Crossed Andreev reflection at spin-active interfaces
With the aid of the quasiclassical Eilenberger formalism we develop a theory of non-local electron transport across three-terminal ballistic normal-superconducting-normal (NSN) devices with spin-active NS interfaces. The phenomenon of crossed Andreev reflection (CAR) is known to play the key role in such transport. We demonstrate that CAR is highly sensitive to electron spins and yields a rich variety of properties of non-local conductance which we describe non-perturbatively at arbitrary voltages, temperature, spin-dependent interface transmissions and their polarizations. Our results can be applied to multi-terminal hybrid structures with normal, ferromagnetic and half-metallic electrodes and can be directly tested in future experiments.
pacs:74.45.+c, 73.23.-b, 74.78.Na
Low energy electron transport in hybrid structures composed of a normal metal (N) and a superconductor (S) is governed by Andreev reflection And () (AR) which causes non-zero subgap conductance BTK () of such structures. AR remains essentially a local effect provided there exists only one NS interface in the system or, else, if the distance between different NS interfaces greatly exceeds the superconducting coherence length . If, however, the distance between two adjacent NS interfaces (i.e. the superconductor size) is smaller than (or comparable with) , two additional non-local processes come into play (see Fig. 1). One such process corresponds to direct electron transfer between two N-metals through a superconductor. Another process is the so-called crossed Andreev reflection BF (); GF () (CAR): An electron penetrating into the superconductor from the first N-terminal may form a Cooper pair together with another electron from the second N-terminal. In this case a hole will go into the second N-metal and AR becomes a non-local effect. Both these processes contribute to the non-local conductance of hybrid multi-terminal structures which has been directly measured in several recent experiments Beckmann (); Teun (); Venkat ().
Theoretically the non-local conductance of NSN hybrids was analyzed within the perturbation theory in the transmission of NS interfaces in Refs. FFH, ; Fabio, where it was demonstrated that in the lowest order in the interface transmission and at CAR contribution to cross-terminal conductance is exactly canceled by that from elastic electron cotunneling (EC), i.e. the non-local conductance vanishes in this limit. Thus, in order to determine the scale of the effect it is necessary to include higher order (in the barrier transmission) terms into consideration. The corresponding analysis was employed in Refs. MF, ; Melin, by means of effective “dressing” of both EC and CAR contributions by higher order processes and more recently by the present authors KZ06 () with the framework of quasiclassical formalism of Eilenberger equations. We note that the results MF () and KZ06 () disagree beyond perturbation theory since “dressing” procedure MF () does not account for all higher order processes. Hence, the approach MF () is in general insufficient to correctly describe non-trivial interplay between normal reflection, tunneling, local AR and CAR to all orders in the interface transmissions. We will return to this issue further below.
Another interesting issue is the effect of disorder. It is well known that disorder enhances interference effects and, hence, can strongly modify local subgap conductance of NS interfaces in the low energy limit VZK (); HN (); Z (). Non-local conductance of multi-terminal hybrid NSN structures in the presence of disorder was recently studied in Refs. BG, ; Belzig, ; Duhot, ; GZ07, . Brinkman and Golubov made use of the quasiclassical formalism of Usadel equations and proceeded perturbatively in the interface transmissions. Duhot and Melin Duhot () discussed the impact of weak-localization-type of effects inside the superconductor on non-local electron transport in NSN structures. Morten et al. Belzig () employed the circuit theory (thereby going beyond perturbation theory in tunneling) and considered a device with normal terminals attached to a superconductor via an additional normal island (dot) FN (). Very recently a similar structure with a superconducting dot was analyzed GZ07 () providing a rather general theoretical framework to study non-local electron transport in multi-terminal NSN structures in the presence of disorder and non-equilibrium effects.
Yet another interesting subject is an interplay between CAR and Coulomb interaction. The effect of electron-electron interactions on AR was investigated in a number of papers Z (); HHK (); GZ06 (). Interactions should also affect CAR, e.g., by lifting the exact cancellation of EC and CAR contributions LY () already in the lowest order in tunneling. A complete theory of non-local transport in realistic NSN systems should include both disorder and interactions which remains an important task for future investigations.
An important property of both AR and CAR is that these processes should be sensitive to magnetic properties of normal electrodes because these processes essentially depend on spins of scattered electrons. One possible way to demonstrate spin-resolved CAR is to use ferromagnets (F) instead of normal electrodes ferromagnet-superconductor-ferromagnet (FSF) structures Beckmann (); MF (); Yam (); Fazio (). First experiments on such FSF structures Beckmann () illustrated this point by demonstrating the dependence of non-local conductance on the polarization of ferromagnetic terminals. Hence, for better understanding of non-local effects in multi-terminal hybrid proximity structures it is desirable to construct a theory of spin-resolved non-local transport. In the lowest order in tunneling this task was accomplished in Ref. FFH, . For FSF structures higher orders in the interface transmissions were considered in Refs. MF (); Melin ().
In this paper we are going to generalize our quasiclassical approach KZ06 () and construct a theory of non-local electron transport in ballistic NSN structures with spin-active interfaces to all orders in their transmissions. Our model allows to distinguish spin-dependent contributions to the non-local conductance and to effectively mimic the situation of ferromagnetic and/or half-metallic electrodes.
The structure of the paper is as follows. In Sec. 2 we introduce our model and discuss the quasiclassical formalism supplemented by the boundary conditions for Green-Keldysh functions which account for electron scattering at spin-active interfaces. In Sec. 3 we employ this formalism and develop a theory of non-local spin-resolved electron transport in NSN structures with spin-active interfaces. Our main conclusions are briefly summarized in Sec. 4.
Ii The model and formalism
Let us consider three-terminal NSN structure depicted in Fig. 2. We will assume that all three metallic electrodes are non-magnetic and ballistic, i.e. the electron elastic mean free path in each metal is larger than any other relevant size scale. In order to resolve spin-dependent effects we will assume that both NS interfaces are spin-active, i.e. we will distinguish “spin-up” and “spin-down” transmissions of the first ( and ) and the second ( and ) SN interface. All these four transmissions may take any value from zero to one. We also introduce the angle between polarizations of two interfaces which can take any value between 0 and .
In what follows effective cross-sections of the two interfaces will be denoted respectively as and . The distance between these interfaces as well as other geometric parameters are assumed to be much larger than , i.e. effectively both contacts are metallic constrictions. In this case the voltage drops only across SN interfaces and not inside large metallic electrodes. Hence, nonequilibrium (e.g. charge imbalance) effects related to the electric field penetration into the S-electrode can be neglected. In our analysis we will also disregard Coulomb effects Z (); HHK (); GZ06 ().
For convenience, we will set the electric potential of the S-electrode equal to zero, . In the presence of bias voltages and applied to two normal electrodes (see Fig. 2) the currents and will flow through SN and SN interfaces. These currents can be evaluated with the aid of the quasiclassical formalism of nonequilibrium Green-Eilenberger-Keldysh functions BWBSZ () which we briefly specify below.
ii.1 Quasiclassical equations
In the ballistic limit the corresponding Eilenberger equations take the form
where , is the quasiparticle energy, is the electron Fermi momentum vector and is the Pauli matrix in Nambu space. The functions also obey the normalization conditions and . Here and below the product of matrices is defined as time convolution.
Green functions and are matrices in Nambu and spin spaces. In Nambu space they can be parameterized as
where , , , are matrices in the spin space, is the BCS order parameter and are Pauli matrices. For simplicity we will only consider the case of spin-singlet isotropic pairing in the superconducting electrode. The current density is related to the Keldysh function according to the standard relation
where is the density of state at the Fermi level and angular brackets denote averaging over the Fermi momentum.
ii.2 Riccati parameterization
The above matrix Green-Keldysh functions can be conveniently parameterized by four Riccati amplitudes , and two “distribution functions” , (here and below we chose to follow the notations Eschrig00 ()):
where functions and are Riccati amplitudes
and are the following matrices
With the aid of the above parameterization one can identically transform the quasiclassical equations (1) into the following set of effectively decoupled equations for Riccati amplitudes and distribution functions Eschrig00 ()
Depending on the particular trajectory it is also convenient to introduce a “replica” of both Riccati amplitudes and distribution functions which – again following the notations Eschrig00 (); Zhao04 () – will be denoted by capital letters and . These “capital” Riccati amplitudes and distribution functions obey the same equations (7)-(10) with the replacement and . The distinction between different Riccati amplitudes and distribution functions will be made explicit below.
ii.3 Boundary conditions
Quasiclassical equations should be supplemented by appropriate boundary conditions at metallic interfaces. In the case of specularly reflecting spin-degenerate interfaces these conditions were derived by Zaitsev Zaitsev () and later generalized to spin-active interfaces in Ref. Millis88, .
Before specifying these conditions it is important to emphasize that the applicability of the Eilenberger quasiclassical formalism with appropriate boundary conditions to hybrid structures with two or more barriers is, in general, a non-trivial issue GZ02 (); OS (). Electrons scattered at different barriers interfere and form bound states (resonances) which cannot be correctly described within the quasiclassical formalism employing Zaitsev boundary conditions or their direct generalization. Here we avoid this problem by choosing the appropriate geometry of our NSN device, see Fig. 2. In our system any relevant trajectory reaches each NS interface only once whereas the probability of multiple reflections at both interfaces is small in the parameter . Hence, resonances formed by multiply reflected electron waves can be neglected, and our formalism remains adequate for the problem in question.
It will be convenient for us to formulate the boundary conditions directly in terms of Riccati amplitudes and the distribution functions. Let us consider the first NS interface and explicitly specify the relations between Riccati amplitudes and distribution functions for incoming and outgoing trajectories, see Fig. 3. The boundary conditions for , and can be written in the form Zhao04 ()
Here we defined the transmission (), reflection (), and branch-conversion () amplitudes as:
Similarly, the boundary conditions for , , and take the form:
Boundary conditions for , , and can be obtained from the above equations simply by replacing .
The matrices , , , and constitute the components of the -matrix describing electron scattering at the first interface:
In our three terminal geometry nonlocal conductance arises only from trajectories that cross both interfaces, as illustrated in Fig. 4. Accordingly, the above boundary conditions should be employed at both NS interfaces.
Finally, one needs to specify the asymptotic boundary conditions far from NS interfaces. Deep in metallic electrodes we have
where - equilibrium distribution function. In the bulk of superconducting electrode we have
where we denoted .
ii.4 Green functions
With the aid of the above equations and boundary conditions it is straightforward to evaluate the quasiclassical Green-Keldysh functions for our three-terminal device along any trajectory of interest. For instance, from the boundary conditions at the second interface we find
Employing the boundary conditions again we obtain
We also note that the relation and makes it unnecessary (while redundant) to separately calculate the advanced Riccati amplitudes.
Let us now evaluate the distribution functions at both interfaces. With the aid of the boundary conditions at the second interface we obtain
Integrating Eq. (9) along the trajectory connecting both interfaces with initial condition for , we arrive at the expression for
Then we can find distribution functions at the first interface. On the normal metal side of the interface we find
The corresponding expression for is obtained analogously. We get
Combining the above results for the Riccati amplitudes and the distribution functions we can easily evaluate the Keldysh Green function at the first interface. For instance, for the trajectory (see Fig. 4) we obtain
The Keldysh Green function for the trajectory is evaluated analogously, and we get
Iii Nonlocal conductance
iii.1 General results
Now we are ready to evaluate the current across the first interface. This current takes the form:
is the normal state nonlocal conductance of our device at fully transparent interfaces, is normal to the first (second) interface component of the Fermi momentum for electrons propagating straight between the interfaces, define the number of conducting channels of the corresponding interface, is the quantum resistance unit.
Here stands for the contribution to the current through the first interface coming from trajectories that never cross the second interface. This is just the standard BTK contribution BTK (); Zhao04 (). The non-trivial contribution is represented by the last term in Eq. (59) which accounts for the presence of the second NS interface. We observe that this non-local contribution to the current is small as (rather than as suggested in Ref. Yam, ). This term will be analyzed in details below.
The functions and are the Keldysh Green functions evaluated on the trajectories and respectively. Using the above expression for the Riccati amplitudes and the distribution functions we find
The correction to the local BTK current (arising from trajectories crossing also the second NS interface) has the following form
while for the cross-current we obtain
In right hand side of Eq. (64) we can distinguish four contributions with different products of -matrices. Each of these terms corresponds to a certain sequence of elementary events, such as transmission, reflection, Andreev reflection and propagation between interfaces. Diagrammatic representation of these four terms is offered in Fig. 5. The amplitude of each of the processes is given by the product of the amplitudes of the corresponding elementary events. For instance, the amplitude of the process in Fig. 5c is