrix. Various special cases for boundary conditions are known in the literature, which are, however, limited to either equilibrium situations or single band systems. The present formulation unifies and extends all these results. In this paper I will present the general theory in terms of coherence functions and distribution functions and demonstrate its use by applying it to the problem of spin-active interfaces in superconducting devices and the case of superconductor/half-metal interface scattering.

Figure 1

(Color online) (a) The coherence function $\gamma \left(t,t^{\prime }\right)$ describes the local probability amplitude for conversion of a hole (dotted line) at time $t^{\prime }$ to a particle (full line) at time $t$. For retarded functions $t>t^{\prime }$ and for advanced functions $t<t^{\prime }$. (b) The corresponding local amplitude for conversion of a particle at time $t^{\prime }$ into a hole at time $t$ is described by the coherence function $\tilde{\gamma }\left(t,t^{\prime }\right)$ (Ref. 54). (c) Diagrammatic representation of Eqs. (7, 11).

Figure 2

(Color online) Identities that hold between the six quantities $\mathcal{F}$, $\tilde{\mathcal{F}}$, $\mathcal{G}$, $\tilde{\mathcal{G}}$, $\gamma$, and $\tilde{\gamma }$ as defined in Eqs. (7, 8).

Figure 3

(Color online) The distribution functions $x$ and $\tilde{x}$ (left) connect incoming advanced and outgoing retarded propagators. The Keldysh components ${\mathcal{X}}^{\text{K}}$, ${\tilde{\mathcal{X}}}^{\text{K}}$, ${\mathcal{Y}}^{\text{K}}$, and ${\tilde{\mathcal{Y}}}^{\text{K}}$ are shown in a diagrammatic representation of Eqs. (14, 15, 16, 17). “R” and “A” refer to “retarded” and “advanced.” The particle distribution function $x$ and hole distribution function $\tilde{x}$ are coherently mixed due to multiple coherent Andreev scattering events with amplitudes given by the renormalized quantities $\mathcal{G}$, $\tilde{\mathcal{G}}$, $\mathcal{F}$, and $\tilde{\mathcal{F}}$ that are sums of terms shown in Fig. 1.

Figure 4

(Color online) Diagrammatic symbols for the elementary scattering events described by Eqs. (47, 48). “p” and “h” refer to “particle” and “hole,” and “R” and “A” to “retarded” and “advanced.” A sum over internal variables according to Eqs. (39, 40, 41, 42, 43, 44) is implied.

Figure 5

(Color online) Diagrammatic representation of Eq. (53) for the retarded functions. In the last line, identity (56) is shown diagrammatically, which defines the diagrammatic expansion for ${\mathcal{G}}^{\text{R}}$. Summation over internal variables is implied.

Figure 6

(Color online) Diagrammatic representation of Eqs. (72, 75) using the identities in Eqs. (56, 60).

Figure 7

(Color online) Critical Josephson current $I_{\text{c}}$ multiplied with the normal state resistance $R_{\text{N}}$ for an S/HM/S Josephson junction with magnetic interfaces (thick lines) and for an S/N/S junction with nonmagnetic interfaces (thin lines). In (a) for both cases, the transmission amplitudes $t_0$ are varied from 0.2 to 0.5. The spin-mixing angle for normal impact is ${\vartheta }_0=0.75\pi$ and the junction length $L$ is equal to the coherence length in the half metal, ${\xi }_{\text{F}}=v_{\text{F}}/2\pi T_c$. In (b) we fix $L=3{\xi }_{\text{F}}$, $t_0=0.5$, and vary for the S/HM/S junction the spin-mixing angle. For (a) and (b) the interface spin misalignment angle is $\alpha =\pi /2$.

Figure 8

(Color online) Local density of states in the center of a current biased high-transmissive symmetric Josephson junction for (a) an S/HM/S junction and (b) an S/N/S junction. In both cases the phase difference over the junction is varied from 0 to $\pi$. The remaining parameters are indicated.

Figure 9

(Color online) Local density of states in the center of a high-transmissive symmetric S/HM/S Josephson junction for (a) a phase difference over the junction of $\pi$ and (b) of 0. In both cases the junction length is varied from the short junction limit to $L=50{\xi }_{\text{F}}$. The remaining parameters are indicated.

Figure 10

(Color online) Local density of states in the center of a high-transmissive symmetric S/HM/S Josephson junction, for (a) a phase difference over the junction of $\pi$ and (b) of 0. In both cases the spin-mixing angle ${\vartheta }_0$ is varied from 0 to $\pi$. The remaining parameters are indicated.

Figure 11

(Color online) Point-contact spectra for an S/HM contact. In (a) the transmission $t_0$ varied from 0 to 1, and in (b) the spin-mixing angle ${\vartheta }_0$ is varied from 0 to $\pi$. Both quantities depend on the quasiparticle impact angle as discussed in the text. The remaining parameters are indicated.