Conductance and current noise of a superconductor/ferromagnet quantum point contact
We study the conductance and current noise of a superconductor/ferromagnet () single channel Quantum Point Contact (QPC) as a function of the QPC bias voltage, using a scattering approach. We show that the Spin-Dependence of Interfacial Phase Shifts (SDIPS) acquired by electrons upon scattering by the QPC can strongly modify these signals. For a weakly transparent contact, the SDIPS induces sub-gap resonances in the conductance and differential Fano factor curves of the QPC. For high transparencies, these resonances are smoothed, but the shape of the signals remain extremely sensitive to the SDIPS. We show that noise measurements could help to gain more information on the device, e.g. in cases where the SDIPS modifies qualitatively the differential Fano factor of the QPC but not the conductance.
pacs:73.23.-b, 74.20.-z, 74.50.+r
Mesoscopic circuits with ferromagnetic elements are generating a growing interest, both for the fundamentally new effects they can exhibit due to the lifting of spin-degeneracy, and for the possibilities of technological advances, e.g. towards nanospintronics(1). The description of the interface between ferromagnetic and non-magnetic elements is crucial to understand the behavior of these devices. Obviously, one first has to take into account the spin-polarization of the electronic transmission probabilities through the interfaces, a property which generates the widely known magnetoresistance effects(2). Importantly, the phases of the scattering amplitudes also depend on the spin in general. This so-called spin-dependence of interfacial phase shifts (SDIPS) is less frequently taken into account. However, the SDIPS has already been found to affect the behavior of many different types of mesoscopic conductors with ferromagnetic leads, like diffusive normal metal islands (3), resonant systems (4); (5), Coulomb blockade systems(6); (13); (5), or Luttinger liquids(8). The SDIPS can also have numerous types of consequences in superconducting/ferromagnetic () hybrid systems (9); (10); (11); (12); (13); (14); (15). It can e.g. produce effective field effects in superconducting electrodes(9); (11); (13), or a phase shift of the spatial oscillations of the superconducting correlations induced by the proximity effect in a diffusive electrode(13); (12). The SDIPS has also been predicted to induce triplet correlations in a electrode in contact with several ferromagnetic insulators with non-collinear polarization directions(14).
Recently, mean current measurements through quantum points contacts (QPCs) have raised much interest as a possible means to determine the polarization of a material. However, the effects of the SDIPS in these devices have not raised much attention so far. In most cases, the experimental results were interpreted in terms of a generalization of the Blonder-Tinkham-Klapwick (BTK) model(16) to the spin-dependent case (17). We would like to emphasize that the description of a interface in terms of a delta-function barrier naturally includes a finite SDIPS but does not allow one to analyze its effects separately. Moreover, the SDIPS included in the generalized BTK model is specific for the delta-function barrier, and for a more realistic interface potential the SDIPS can be different. This is the reason why the influence of the transmission probabilities and of the SDIPS have to be investigated separately. The scattering approach seems perfectly adapted for this purpose since it allows to account explicitly for scattering phases(18). In a few works, current transport through QPCs was described with a single-channel scattering description(19); (20). However, the SDIPS was not taken into account in these works.
In this article, we study how the conductance and the current noise through a single channel QPC are affected by the SDIPS, using a scattering approach. We would like to emphasize that the current noise in hybrid systems has draw little attention so far(21), although it can provide more information on the system considered. We find that the behavior of the QPC depends on its spin-dependent normal state transmission probabilities , but also on the difference between the normal state reflection phases of up spins and down spins against . The device is thus more difficult to characterize than in the spin-degenerate case since there are two more parameters to determine, i.e. and . We show that a finite SDIPS can strongly modify the behavior of the QPC, for instance by shifting the Andreev peaks appearing in the QPC conductance to subgap voltages, as found by Zhao et al. for quantum point contacts(15), or by producing subgap peaks or dips in the differential Fano factor curves. At finite temperatures, we find that the SDIPS-induced shift of the Andreev peaks in the conductance can be difficult to distinguish from a gap reduction. We show that, in this context, a noise measurement can be very useful to characterize better the contact properties. This is particularly clear in cases where the QPC differential Fano factor versus bias voltage is qualitatively modified by the SDIPS while the conductance curve remains similar to the curves obtained without a SDIPS.
This paper is organized as follows: section II.A introduces expressions of the conductance and noise of a interface in the scattering formalism, section II.B presents an explicit expression of the scattering matrix of a single channel QPC, section II.C discusses the conductance and noise of this QPC at zero temperature, and section II.D discusses the finite temperature case. Section III concludes.
Ii Conductance and noise of a single channel Qpc
ii.1 Current and noise through a interface in the scattering approach
We consider a superconducting/ferromagnetic () interface with a conventional BCS conductor with a s-wave symmetry. We assume that the and leads can be considered as ballistic, so that the interface can be described with the scattering approach(18) and the leads with Bogoliubov-De Gennes (BdG) equations(24). We use a gauge transformation to set the origin of the quasiparticles energies to the Fermi level of , i.e. (25). Due to the bias voltage applied to the QPC, the Fermi level of is , with the absolute value of the electron charge. In the following, we will denote by electron states with spin and by hole states in the spin band, with a spin component collinear to the polarization of . A hole corresponding to an empty electronic state at an energy carries the same energy as an electron occupying a state with energy . These two types of quasiparticles are coupled by Andreev reflection processes occurring at the interface. One can thus write
for , with and the scattering matrix of the interface for quasiparticles carrying an energy . Here, we consider the single channel case, so that  refers to the annihilation operator associated to the incident [outgoing] state for a particle with type and energy of lead . In the following, we use a picture consisting of both positive and negative energy states. The operator associated to the total current flowing through the device at time can be calculated as the sum of the electron and hole currents of the space, which replaces the summation on the spin direction(25). More precisely, one has with
for . Here, is the coordinate along the leads and is the transverse coordinate. The field operator associated with particles of type in lead is defined as
with , We have introduced above different quantities which characterize the conduction channel of the QPC, i.e. the transverse wave function , the wavevector , and the velocity of carriers . The operator conjugated to is denoted . Neglecting the energy dependence of (see e.g. Ref. (18)), one finds
In the above Eqs., capital Latin indices correspond to the lead or , Greek indices correspond to the electron or hole band of the space , and is the identity matrix in the subspace of states of type of lead . In this paper, we study the average current flowing through the interface and the zero-frequency current noise . Equations (1) and (2) lead to the expressions (22); (26)
In the limit , simplified expressions of and can be obtained. In order to account for the two spin bands in the same way, we use a symmetry property stemming from the structure of the BdG equations, i.e. , with , , and (see derivation in Appendix A). Equations (3) and (4) combined with the unitarity of then lead to
with . Note that Eq. (5) is valid only if , with , can be considered as independent from . The integration domain in Eq.(7) is for and for , similarly to the fact that the elements contributing to in Eq. (5) must be taken at depending on the sign of .
Before concluding this section, we note that in the multichannel case, the elements would be matrices relating the incoming and outgoing states of the different channels of leads and . In this case, one could generalize straightforwardly Eqs. (3), (4), (5) and (7) by applying to their right-hand sides a trace on the channels space and using . In section II, we focus on the single channel case. We briefly discuss possible extensions of this work to the multichannel case in section III.
ii.2 Scattering matrix of a single channel Qpc
We consider a interface which is narrow compared to the coherence length of , so that it can be modeled as a specular interface in series with a (possibly) dirty interface, with the length of tending to zero(28). We would like to emphasize that the layer is introduced artificially, it is merely a trick to describe the superconducting interface on a scale much shorter that the coherence length. The matrix of the interface can expressed in terms of the scattering matrix of electrons with spins on the interface and of the amplitude of the Andreev reflections at the interface. One finds in particular (29),
From Eqs. (5) and (7), at , calculating the current and the noise through the interface only requires one to know the two elements of given above. For , one must use Eqs. (3) and (4), so that the whole matrix is necessary (the other elements of are given in appendix B).
There remains to introduce explicit expressions for and . First, the amplitude can be calculated using the ballistic BdG equations to model the interface(16), with a step approximation for the gap of . This gives for and for . Secondly, in the single channel case, the unitary of leads to
with and . In principle, the scattering phases , with , are spin-dependent because electrons are affected by a spin-dependent scattering potential at the interface. For simplicity, we will assume that , and are independent from and , which ensures the validity of Eq. (5) (see Ref. (30)). Note that the expression that we have introduced in this section for implies and , which is not a general property of superconducting hybrid devices (see e.g. Ref. (31)).
ii.3 Conductance and noise of the QPC at zero-temperature
This section discusses the conductance and current noise of the single channel QPC at zero temperature (). For , no quasiparticle propagates in , so that the unitarity of leads to . One can thus calculate and from Eqs. (5) and (7) by using
for . The expression of is too complicated to be given here. The above Eqs. describe a phenomenon analogous to the one predicted for a quantum point contact(15). The denominators of Eqs. (12) and (13) contain interference terms which describe iterative reflection processes between the and interfaces (Andreev bound states). A quasiparticle can interfere with itself after two back and forth travels between and , one as an electron and one as a hole . The conductance and noise depend on the phase difference because the scattering matrix for holes is . This picture is in fact valid at any temperatures and voltages. In the general case, the whole matrix is necessary to calculate the current and noise, but all the elements of have the same denominator as the one appearing at the right hand side of Eq. (12) and (13) for and respectively [see Appendix B]. In addition, we have checked analytically that, in the general case, and depend on the phases of the scattering matrix through the parameter only (they are actually periodic functions of with a -periodicity).
In an experimental context, one may hope to determine the value of the polarization from the zero voltage conductance and the high voltage conductance , with . We define
For , the polarizations with
will be solutions of the problem provided is real and . Thus, depending on the values of , and , there can be either 0, 1 or 2 solutions for . Importantly, the inferred values of depend not only on the zero voltage and normal conductances, but also strongly on . For simplicity, we will consider in the following the situation , for which one has and , so that there is either no solution for or one solution if . This situation is illustrated in Fig. 1, whose left panel shows the calculated as a function of for and different values of , with (the use of is due to the fact that is an even function of ). For the largest values of used in Fig. 1, left panel (see pink solid curve and green double dot-dashed curve), a finite is found for (see pink and green open squares). However, the calculated can also be larger if one uses a finite . This can be understood by noting that, for the relatively low values of used in this Figure and , decreases monotonically with both and . An increase in thus compensates an increase in (see upper right panel of Fig. 1). For the lowest values of used in Fig. 1, left panel (see blue dotted curve and red dot-dashed curve), there is no solution if , because the zero-bias conductance of the device cannot reach the considered value . However, as long as is not too large, it is possible to find a solution at finite because the zero bias conductance of the QPC increases with for (see e.g. blue triangle in the bottom right panel of Fig. 1). In this case, the calculated value of also increases with , for the same reason as previously. Note that due to the continuity of the equations, there exists a limiting case where a finite can explain the considered values of and in absence of polarization (these points are indicated with solid circles in Fig. 1)(32). For , is not always a monotonic function of , so that there can be 2 solutions and for in some situations (not shown). We conclude that even in the single-channel case, knowing the zero and high voltage conductances of the QPC is not sufficient to determine (33). We will thus consider below the full voltage dependence of , which brings more information on the system. We will also show that noise measurements prove to be very useful to characterize unambiguously the properties of the QPC.
The top panels of Figs. 2 and 3 show the voltage dependence of for low and high values of , respectively. In this paragraph, we comment on the results for only (black full lines). For , the conductance shows peaks at , because for , one has , so that the multiple reflection paths between and interfere constructively. Between these peaks, the conductance reaches a minimum of at . The existence of a finite SDIPS can strongly modify this behavior. Indeed, for and a small enough value of , the resonance peaks of are shifted to lower voltages with (Fig. 2, top middle panel). For , these peaks are merged into a single peak and is maximum at (Fig. 2, top right panel). For high values of , no clear subgap conductance peaks occur, but the curvature of the sub-gap characteristic can be inverted by a finite SDIPS (Fig. 3, top panels). The curve is independent from only if interferences are suppressed for one spin direction, i.e. or . Above the gap, in any case, the conductance of the device drops to its normal-state value which does not depend on the SDIPS.
The current noise through the QPC can provide more information on its properties, which is of particular interest since this device is characterized by a larger number of parameters than in the spin-degenerate case. According to Eqs. (5) and (7), the conductance and the noise are not mathematically equivalent and the noise can thus provide information complementary to the conductance. The bottom panels of Figure 2 and 3 show the voltage dependence of the differential Fano factor for low and high values of , respectively. This quantity can be measured directly (see e.g. Ref. (35)) or obtained from a measurement. In this paragraph, we comment on the results for only. One can first note that is an odd function of due to . For a low and , shows subgap plateaus at values for , with . For , a dip/peak appears on these plateaus, at the resonance voltages , again due to constructive quasiparticle interferences (Fig. 2, bottom middle panel). For higher values of , the dips and peaks are smoothed, but the shape of remains sensitive to as long as (Fig. 3). Above the gap, in any case, drops to the normal state value which does not depend on the SDIPS. One can thus determine the polarization of the tunnel rates by using
Then, the SDIPS parameter and the BCS gap can be determined from the voltage dependence of the conductance and noise curves.
ii.4 Conductance and noise of the QPC at finite temperatures
We now comment on the finite temperature and curves, obtained by differentiating Eqs. (3) and (4) with respect to (see blue dashed lines in Figs. 2 and 3). At first glance, these curves simply seem to be thermally rounded. However, the finite temperature expressions of and involve elements of the matrix which do not appear in the zero-temperature expressions, thus the finite temperature curves sometimes display features not predictable from the zero temperature curves. For instance, in the bottom left panel of Fig. 3, a slight dip[peak] appears in the curve at for , an effect not present for .
In the case of a weakly transparent contact, since the Andreev resonance peaks of the conductance are shifted to lower voltages for , one can obtain, for , conductance curves similar to those obtained for and a reduced value of . The determination of the quantum point contact properties from the curves alone can thus be difficult at finite temperatures. Figure 4 shows two examples where measuring the voltage dependence of the noise can clearly bring more information on the system. The left panel presents two cases where the curves are extremely close, one case with and one case with and a reduced gap value. The corresponding curves have a strong quantitative difference, which can help to discriminate the two cases. The right panels of Fig. 4 present two cases where the curves are qualitatively similar, one case with and one case with and a reduced gap value. The corresponding curves have a strong qualitative difference, which can help to discriminate the two cases again. In the case , the differential Fano factor shows a secondary peak[dip] at [ ]. This effect, which can also be seen in the bottom middle panel of Fig. 2, never occurs for .
Before concluding this section, we point out that in most cases, experimental results on QPCs were interpreted with various models inspired from the BTK approach(17). The effect of the SDIPS was not studied in these works. However the description of a interface with a delta-function barrier naturally takes into account the SDIPS. We have checked that the spin-dependent BTK-like model corresponds to a particular case of the scattering model of Sec. II (generalized to the multichannel case), with parameters given in appendix C.
Iii Conclusion and discussion
In this paper, we have studied the conductance and noise of a superconducting/ferromagnetic () single channel Quantum Point Contact (QPC) as a function of the QPC bias voltage , using a scattering approach. We have shown that the Spin-Dependence of Interfacial Phase Shifts (SDIPS) acquired by electrons at the interface strongly modifies these signals. In particular, for a weakly transparent contact, the SDIPS produces unusual sub-gap resonances or dips in the and curves. We have shown that measuring the noise should help to gain information on the system. This fact is well illustrated e.g. by cases where the SDIPS modifies qualitatively the zero-frequency noise but not the conductance.
One should note that, so far, experiments on quantum point contacts were performed in the multichannel limit. Although single-channel contacts were already realized in the case(36), this seems more difficult in the case with the present fabrication techniques. A multichannel theory should thus be very useful. For a multichannel disordered QPC, Ref. (28) has shown that, due to time reversal symmetry, the eigenstates the QPC normal state scattering matrix are also those of . As a consequence, the multichannel generalization of and just requires a summation on the normal channels index. In the case, the calculation should be more complicated because time reversal symmetry breaking suppresses the normal channels independence (20); (29). So far, the multichannel case has been addressed with the BTK model, in which the hypothesis of a specular interface allows one to consider the normal channels as independent in spite of the time reversal symmetry breaking(22); (37). However, this approach lacks of generality since it imposes a particular relation between the transmission probabilities and the SDIPS parameters, and it is not valid for disordered interfaces. Interestingly, Refs. (29) reports on the ab-initio modelization of a particular type of disordered multichannel interface. However, to our knowledge, the multichannel case has not been studied with a more general approach so far. Our work suggests that such a study should take into account the SDIPS. Considering that, even in the single channel case, the conductance and noise of the device vary nonlinearly with the SDIPS parameter (see e.g. Eq. 12), there is no reason to expect that the effects of the SDIPS average out in the multichannel case, even in the extreme case of a SDIPS parameter randomly distributed with the channel index .
We would like to acknowledge useful discussions with C. Bruder, B. Douçot and T. Kontos. This work was financially supported by the Swiss National Science Foundation, the DFG through SFB 513 and SFB 767, the Priority Programm Semiconductor Spintronics, and the Landesstiftung Baden-Württemberg through the Kompetenznetzwerk Funktionelle Nanostrukturen.
Iv Appendix A : A symmetry property of the scattering matrix
We first consider the eigenstates of the BdG equations in a bulk S. An eigenstate with energy in the subspace has electron and hole components and such that
with the normal state hamiltonian for electrons with spin . This equation can be rewritten as
This shows that is another solution of the BdG Equations, with energy in the subspace. This property is also valid in a bulk (in this case and ).
We now consider the scattering processes at a interface. From the definition of , and outgoing wave with energy in lead writes
with the incoming state with energy in the space of lead . This induces
with for . From the previous paragraph, corresponds to an outgoing [incoming] state with energy in the subspace of lead . Equation (18) thus gives and , which generalizes Eq. (C7) of Ref. (25) to the spin-dependent case. The relation follows straightforwardly.
V Appendix B : Coefficients of the scattering matrix
When a interface can be modeled as a ballistic interface in series with a dirty interface, with the thickness of tending to zero, the scattering matrix of the interface can expressed in terms of the scattering matrix of electrons with spins on the interface and of the Andreev reflection amplitude defined in Section II.B. One finds Eqs. (8-11), and
with the Andreev transmission amplitude. The eight missing elements of can be obtained from the above Eqs. by replacing by and vice versa in the upper indices of the elements and by doing the permutations and at the right hand sides of the equations.
Vi Appendix C : Equivalent parameters of the BTK model
In some works about QPCs (17), the data were interpreted in terms of a generalization of the BTK model to the case. In this approach, no fictitious layer is used. The and leads are described with BdG equations(24), and the interfacial scattering is attributed to a delta-function barrier . A multichannel description is generally used, where channel corresponds to the transverse mode of the device, for which quasiparticles have a spin-dependent wavevector in and a spin-independent wavevector in