Tunneling conductance in superconductor/ferromagnet junctions: Self-consistent approach

We evaluate the tunneling conductance of clean ferromagnet/superconductor junctions via a fully self-consistent numerical solution of the microscopic Bogoliubov-de Gennes equations. We present results for a relevant range of values of the Fermi wave-vector mismatch (FWM), the spin polarization, and the interfacial scattering strength. For nonzero spin polarization, the conductance curves vary nonmonotonically with FWM. The FWM dependence of our results is stronger than that previously found in non-self-consistent calculations since, in the self-consistent case, the effective scattering potential near the interface depends on the FWM. The dependence on interfacial scattering is monotonic. These results confirm that it is impossible to characterize both the FWM and the interfacial scattering by a single effective parameter and that analysis of experimental data via the use of such one-parameter models is unreliable.

Figure 1

(Color online) Scattering at an SF interface. An electron is incident from the ferromagnet at an angle ${\theta }_{\sigma }$ with spin $\sigma$ and momentum ${\mathbf{k}}_{\sigma }$. Ordinary scattering leads to a partially reflected electron with the same spin and wave vector. For subgap scattering, excitations in the superconductor are Cooper pairs, consisting of an electronlike quasiparticle (ELQ) and a holelike quasiparticle (HLQ) carrying a net spin of zero and charge $2e$. To conserve charge, a hole is retroreflected (Andreev reflection) at an angle ${\theta }_{\overline{\sigma }}$ into the opposite spin band denoted by $\overline{\sigma }$.

Figure 2

(Color online) Example of the fitting procedure for the normalized current, as described in the text in connection with Eq. (2.21). The example shown here corresponds to the case $\Lambda =1$ and $I=0.866$, which will be discussed below (Figs. 4, 5). The crosses are the numerical results obtained as explained in the text and the curves are the fits. The units are explained in the text.

Figure 3

(Color online) Results for the dimensionless conductance $G\left(V\right)$ (see text) as a function of bias voltage $V$ given in units of ${\Delta }_0/e$. The top panel shows the forward-scattering case for $I=0.2$ and various $\Lambda \le 1$. The forward conductance is calculated by averaging over an incident-particle flux with angles smaller than $\pi /30$. The bottom panel shows the angularly averaged case at the same values of $I$ and $\Lambda$. The conductances are averaged over angles up to the critical angle ${\Omega }_{\sigma }$ for spin-up and spin-down particles.

Figure 4

(Color online) Results for $G\left(V\right)$ at a larger value of $I$ than in Fig. 3, $I=\sqrt{3}/2\approx 0.866$, and the same range of $\Lambda$. The top panel shows the forward-scattering results and the bottom panel the angularly averaged values. The effects of a strong exchange field are readily apparent when this figure is compared with Fig. 3 (see text).

Figure 5

(Color online) Results for the same case presented in Fig. 4 and with the same panel arrangements but for values of $\Lambda$ in the range $\Lambda \ge 1$. See text for discussion.

Figure 6

Angularly averaged values of $G\left(V\right)$ for various $I$ at the same $\Lambda =1.0$. This figure shows (see text) that stronger $I$ suppresses both ordinary scattering and Andreev reflection. Note that $G\left(0\right)/G\left(3\right)\approx 2$ for all values of $I$ shown here and that $G\left(0\right)/G\left(1\right)$ increases with increasing $I$.

Figure 7

(Color online) Plots of $G\left(I;V=1\right)$, the value of the dimensionless conductance at $V=1$, in the forward-scattering case for $I=0.2$ (top panel) and $I=0.866$ (bottom panel). The horizontal lines labeled by $G\left(V,I\right)$ show the result from the non-self-consistent formula [Eq. (3.1)]. The points represent the numerical results from the present self-consistent study at the values of $I$ indicated in the legend. The error bars represent the uncertainty in the numerically obtained values (see text). Note the different horizontal range included in the upper and lower panels. This figure shows that there is an increase in $G\left(V=1\right)$ with increasing $\Lambda$, a trend which is not obtained without a fully self-consistent pair amplitude.

Figure 8

(Color online) Illustration of the large difference between the value of the $I$ parameter that would be obtained from analyzing experimental data using the non-self-consistent results (Ref. 9) and those obtained from the correct self-consistent analysis. The inferred value of $I$ is plotted vs the measured value of $G\left(1\right)$. The non-self-consistent (analytical) results are the solid curve and the self-consistent ones the data points. See text for details.

Figure 9

(Color online) The conductance at interface scattering value $H_B=1$ near the tunneling regime. All results correspond to the forward-scattering case. The top panel shows $G\left(V\right)$ for three values of $I$ at $\Lambda =1$, while the bottom panel shows $G\left(V\right)$ at $I=0.866$ for three values of $\Lambda$.