Odd-frequency superconducting pairing in topological insulators
We discuss the appearance of odd-frequency spin-triplet -wave superconductivity, first proposed by Berezinskii [JETP 20, 287 (1974)], on the surface of a topological insulator proximity coupled to a conventional spin-singlet -wave superconductor. Using both analytical and numerical methods we show that this disorder robust odd-frequency state is present whenever there is an in-surface gradient in the proximity induced gap, including superconductor-normal state (SN) junctions. The time-independent order parameter for the odd-frequency superconductor is proportional to the in-surface gap gradient. The induced odd-frequency component does not produce any low-energy states.
pacs:74.45.+c, 74.20.Rp, 74.50.+r
Topological insulators (TIs) are a new class of materials Hasan and Kane (2010); Qi and Zhang (2011) with an insulating bulk but with a conducting surface state. The surface state has its spin locked to the momentum in a Dirac-like energy spectrum. Superconducting TI surfaces have received a lot of attention recently, Beenakker (); Alicea () since it was predicted that Majorana modes appear in e.g. superconducting vortex cores and at superconductor-ferromagnet (SF) interfaces.Fu and Kane (2008, 2009) Experimentally, both superconducting transportSacépé et al. (2011) as well as the Josephson effectVeldhorst et al. (2012) have already been demonstrated in TIs proximity-coupled to conventional superconductors. Despite this large interest, relatively little attention has been paid to the superconducting state itself. In addition to the standard proximity effect, one could expect the spin-orbit coupling in TIs to lead to significant modifications and produce novel superconducting states that are not easily accessible in conventional superconductors. For example, it has already been demonstrated that an effective -wave pairing is induced when the TI is proximity-coupled to a conventional -wave superconductor.Fu and Kane (2008); Stanescu et al. (2010); Black-Schaffer (2011)
Quite generally, the superconducting pair amplitude, being the wave function of the Cooper pairs, needs to obey Fermi-Dirac statistics. This leads to the traditional classification into even-parity (, , …) spin-singlet and odd-parity (, , …) spin-triplet pairing. It has also been shown that the pair amplitude can be odd in frequency.Berezinskii (1974); Balatsky and Abrahams (1992) Odd-frequency spin-triplet -wave pairing has been found in spin-singlet -wave SF junctions due to spin-rotational symmetry breaking and it explains the long-range proximity effect in these junctions.Bergeret et al. (2001, 2005) Very recently, the same magnetic field induced odd-frequency pairing has also been found in TIs.Yokoyama () In superconductor-normal metal (SN) junctions translational symmetry breaking instead generates odd-frequency spin-singlet odd-parity components Tanaka et al. (2007a, b). However, the odd-parity limits the odd-frequency pairing to ballistic junctions.Tanaka and Golubov (2007)
In this article we show with a simple analysis that the effective spin-orbit coupling on the TI surface immediately induces odd-frequency spin-triplet -wave correlations, even in the absence of a magnetic field. The -wave nature of the odd-frequency component makes it robust against disorder, in sharp contrast to normal SN junctions. The odd-frequency correlations appear whenever there is an in-surface gradient in the proximity-induced spin-singlet -wave pairing, with the odd-frequency order parameter directly proportional to the in-surface gradient. We numerically calculate the odd-frequency response in several superconducting two-dimensional (2D) TI systems, including SN, SS’ junctions, and in the presence of surface supercurrents. These results point to an important missing component in the discussion on the role of proximity induced superconductivity in TIs and the odd-frequency component ought to be included in the study of low energy states and Majorana fermions in TIs. We also discuss experimental consequences of the odd-frequency pairing. We find that the analytic form of the odd-frequency response does not result in low energy states, which previously has been intimately linked to the appearance of odd-frequency components,Tanaka et al. (2012); Asano and Tanaka () but the predicted gapped spectrum could allow detection of the odd-frequency component with local tunneling probes. The spin-triplet pairing will further produce a finite Knight shift in nuclear magnetic resonance (NMR) or muon spin-rotation measurements.
Ii Analytic derivation
We start with an analytic calculation that illustrates the appearance of an odd-frequency component. The Hamiltonian that describes the contact between a TI and a conventional -wave spin-singlet superconductor (SC), see Fig 1(a), can be written as :
is the Hamiltonian describing the TI surface state at , is the Hamiltonian for the SC electronic states at , and describes the tunneling between the SC and TI. We use the low energy approximation around the Dirac point for the TI dispersion. is of the standard form, where we explicitly allow for position dependence through the site index , where then is the matrix in spin space, which we choose to be spin-singlet. We assume that the kinetic energy in the SC is a simple band and that the tunneling matrix element is nonzero only for neighboring sites at the TI-SC interface. We further assume that is dependent on the in-surface -coordinate and approximate it by a linear slope where is the unit cell size. Superconducting correlations in the TI will be induced by pairs tunneling into the TI from the superconducting side. Therefore, any induced pair amplitude in the TI will be proportional to and its derivatives at the interface. To derive the induced pair component on the TI surface, we evaluate the anomalous Green’s function . Using standard methods we find it to be proportional to
where is the Matsubara Green’s function for the electrons in the TI, is the anomalous Green’s function for a conventional superconductor, where we assume slow variations of . By going to momentum space and using the -space expansion , we can rewrite the nontrivial part of the induced pair amplitude on the surface of the TI as
This result indicates that the Josephson tunneling into the TI will immediately induce an odd-frequency spin-triplet even-momentum superconducting component.Berezinskii (1974) It is the characteristic spin-momentum locking in the Dirac surface spectrum that induces spin-triplet odd-frequency pairing in the presence of translational symmetry breaking. The -wave nature of this pairing guarantees robustness against disorder. The situation here is different from the normal metal case where only odd-frequency spin-singlet odd-parity correlations are induced by translational symmetry breaking.Tanaka et al. (2007a, b) To evaluate the on-site odd-frequency component we will assume that of the conventional supercoductor is the dominant scale and we ignore higher order terms in . The local on-site amplitude on the TI surface is given by an integral over the momenta and is proportional to . Interestingly, dependence has also been reported in heavy fermion compounds.Coleman (1993) The particular form of the gap function allows us to introduce an order parameter, i.e. the inherent parameter that is constant at equal time in the odd-frequency superconductor, see e.g. Dahal et al., 2009. The odd-frequency order parameter in our case is proportional to
One of the observable consequences of odd-frequency superconductivity is usually the appearance of sub-gap states,Tanaka et al. (2007a, b); Yokoyama et al. (2007); Tanaka et al. (2012); Asano and Tanaka () or even a low-energy continuum associated with a gapless nature of the state.Balatsky and Abrahams (1992); Dahal et al. (2009) However, we find here that the particular structure of the odd-frequency gap results in no intragap states at the lowest energies and thus this odd-frequency gap state is fully gapped. Indeed, after analytic continuation from the Matsubara axis, the local density of states (LDOS) is proportional to , which vanishes at energies below the minigap induced by the tunneling . One has to keep in mind that these features will occur in addition to the LDOS features introduced by the induced conventional BCS pairing.
Iii Numerical results
The analytical results in Eqs. (II-4) are derived for a 3D TI, but are equally valid for a 2D TI. To complement these results we explicitly calculate the odd-frequency spin-triplet superconducting correlations in the Kane-Mele 2D TI:Kane and Mele (2005)
where is now the fermion creation operator on site in the honeycomb lattice with the spin-index suppressed. Furthermore, and denote nearest neighbors and next-nearest neighbors respectively, is the nearest neighbor hopping amplitude, the chemical potential, the spin-orbit coupling, and if the electron makes a left (right) turn to get to the second bond. We set the energy and length scales by fixing and , respectively and choose , which gives a bulk band gap of . The influence of the SC can be described by an effective on-site attractive potential acting at the TI edge when it is in proximity to a SC:Black-Schaffer (2011); Sup ()
We solve self-consistently for the spin-singlet -wave mean-field order parameter . We further introduce the odd-frequency spin-triplet -wave pairing correlations:Halterman et al. (2007)
where is the time coordinate (at zero temperature). Eqs. (7-8) contain all space and time information of the odd-frequency spin-triplet response. As required by the Pauli principle, always vanishes for a self-consistent solution of , whereas the time derivative at equal times defines the odd-frequency order parameter, in analogy with Eq. (4). Also, since commutes with , the spin-triplet projection is identically zero. We also define a time-dependent quantity for -wave spin-singlet pairing: , which is related to the order parameter through .
iii.1 SN junction
From Eq. (II) we know that non-zero odd-frequency spin-triplet correlations require a gradient in the superconducting order parameter along the TI edge, i.e. needs to be finite. This is the case e.g. at any step edge in a TI proximity coupled to a SC, but a more striking example is a SN junction along the TI edge, as schematically pictured in Fig. 1(a). In the S region of the TI, we apply a constant , and, since the SC provides an ample source of charge, we also set .
In Fig. 1(b) we plot the magnitude of the self-consistent spin-singlet pairing amplitude , which shows proximity-induced superconducting pairing in the N region, as well as an inverse proximity effect (depletion of Cooper pairs) on the S side of the junction. The odd-frequency response is shown in Figs. 1(c-e), with only non-zero in the SN interface region where the gradient of is finite. For small , is sharply peaked at the interface with the peak height rapidly increasing with , as seen in Fig. 1(e). For (small frequencies ) the height is approximately constant, but the peak instead becomes broader, with the average width , defined as the ratio of the total weight of the peak to its height , increasing roughly linearly with . In Fig. 1(e) we also plot the height and width of the gradient of the spin-singlet response for a direct comparison. For large , is directly related to , supporting the analytic result in Eq. (II), whereas at small the direct relation breaks down for the peak height. In Fig. 1(f) we finally compare the odd-frequency order parameter with . As predicted in Eq. (4), we find these two quantities to be directly proportional.
The odd-frequency pairing in a SN TI junction is strikingly different from that of a conventional SN junction. The odd-frequency response in the TI case has even-parity spin-triplet symmetry, which is robust against impurity scattering and can thus survive even in the diffusive regime,Tanaka and Golubov (2007) in contrast to the odd-parity spin-singlet symmetry in a regular SN junction.Tanaka et al. (2007a, b) Related to this disorder robustness, we also find that is insensitive to any Fermi level mismatch at the SN interface, created by using different chemical potentials in N and S. In a normal SN junction, the odd-frequency pairing quickly disappears when the transparency of the interface is reduced.Tanaka et al. (2007b) Finally, to investigate the influence of the odd-frequency pairing on the low-energy spectrum, we study SS’ junctions, where the two S regions have different pair potentials . We find no evidence for a reduced gap or intragap states in both sharp and extended interface junctions.Sup () This expands our analytical low-energy result to exclude any intragap states, and is again in sharp contrast to conventional SN and SF junctions.Tanaka et al. (2007a, b); Yokoyama et al. (2007); Tanaka et al. (2012); Asano and Tanaka ()
Odd-frequency correlations can also be generated in a homogenous TI-SC system if a supercurrent is applied in-surface, since the current is proportional to the gradient of the phase of the superconducting order parameter. We model such a system by setting , with being the (fixed) phase winding which is proportional to the current, and solve self-consistently for . We find that and , i.e. both and have the same phase winding for all , but the amplitudes and phase off-sets are dependent on .
In Fig. 2(a) we plot the (position-independent) magnitudes of and as function of for two different values of . Similar to the SN junction, increases rapidly for small . For small currents continues to increase even for large , even though less steeply. This is in contrast to which, apart from small oscillations, decreases with increasing time. For larger currents we see a sharp downturn in at intermediate time values. Before this downturn we find that is directly proportional to , as expected from its relation to in Eq. (II). However, for times past the downturn, instead decreases with increasing . The phase off-set parameters and also tracks each other (with the expected shift) before the downturn in , but past this time the correlation between them is lost. Thus, we find that and are only tracking each other in a finite -range, which set by the current. In Fig. 2(b) we focus on the behavior at . Both the odd-frequency order parameter and are linear in , with their ratio being a constant for all currents. This shows that the odd-frequency order parameter is always directly proportional to the current (phase winding), in agreement with Eq. (4), and thus applying a supercurrent offers an experimentally easy way to tune the strength of the odd-frequency pairing.
iii.3 Rashba spin-orbit coupling
So far we have only discussed spin-triplet pairing. Equal spin-triplet pairing ( is produced at the TI edge if a term misaligned with the spin-quantization axis is present in the Hamiltonian. This happens for Rashba spin-orbit coupling Kane and Mele (2005):
which is present when symmetry is broken. Here is the unit vector along the bond between sites and . is still in the non-trivial topological phase with a single Dirac cone at the edge for .
We find that decreases only slightly when introducing a finite , and even for , we have . For a SN junction we find that is localized to the interface region, having a similar dependence as , see Figs. 3(a-c). Here we also see that extends somewhat farther into the TI, especially for larger . We further analyze by studying its two spin parts . Surprisingly, is not only non-zero at the SN interface, but in the whole S region, as shown in Fig. 3(d-e). In fact, we find that a non-zero is generated whenever is non-zero, i.e. for a finite order parameter gradient perpendicular to the surface of the TI. Since the low-energy density of states varies dramatically between the surface and the bulk of a TI, there will always be a strong such gradient for proximity-induced superconductivity in a TI. In the bulk of the S region , and thus the criterion for to be non-zero is the same as for , i.e. a finite gradient in-surface gradient . In terms of the -vector for the odd-frequency pairing, we find that it is always real, yielding a unitary spin-triplet state.
In summary, we have shown that odd-frequency spin-triplet -wave pairing is present in a TI proximity-coupled to a conventional -wave spin-singlet superconductor in zero magnetic field. The time-independent order parameter for the odd-frequency pairing is proportional to the in-plane gradient of the induced -wave gap. This disorder robust odd-frequency response is an immediate consequence of the spin-momentum locking in the TI surface state. We have explicitly demonstrated the occurrence of odd-frequency correlations not only in SN and SS’ junctions at a 2D TI edge, but also when a supercurrent is applied along the edge. In terms of experimental observables, we find no evidence of subgap states in the presence of odd-frequency pairing, due to its particular frequency dependence. The gapped LDOS could allow the detection of the odd-frequency component with local tunneling probes. Furthermore, the spin-triplet component could produce a Knight shift in NMR or muon spin-rotation measurements.
Acknowledgements.We are grateful to E. Abrahams, M. Fogelström and J. Linder for discussions. AMBS was supported by the Swedish research council (VR) and thanks LANL for hospitality where this work was initiated. Work at LANL was supported by US DoE Basic Energy Sciences and in part by the Center for Integrated Nanotechnologies, operated by LANS, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under contract DE-AC52-06NA25396.
Appendix A Supplementary material
In this supplementary material we provide numerical data showing the absence of subgap states in the presence of odd-frequency pairing in a topological insulator (TI) induced by a gradient in the proximity-induced conventional -wave superconducting state. In order to do so accurately we explicitly model the microscopic interface between a superconductor (SC) and a two-dimensional (2D) TI displayed in Fig. 4.
The total Hamiltonian is , where defines the TI and is given by Eq. (5) in the main text, whereas
defines the SC and the coupling between the TI and the SC, respectively. The SC is defined on a square lattice with nearest neighbor hopping and an on-site spin-singlet -wave pairing from an attractive Hubbard term. The coupling between the TI and the SC is by a tunneling element acting between nearest neighbors across the interface. We treat Eq. (10) self-consistently within mean-field theory by using the self-consistency condition .
In Fig. 5 we show a typical interface when the pairing potential is set to vary along the interface, i.e. in the -direction, in order to produce a gradient along the TI edge, but is constant along the -direction. Region A has a constant giving in the bulk of the SC and region C has a similarly constant giving . In the intervening region B there is a linear rise of between these two values. Finally, the interface is put on a cylinder, making a sharp S-S interface at D. In Fig. 5(a) we clearly see how the magnitude of the induced -wave pairing in the TI reflects the change in along the interface. Figure 5(b) displays the magnitude of the odd-frequency pairing order parameter , which is only non-zero in the B and D regions. The odd-frequency pairing leaking into the SC is at least an order of magnitude smaller and we can not deduce any physical consequences in the SC from this back action. Figures 5(c)-(f) shows the corresponding local density of states (LDOS) in the A-D regions. Starting from the far left, these LDOS plots show the unperturbed left TI edge, with a constant DOS due to the one-dimensional surface Dirac cone. When we see the constant TI bulk gap of 1 before the right TI edge which is gapped by the induced superconductivity from the SC. The gap in the bulk of the superconductor equals which varies significantly in , but the induced gap in the TI surface state varies much less. Most notably, we see no evidence of any subgap states in the C and D region where odd-frequency pairing is present. In fact, the right TI edge in the C and D regions looks remarkably similar to a simple interpolation between the edge states in regions A and B.
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