

Spin Polarization Measurements


Andreev reflection was introduced in 1964 by Alexander Andreev to explain the effects of heat transfer in the intermediate state of typeI superconductors. He argued that the origin of the enhanced thermal resistance in the intermediate state compared to that in the Meissner state is related to a completely unexpected type of transport between normal metal (N) and superconducting (S) parts of the intermediate state. Andreev considered quasiparticle transfer across a clean (nobarrier) interface between a normal metal (N) and a superconductor (S). Obviously, one of the two things must happen to an electron when it approaches the NS interface: either it goes through the interface into the superconductor or it gets reflected. While a single electron with the energy below a superconducting gap Δ can not propagate from a normal metal into the superconductor, it also can not get reflected by the usual process, in which one component of its momentum (in the direction perpendicular to the interface) changes sign (see Fig.1a), as this requires a sizeable barrier to reverse the (large) Fermi momentum of the electron. It is possible, however, for the electron to get reflected via the Andreev reflection process, often called retroreflection (see Fig. 1b). In such a process the reflected particle (the hole) has the opposite charge and spin as the incident particle (the electron) and is reflected along the same trajectory as the incident electron (hence retroreflection). This is equivalent to two single electrons with opposite momentum and spin pairing up to form a Cooper pair which will travel inside the superconductor. This process is elastic as the quasiparticle (excitation) energy is the same for the incident electron and the reflected hole. Momentum is also conserved to the first order of Δ/E_{F}, which holds in most cases (for conventional superconductors!) as the superconducting gap is usually much smaller than the Fermi energy.




Fig. 1a  Normal (elastic) reflection is not feasible in the absence of a barrier. 





Fig. 1b  Andreev (retroreflection): a hole is retracing electron's trajectory (the process has timereversal symmetry). 

One can also describe the Andreev process as a continuous charge conversion near the interface of a single excitation (electron) in the normal phase into a supercurrent (carried by Cooper pairs) in the superconducting phase. As an electronlike quasiparticle is moving farther into the S region it becomes more and more holelike. The charge conservation requires that the charge difference should then be transferred to the condensate. Due to this charge conversion Andreev reflection doubles the charge carried across the interface which manifests itself in a reduction of the resistance of an NS junction below the gap by a factor of two. An electron coming from the N side can not propagate inside S and is reflected as a hole (at energies smaller than Δ). A charge of 2e then flows across the interface, which corresponds to an increase in the conductance of the contact by a factor of two compared to that of the normal state (i.e. at biases much larger than the gap). The probability of the Andreev process is energy dependent  decreasing fairly quickly for higher energy electrons above the gap. In SNS structures multiple Andreev reflections may take place, leading to the socalled Andreev bound states.




The quasiparticle transmission probability was calculated by Griffin and Demers and later by Blonder, Tinkham, and Klapwijk (BTK), who generalized the problem to describe a junction of arbitrary transparency with nonzero normal reflection at the interface, which occurs in the presence of an oxide layer or any other nonuniformity. BTK considered scattering of a onedimensional plane wave with the interface described by an idealized deltafunction barrier U=Z·δ(x). The strength of the barrier is determined by a dimensionless parameter Z, with Z=0 corresponding to a clean, nobarrier interface, for which only Andreev reflection is possible below the gap, and Z→∞ corresponding to a tunneling case for which Andreev reflection is negligible. BTK solved a one dimensional Bogoliubov  de Gennes equation with fixed boundary conditions at the interface (a step function for order parameter) for a ballistic (Sharvin) point contact (in which electron mean free path is larger than the size of the contact).




In a nonmagnetic metal the Andreev process is always allowed, because every energy state in a normal metal has both spinup and spindown electrons. However, in a magnetic metal this is no longer true and Andreev reflection is limited by a minority spin population. The uniqueness of this situation was first emphasized by de Jong and Beenakker in 1995, when they discussed Andreev reflection at the interface between a halfmetal (which is 100% spin polarized) and a conventional superconductor. Andreev reflection is then forbidden, as there are no states for a hole to get reflected to, resulting in zero conductance across the interface below the gap.




The experiments using point contacts as well as lithographically defined samples, confirmed this prediction and have made it possible to introduce a new quantitative technique allowing one to measure spin polarization of metals. The technique which is now often called PCAR (Point Contact Andreev Reflection) Spectroscopy can be applied to both thin films and bulk crystals in a wide variety of ferromagnets. It makes use of a correlation between the degree of suppression of Andreev reflection at a ferromagnetsuperconductor interface and the spin polarization of the material. If a magnetic metal which is not 100% spin polarized, the total spin current I through the interface can be decomposed as the sum of the spin polarized current λI and unpolarized current (1λ)I, giving rise to a family of curves shown in Fig.2. The spinpolarized fraction λ of the current can not Andreevreflect and thus does not contribute to the overall conductance below the gap, whereas the unpolarized fraction of the current contributes 2(1λ)I to the overall conductance, as in the case of a normal metal (see Fig.2). Conversely, if one is able to measure a conductance of an Andreev contact of an unknown material, one can extract the spin polarization of this material from the overall shape of the conductance curve, by fitting an experimental data with an appropriately modified weak coupling (BTK) theory, which takes into account both nonmagnetic and halfmetallic channels.






Fig. 2  Normalized conductance, G for different values of the spin polarization. Nonmagnetic metals have G = 2 at zero bias, while halfmetals have G = 0 (calculations are done for a clean interface, Z = 0 and finite temperatures). 
