Theory of superconductor-ferromagnet point contact spectra:
the case of
strong spin polarization
We study the impact of spin-active scattering on Andreev spectra of point contacts between superconductors(SCs) and strongly spin-polarized ferromagnets(FMs) using recently derived boundary conditions for the Quasiclassical Theory of Superconductivity. We describe the interface region by a microscopic model for the interface scattering matrix. Our model includes both spin-filtering and spin-mixing and is non-perturbative in both transmission and spin polarization. We emphasize the importance of spin-mixing caused by interface scattering, which has been shown to be crucial for the creation of exotic pairing correlations in such structures. We provide estimates for the possible magnitude of this effect in different scenarios and discuss its dependence on various physical parameters. Our main finding is that the shape of the interface potential has a tremendous impact on the magnitude of the spin-mixing effect. Thus, all previous calculations, being based on delta-function or box-shaped interface potentials, underestimate this effect gravely. As a consequence, we find that with realistic interface potentials the spin-mixing effect can easily be large enough to cause spin-polarized sub-gap Andreev bound states in SC/sFM point contacts. In addition, we show that our theory generalizes earlier models based on the Blonder-Tinkham-Klapwijk approach.
The proximity effect near interfaces between superconductors and ferromagnetic materials has been a field of intense research in recent years. bergeret05 (); A. I. Buzdin (2005); eschrig08 (); cuoco08 (); galak08 (); haltermann08 (); volkov08 (); P. M. R. Brydon et al. (2008); zhao08 (); linder09 (); grein09 (); brydon09 (); kalenkov09 (); beri09 (); Barsic (); eschrig09 () This interest is mainly triggered by the observation that exotic types of pairing symmetries that are difficult (or impossible) to be observed in bulk materials can be created in such heterostructures.eschrig07 (); Y. Tanaka and A. A. Golubov (2007); eschrig08 () Examples are the recent revival of pairing states that exhibit a sign change under the exchange of the time coordinates of the particles that constitute a Cooper pair (“odd-frequency pairing”),bergeret05 () or mechanisms for the creation of long-range equal-spin pairing components in half-metallic ferromagnets.eschrig03 (); volkov03 (); kopu04 (); eschrig04 () Supercurrents in half-metals have subsequently been observed,keizer06 () which ignited a strong activity in further theoretical modeling of this effect. eschrig07 (); braude07 (); asano07 (); Linder and Sudbø (2007); linder07 (); takahashi07 (); cuoco08 (); galak08 (); haltermann08 (); volkov08 (); kalenkov09 (); beri09 (); Barsic (); eschrig09 () Spin triplet pairing has proven to be at the heart of new physical phenomena, like --transitions in Josephson junctions with FM interlayersA. I. Buzdin (2005); lofwander05 (); pajovic06 (); champel08 (); P. M. R. Brydon et al. (2008); brydon09 () or the interplay between magnons and triplet pairs.M. Houzet and A. I. Buzdin (2007); takahashi07 ()
So far, transport calculations in SC/FM hybrids have mostly been concentrated on either fully polarized FMs, so-called half metals (HM), or on the opposite limit of weakly polarized systems. However, most FMs have an intermediate exchange splitting of the energy bands of the order of 0.2-0.8 times the Fermi energy , which we here refer to as strongly spin-polarized FMs (sFM). As alternative to solving full Bogoliubov-de Gennes equations,Linder and Sudbø (2007); Valls (); Barsic (); cottet08 (); cottet08b () we have recently presented a quasiclassical theory appropriate for this intermediate range of spin-polarizations, which is of considerable importance for applications.grein09 ()
For such strongly spin-polarized materials, it has been argued that Andreev point contact spectra can be used to obtain the spin-polarization of the FM,S. K. Upadhyay et al. (1998); soulen98 (); Beenakker (); Mazin () which is an important information for spintronics applications. Experimental studies of point contact spectra with strongly spin-polarized systems have been performed for a number of systems.desisto00 (); ji01 (); angu01 (); parker02 (); woods04 (); F. Pérez-Willard et al. (2004); dyachenko06 (); yates07 (); krivoruchko08 (); bocklage07 () However, Xia et al.K. Xia et al. (2002) have objected rightfully, that without taking into account a realistic description of the interface region, the results obtained with this method are questionable.
In the quasiclassical approach to superconducting hybrid structures, interfacial scattering is taken into account by the interface scattering matrix of the structure in its normal state. This is ideal for discussing microscopic models of interfacial scattering which go well beyond the standard Blonder-Tinkham-Klapwijk (BTK) approach.blonder82 () The latter has been employed to fit experimental data of SC/FM point contact spectra,Beenakker () with the interface being described by a single parameter related to its transparency and the ferromagnet by its spin-polarization . The modification of the Andreev point contact spectrum compared to a normal metal contact is then uniquely related to the spin-dependent density of states (DOS) in the FM bulk. This model allows for good fits to experimental data, however, comparing different probes with varying interface transparency, a systematic dependence was found by Woods et al.woods04 () This shows that the extracted spin polarization is not a bulk property, as was originally assumed, but at least partially an interface property. This important difference has been emphasized also in Ref. F. Pérez-Willard et al., 2004.
From the theoretical point of view, it is obvious that if scattering is spin-active, i.e. the scattering event is sensitive to the spin of the incident electron, this may not only imply a spin-dependent transmission probability (spin-filtering)meservey94 () but also a spin-dependent phase shift of the wavefunction.tokuyasu88 () The latter is called the spin-mixing effect and it has been shown to be of crucial importance for the creation of exotic pairing correlations.tokuyasu88 (); fogelstrom00 (); huertas02 (); eschrig03 (); eschrig08 (); cottet05 (); bobkova07 ()
So far, estimates of the magnitude of this effect and its dependence on physical parameters including not only the structure of the interface but also the Fermi surface geometry of the adjacent materials and the FM exchange splitting are still lacking. Instead, phenomenological models have been adopted that introduce a free parameter to account for it.fogelstrom00 (); eschrig03 (); eschrig08 (); Zaikin ()
The main point of this paper is to provide a microscopic analysis of the characteristic interface parameters. In the following we adopt a simple model of the interface region consisting of a spin-dependent scattering potential whose quantization axis may be misaligned with that of the adjacent FM. We allow for an arbitrary shape of this scattering potential and illustrate that this may enhance the spin-mixing effect considerably compared to the previously used box-shaped or delta-function potentials. We also study in detail the relation between spin-mixing angle and impact angle of the quasiparticle, showing that this relation can be non-trivial for transparent interfaces. Furthermore, we provide a very general mathematical discussion of suitable parameterizations and representations of the scattering matrix in this context.
Andreev bound states have proven invaluable for studying the internal structure of the superconducting order parameter.saint64 (); deutscher05 () Andreev states are also induced at spin-polarized interfaces by the spin-mixing effect.fogelstrom00 () In fact, the measurement of such bound states at spin-active interfaces would be an elegant method do determine the spin-mixing angle of the interface. To date this quantity has never been determined in experiment. Our results show, that a measurable effect is more likely to appear when leakage of spin polarization into the superconductor takes place, for example due to diffusion of magnetic atoms. Our theory can discriminate between conventional Andreev reflection processes (AR) and spin-flip Andreev reflection (SAR), the latter being responsible for the long-range triplet proximity effect. We discuss the Andreev bound state associated to the spin-mixing effect and show that it may be observable in experiment. Furthermore we show that for highly polarized FMs, spin-flip scattering can bias the spectra considerably, proving that such processes must be precluded if one wishes to extract the FM spin-polarization from such spectra.
The paper is organized as follows. In Section II, we discuss quasiclassical theory to describe transport through a point contact. In Section III we turn to interface models and discuss the spin-mixing effect and the scattering matrix. In Section IV we present results for Andreev conductance spectra of SC/FM point contacts. We dicuss analytical results, focusing on the Andreev bound state spectrum, as well as numerical results. In Subsection IV.3 we establish the connection to earlier transport theories for such systems which are based on the BTK approach. We prove analytically that they are contained as limiting cases in our formalism. Eventually, in Section V, we conclude on our results.
Ii Quasiclassical theory
We make use of the quasiclassical theory of superconductivitylarkin68 (); eilen (); Serene (); schmid75 (); schmid81 (); rammer86 (); Larkin86 (); FLT () to calculate electronic transport across the SC/FM interface. This method is based on the observation that, in most situations, the superconducting state varies on the length scale of the superconducting coherence length , with the normal state Fermi velocity . The appropriate many-body Green’s function for describing the superconducting state has been introduced by Gor’kov,gorkov58 () and the Gor’kov Green’s function can then be decomposed in a fast oscillating component, varying on the scale of the Fermi wave length , and an envelop function varying on the scale of . The quasiclassical approximation consists of integrating out the fast oscillating component:
where is the inverse quasiparticle renormalization factor (due to self-energy effects from high-energy processes),Serene () a “check” denotes a matrix in Keldysh-Nambu-Gor’kov space,keldysh64 () a “hat” denotes a matrix in Nambu-Gor’kov particle-hole space (with the third Pauli matrix), is the Fermi momentum, the spatial coordinate, the quasiparticle energy, the time, and . The quasiclassical Green’s function obeys the transport equationlarkin68 (); eilen ()
Here, is the superconducting order parameter, contains external fields and self-energies due to impurities etc, and denotes the commutator with respect to a time convolution product (for details see Ref.Serene, ). Eq. (2) must be supplemented by a normalization conditionlarkin68 (); shelankov85 () . The current density is related to the Keldysh component of the Green’s function via:
where is the density of states at the Fermi level in the normal state, and denotes a Fermi surface average which is defined as follows:
The direct inclusion of an exchange energy of order of 0.1 or larger in the quasiclassical scheme violates the underlying assumptions of quasiclassical theory. As we aim to describe a strongly spin-polarized FM, which means that its exchange field will be of the order of the Fermi energy, we cannot include it as a source term (with the vector of Pauli spin matrices) in the quasiclassical equation of motion. Such an approach would neglect terms of order of compared to . The resulting condition , assuming e.g. a gap of 1 meV and eV, would imply meV. In general, the condition for the possibility to include in the quasiclassical low energy scale is violated for most SCs if .
To deal with the strong exchange splitting, we make use of the fact that it results in a rapid suppression of superconducting correlations between quasiparticle states with opposite spin, i.e. singlet () or triplet () correlations. They decay on the short length scale . Here , are the Fermi-momenta of the two spin-bands (2 and 3) in the sFM and with the coherence length in the respective band. Consequently, only equal-spin triplet correlations can penetrate the FM-bulk. Hence we pursue the following approach to model a strongly polarized FM in the frame of QC theory. We define independent QC Green’s functions (QCGF) for each spin-band which are scalar in spin-space, i.e. describe correlations with , respectively spin-wavefunction. The boundary conditions must now match three QC propagators at the SC/FM interface, which we label with , -band and -band (see Fig. 1). These three QCGFs are formally obtained from:
with , and being the respective Fermi-momenta/velocities of the bands. Consequently, the current must then be evaluated for each band separately
Here, is the partial density of states at the Fermi level in band , and denotes the corresponding Fermi surface average
In addition, the system’s properties vary on the atomic length scale in the interface region between the two materials. Thus the QC theory is also not applicable in the immediate proximity to the interface (on the scale of the Fermi wavelength). This is a general problem in the quasiclassical description of heterostructures, which can be circumvented by deriving appropriate boundary conditions for matching the QC propagators on both sides of the interface.zaitsev84 () The full boundary conditions for the present problem have been developed only recently.eschrig09 () Earlier works on Andreev spectra using QC theory were restraint to either SC/normal metal contacts with spin-active interfaces,F. Pérez-Willard et al. (2004); fogelstrom00 (); barash02 (); zhao04 () or contacts with weak ferromagnets. We refer to Ref.eschrig09, and references therein for a detailed discussion of this problem. In the following subsection we discuss a parameterization of the QC propagator, and return to the problem of boundary conditions at the interface in Subsection II.2.
ii.1 Riccatti parameterization
For our calculations we choose a representation of the quasiclassical Green’s function (QCGF) that has proven very useful in the past and is standard by now. In this representation, the Keldysh QCGF is determined by six parameters in particle-hole space, , of which and are the retarded () and advanced () coherence functions, describing the coherence between particle-like and hole-like states, whereas and are distribution functions, describing the occupation of quasiparticle states.eschrig00 (); cuevas06 () The coherence functions are a generalization of the so-called Riccatti amplitudesnagato93 (); schopohl95 () to non-equilibrium situations. All six parameters are 22 spin-matrix functions of Fermi momentum, position, energy, and time. The parameterization is simplified by the fact that, due to symmetry relations, only two functions of the six are independent. The particle-hole symmetry is expressed by the operation , which is defined for any function of the phase space variables by
where is real for the Keldysh components and is situated in the upper (lower) complex energy half plane for retarded (advanced) quantities. Furthermore, the symmetry relations
hold. As a consequence, it suffices to determine fully the parameters and .
The QCGF is related to these amplitudes in the following way [here the upper (lower) sign corresponds to retarded (advanced)]:eschrig09 ()
with the abbreviations and , and
Note that all multiplication and inversion operations include 22 matrix algebra (and, more general, for time-dependent cases also a time convolution).
From the transport equation for the quasiclassical Green’s functions one obtains a set of 22 matrix equations of motion for the six parameters above.eschrig99 (); eschrig00 () For the coherence amplitudes this leads to Riccatti differential equations,schopohl95 () hence the name Riccatti parameterization. As we are interested in this paper only in the interface problem in relation to a point contact, the transport equations are not relevant for the problem at hand. For a point contact, the superconductivity is modified only in a very small spatial region, and this modification can be neglected consistent with quasiclassical approximation. We assume that the half-space problem is solved and calculate the conductance across the point contact. For this, we turn now to the problem of solving the boundary conditions for the point contact.
ii.2 Boundary conditions
ii.2.1 General case
The QCGF mixes particlelike and holelike amplitudes, and as a result the transport equations are numerically stiff, with exponentially growing solutions in both positive and negative directions along each trajectory, which must be projected out. A particular advantage of the coherence and distribution functions is that, in contrast to the QCGF, they have a stable integration direction for each trajectory. This direction coincides with their propagation direction, and is opposite for holelike and particlelike amplitudes as well as advanced and retarded ones. This allows to distinguish between incoming and outgoing amplitudes at the interface. We adopt the notationeschrig00 () that incoming amplitudes are denoted by small case letters and outgoing ones by capital case letters. Boundary conditions express outgoing amplitudes as functions of incoming ones and as functions of the parameters of the normal-state scattering matrix. They are formulated in terms of the solution of the equationeschrig09 ()
for , where the trajectory indices run over outgoing trajectories involved in the interface scattering process, and the scattering matrix parameters enter only via the “elementary scattering event”
(the trajectory index runs over all incoming trajectories). It is useful to split the quantity into its forward scattering contribution, which determines the quasiclassical coherence amplitude,
and the remaining part
which is relevant only for the Keldysh components. Analogous equationseschrig09 () hold for the advanced and particle-hole conjugated components, , , and . The boundary conditions for the distribution functions readeschrig09 ()
which depend on the scattering matrix parameters only via the elementary scattering event
Analogous relations hold for .
ii.2.2 Special case for point contact
In the case under consideration the trajectory labels and run from 1 to 3, with 1 denoting (spin-degenerate) trajectories on the superconducting side, and 2 and 3 trajectories for the two spin directions on the ferromagnetic side. We use the following notation for the (unitary) scattering matrix:
The current across the interface is conserved (this is ensured by our boundary conditions), so that it suffices to calculate the current density at the FM side of the interface. We proceed with expressing the outgoing amplitudes for bands and in terms of the incoming amplitudes and the scattering matrix.
For a point contact with semi-infinite SC and FM regions (assuming that the Thouless energy related to the geometry of the system is negligibly small), there are no incoming correlation function from the FM side, , whereas on the SC side we can use the bulk solutions. For a singlet order parameter the bulk solutions of the coherence functions read
with the singlet superconducting order parameter . Taking into account these facts, we obtain from Eq. (14)
with for . The first equation, Eq. (22), can be solved,
It appears useful to introduce the notation
Note that the identity holds. The corresponding solutions for band 3 are simply obtained by replacing . Amplitudes and are obtained using Eq. (10), with . The required advanced amplitudes can be obtained from the fundamental symmetry relations of this formalism, which imply and .
For the distribution functions, we use a gauge in terms of anomalous components.eschrig09 () Taking the electrochemical potential equal to zero in the SC, and equal to in the ferromagnet, these are and
Note that in our notation . From Eq. (18) we arrive at the following expressions for the outgoing Keldysh amplitudes for band :
with for . Introducing what has been obtained before, we arrive at
Again, the corresponding solution for band 3 is obtained by replacing .
Iii Interface model
We consider a point contact with a diameter much smaller than the superconducting coherence length but still larger than the Fermi-wavelength, as shown in Fig. 2 a.
A larger contact would result in a perturbation of the SC state, a smaller one would invoke conductance quantizationGolubovRMP (). This also allows for the decisive assumption of translational invariance on the scale of . The region in the immediate vicinity of the interface (I) cannot be described within QC theory. Instead, the normal state scattering matrix of the interface must be obtained from microscopic calculations and then enters the QC theory through boundary conditions as outlined above.
The mechanism giving rise to spin-active scattering at the interface is the ferromagnetic exchange field in both the adjacent ferromagnetic material and in the interface itself. The interface will in general carry a magnetic moment, that in the simplest case is induced by the magnetization of the bulk ferromagnetic material; however, there might be cases where an extra interface magnetic moment develops, either manufactured by using a thin magnetic layer, or due to spin-orbit coupling, and related to that, magnetic anisotropy. The interface magnetic moment can be misaligned with the one of the bulk sFM. We characterize this misalignment by two spherical angles and , as indicated in Fig. 2 b. While the spin-activity of interfaces has been discussed extensively in the theory of superconducting heterostructures, most of this work so far considered a set of phenomenological parameters for characterizing the interfacial scattering. Notably, one of these parameters, the so-called spin-mixing angle, or spin-dependent phase shift, turned out to be of decisive importance for the creation of unconventional superconducting correlations in proximity to the interface. The spin-mixing angle is essentially a relative phase difference between and electrons acquired upon scattering. Obviously, an exchange field in the interface region will provide such an effect, but other mechanisms, like for instance spin-orbit coupling are also candidates.
So far, estimates of the possible magnitude of this effect based on a physical model of the interface region are still lacking. Here, we will provide such an analysis based on wavefunction matching techniques. In particular, we will discuss the dependence of the spin-mixing effect on the shape of the barrier. To this end, we consider a spin-split potential barrier which is assumed to conserve the momentum component parallel to the interface upon scattering. For the system we deal with, this gives rise to two types of transmission events (see Fig. 1). Depending on the impact angle the parallel momentum conservation constraint either allows for or prohibits scattering into/from the minority spin-band of the sFM. For a half metal, where the -band is completely insulating, only the latter case occurs.
iii.1 Interface scattering matrix
At this point we mention some general considerations concerning the scattering matrix of a spin-active interface. Such a matrix is unitary and of dimensions in the FM and in the HM case. The maximum number of free parameters is 16 or 9 respectively. However, not all of these parameters will be relevant for the physical problem at hand. For instance, spin-scalar phase factors do only matter for two or more interfaces. Furthermore, since a singlet SC is spin-isotropic, one is free to choose the spin-quantization axis in the SC conveniently. To clearly identify these irrelevant parameters we use a special parameterization of a general unitary matrix with the aforementioned dimensions, as discussed in App. A. The most important result of these considerations is that the spin-mixing effect can be fully described by only one parameter in the HM case, but 3 are required in the FM case.
Neglecting irrelevant spin-scalar phases and using the gauge freedom in the SC the scattering matrix reads for the first type of scattering
The scattering matrix for the second, HM type, scattering is
There is also the possibility of total reflection with no transmission on either side, in which case the scattering matrix consists of the reflection parts only. In writing the scattering matrices (32) and (33) we have put the -phase that appears in Fig. 2(b) to zero, since the problem we consider is invariant with respect to rotation of the interface magnetic moment around the bulk magnetization; the scattering matrix is symmetric in this case, . We also omitted possible complex phases in the reflection part on the FM-side, i.e. , and , as they are irrelevant to the problem at hand. The requirement of unitarity leads to additional relations between the reflection and transmission parameters. The phases that we wrote explicitly in Eqs. (32) and (33) are crucial, since they account for the spin-mixing effect. In the following section, we will discuss their magnitude as a function of various interface parameters.
Using the set of independent parameters described in the appendix we have:
The angle defines a rotation in spin-space to the interface eigenstates, characterized by transmission and reflection eigenvalues. Its precise definition is given in the appendix. Most importantly, it is in general not identical to the interface misalignment angle , however approaches it in the limit of thick interfaces. For thin interfaces it is renormalized by the influence of the exchange field of the adjacent FM. and are the singular values of the reflection block . In the tunneling limit, , and the off-diagonal elements vanish even for . This is easily understood from a physical point of view, since spin-flip reflections on the SC side requires that the reflected quasiparticles “feel” both misaligned exchange fields and not just that of the interface. It is possible to provide analogous expressions for the remaining parameters of the scattering matrix, however in the sFM case they are rather cumbersome and also not needed for the following analytical discussion. For the half-metallic case, the only relevant phase parameter is the spin-mixing-angle , and for the remaining parameters we have and
In the following we will discuss the influence of the shape of the scattering potential, and will show that the widely used box shaped or delta-function shaped potentials gravely underestimate the magnitude of the spin-mixing effect.
iii.2 Box-shaped scattering potential
In this section we consider spin-dependent box potentials, for which analytical solutions can be obtained. In particular, we discuss here the dependence of on the impact angle of the incoming quasiparticle which is parameterized by the momentum component parallel to the interface, . The model parameters are the misalignment angle (see Fig. 2 b), the energies of the band minima in the FM with respect to that in the SC (), the spin-dependent height of the potential (), and the width of the potential (see Fig. 3). All energies are given in units of and in units of .
The scattering matrix is defined with respect to the chosen spin-quantization axes on both sides of the interface. Naturally, on the FM side we use the bulk sFM magnetization axis. On the SC side we use that of the interface magnetic moment. To obtain an S-matrix with the structure defined above, one must subsequently calculate and apply a rotation of the quantization axis in the SC:
where is a spin rotation matrix acting on spins in the superconductor. We describe this procedure in App. A.1. All the quantities plotted are calculated in this rotated frame, the point being that otherwise one does not have an unambiguous definition of the mixing-phases. Naturally, the Andreev spectra are invariant under these transformations. We obtain the scattering matrix by matching wave functions as described in App. A.2.
In Fig. 4 a,b we show the spin-mixing angle for different values of the interface potential width . The band minima in the FM are and , which implies that at the minority band becomes insulating and the scattering matrix reduces to a matrix. In the tunneling limit () the spin-mixing angle behaves as expected: it is approximately given by the value (see App. A.2)
which appoaches zero for grazing impact (), and for normal impact ( for Fig. 4). Here, is the component of the wavevector perpendicular to the interface in the superconductor, and are the exponential decay factors for the spin-up/down wave function in the barrier. For thin (highly transparent) interfaces the mixing-angle is a more complicated function of the quasiparticle impact angle. In this regime, is predominantly controlled by the Fermi-surface geometry indicated in Fig. 3. There is a local minimum at , and for very thin interfaces is largely enhanced for grazing impact ( in Fig. 4). This enhancement can be understood from the limit, i.e. the case where the interface barrier is absent. In this case,
where, corresponds to the imaginary wave vector in the insulating band 3, which controls the exponential decay of the spin-down wave function into the ferromagnet. In the particular case we show here, see Fig. 3, takes a finite value for all trajectories that contribute to the current, while increases monotonously from at to some finite value at . This is because the effective height of the potential for tunneling into the insulating band increases with . For Fermi-surface geometries with (not shown here) the wave vector drops to zero for grazing impact, and the spin-mixing angle approaches .
In the present case, the situation is complicated by the fact that we consider both a finite interlayer and a broken spin-rotation symmetry. This leads to a finite spin-mixing angle even for and below, which leads to the non-trivial behavior with a minimum for intermediate impact angles. This illustrates that not only the scattering potential itself but also the Fermi-surface geometry is highly important for spin-active scattering beyond the tunneling limit.
As for the magnitude of the mixing effect, we stress that for a realistic choice of parameters, it is hardly possible to achieve mixing-phases above in this model. In Fig. 4 we use an exchange field of , which is close to the half-metallic limit. Using smaller exchange energies naturally leads to a smaller effect, as can be seen in Fig.5 a,b, where we plot for different values of the exchange field .
In Fig. 6 we show the spin-mixing phases associated to transmission and . One can see that for . This relation one would expect for a SC contacted with a half-metallic ferromagnet; the finding in Fig. 6 is consistent with this and the discussion presented above, since the trajectories under consideration effectively correspond to the HM case. For , the mixing phase is considerably enhanced above the value of . The plots also illustrate that and are different in magnitude and also vary differently with . As we show in the appendix, the mixing-phases and are correlated with but in general also depend on a number of other free parameters. Their magnitude is decisive for the creation of triplet correlations in the corresponding band, as we will show below.
In Fig. 7 we present the product (which controls the magnitude of long-range SAR). We plot this quantity for both the majority (upper row) and minority (lower row) band of the FM. Apparently there is a non-monotonous dependence on the interface width , which is related to the fact that spin-flip scattering becomes more effective as the interface region becomes larger. For even larger the global suppression of transmission intervenes and we approach the tunneling limit. Again, we note that for thin interfaces the dependence on trajectory impact angle is non-monotonous, showing maxima for non-perpendicular impact. These maxima coincide exactly with the minima of the spin-mixing angle. Note that a nonzero requires a non-vanishing misalignment angle .
To conclude on this section, we have shown that the magnitude of the spin-mixing effect is limited to rather small values in the box potential case if one assumes and . Moreover, both spin-mixing effect and spin-flip scattering are very sensitive to trajectory impact, interface thickness, exchange field of the interface and the Fermi surface geometry of the adjacent materials.
iii.3 Delta-function scattering potential
A special case of the box-shaped potential is that of the delta-function potential, that is widely used in describing interfaces within the BTK paradigm. Here, we show that the situation is in this case comparable to that of the box potential. Delta-function models introduce a weight factor of the Delta-function which enters the matching condition for wavefunction derivatives:
A spin-dependent potential can simply be modeled by choosing a spin-dependent weight factor . This weight factor effectively corresponds to the area under the scattering potential, i.e. we have , to connect with the notation above. In Fig. 8 we plot as a function of for perpendicular impact and two different choices of the Fermi-surface geometry. Since we do not calculate any spectra for this model, we choose for simplicity. Generically, spin-mixing angles can only be reached for , which requires either to be very small or an interface exchange field exceeding the Fermi-energy.
iii.4 Scattering potentials with arbitrary shape
The box potential actually constitutes a high degree of idealization. The most obvious generalization is to consider a potential that varies smoothly on the scale of a few interatomic distances, or on the scale of the Fermi wavelength in metals.smooth () This is quite realistic taking into account that metallic screening of charges takes place only on the Thomas-Fermi wavelength scale. In addition, some magnetic ions might penetrate the superconductor from the ferromagnet, leading to a spin-dependent potential that decays in the bulk of the superconductor. In the latter case a certain degree of disorder is introduced. However, we will assume that any such disorder is weak, so that the momentum component parallel to the interface is still a good quantum number. A truly realistic description would have to drop the assumption of translational invariance and consider disorder on a microscopic level. In principle our theory can be extended to this regime, but this is beyond the scope of this paper. If one is only interested in transmission and reflection amplitudes, the difference between the box-shape and a smoothened potential is negligible. But when scattering phases are important, as in our case, this is not true, as we will show in the following.
For definiteness, we consider a potential shape as shown in Fig. 9, with Gaussian “slopes”. The ‘’smoothness” of the interface barrier is then controlled by the standard deviation of the Gaussian. Hence, we have the spin-dependent potential:
In the limit of a very smooth potential, one may resort to the Wentzel-Kramers-Brillouin (WKB) approximationWKB () to calculate the scattering problem. An interface that complies to the requirements of WKB would have to be much larger than the Fermi-wavelength however, which is unrealistic. For this reason we resort to a numerical method for calculating the scattering problem. We use a recursive Green’s function techniqueGeorgo () to calculate the single particle Green’s function of the interface Hamiltonian and obtain the scattering matrix from it using the Fisher-Lee relations.Fisher () To study the effect of the potential shape on the spin-mixing angle, we plot the angle in Fig. 10 b for different values of . To avoid a large variation of the interface transmission when varying , we keep (see Fig. 10 a).
Furthermore, we use here, i.e. both the FM-bands have a larger Fermi-surface than the SC. As we will see later on, this Fermi surface geometry and the scattering constraints it implies can have an important effect on the shape of the spectra, and in particular on features which are related to the spin-mixing effect.
The main result of considering a variation of the potential shape is however, that it has a tremendous effect on the spin-mixing angle, as clearly seen in Fig. 10 b. Its magnitude can exceed for a smooth potential that for a box potential of similar transmission easily by a factor of 3-4 or more. This is sufficient to observe some exotic features related to this effect in the Andreev spectra of point contacts, as discussed in the next section. The physical reason for this is that, unlike in the box potential case, electrons with opposite spin acquire a phase difference while they are still propagating, which implies that a larger mixing phase is not inevitably tied to a strongly reduced transmission. This can be best seen in the WKB limit, where the mixing angle is exclusively given by this dephasing:
Here and are the classical return points for the respective spin band (see Fig. 3 for the notation). In the intermediate case, that we consider here both the different wavevector mismatches and the dephasing of propagating modes will add to the mixing effect.
The discussion in terms of scattering matrix parameters presented here is flexible enough to be extended, e.g. to other Fermi surface geometries, or adiabatic variation of the interface magnetization. Furthermore, instead of insulating interfaces one could consider interfaces where one or even both channels are conducting. The latter case has been considered by Béri et al.beri09 ()
Iv Andreev conductance spectra of SC/FM point contacts
In the remaining part of the paper we discuss Andreev spectra that result from our model. We use a definition for the FM’s spin-polarization given by
For parabolic bands, the density of states is proportional to the Fermi-momentum, , assuming equal effective masses.
The current density in terms of the distribution functions and coherence functions is given by
where the expression for is given by
and an analogous expression is obtained for by interchanging 2 with 3. Here, means a Fermi-surface average over one half of the Fermi surface (positive momentum directions, pointing into the FM, for the first and third term of Eq. (44), negative directions for the second term). To derive this expression, we used the universal symmetry relation (10). Furthermore as defined in (29), is defined in (26) and the scattering matrix parameters in (20). Equations (43)-(44) are the main result of this paper.
The interpretation of Eqs. (II.2.2) and (43) allows for identifying two types of Andreev reflection, shown in Fig. 11, one of them giving rise to a long-range proximity effect in the FM. The terms in Eq. (43) and entering in Eq. (30) both describe current contributions from Andreev reflected holes to the current in band . The first term is proportional to the incoming distribution function in the same band. Thus we refer to it as spin-flip Andreev reflection (SAR), as it requires a spin-flip to transmit a singlet pair into the SC. The second term corresponds to normal Andreev reflection, since it reflects a hole in the opposite band. While SAR is related to the outgoing (equal-spin) triplet correlation function in the respective band, AR is described as a term renormalizing the outgoing distribution function. Unlike SAR, AR does not contribute to the coherence functions in the FM spin-bands.
Using the scattering matrix parameterization introduced in Sec. III, we can obtain explicit analytical solutions for the coherence functions:
Here and we omitted the index for the incoming coherence functions. is related to in (21) by . The advanced component is obtained via .
Note that the -functions differ only by the transmission vectors but since the numerator is a matrix product, this still gives expressions that differ markedly. In any case, we have if or and . We focus on the denominator which arises from the matrix inversion in Eq. (25) and is the same for all coherence functions. It is of particular interest since it leads to the emergence of conductance peaks in the Andreev spectrum.
iv.1 Andreev bound state spectrum
The appearance of the Andreev conductance peaks can be seen most clearly in the tunneling limit. Here and which simplifies the expressions above considerably. The full solutions read
This pole corresponds to an Andreev bound state induced by the spin-mixing effect at the superconducting side of the sample. Following Fogelström,fogelstrom00 () one can show that these bound states appear in the DOS of the superconductor close to the interface and are actually associated to different spins. The bound state for appears in the DOS of -quasiparticles and that for in that of -quasiparticles if . This is why the appearance of the sub-gap peak is only tied to the spin-mixing angle . It does not depend on spin-flip scattering or the mixing phases associated to transmission. However, we shall see that a high mixing angle of