Detecting Topological Superconductivity with \varphi_{0} Josephson Junctions

Detecting Topological Superconductivity with Josephson Junctions

Constantin Schrade, Silas Hoffman, and Daniel Loss Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
July 5, 2019
Abstract

The interplay of superconductivity, magnetic fields, and spin-orbit interaction lies at the heart of topological superconductivity. Remarkably, the recent experimental discovery of Josephson junctions by Szombati et al. bib:Kouwenhoven2016 (), characterized by a finite phase offset in the supercurrent, require the same ingredients as topological superconductors, which suggests a profound connection between these two distinct phenomena. Here, we theoretically show that a quantum dot Josephson junction can serve as a new qualitative indicator for topological superconductivity: Microscopically, we find that the phase shift in a junction of wave superconductors is due to the spin-orbit induced mixing of singly occupied states on the qantum dot, while for a topological superconductor junction it is due to singlet-triplet mixing. Because of this important difference, when the spin-orbit vector of the quantum dot and the external Zeeman field are orthogonal, the -wave superconductors form a Josephson junction while the topological superconductors have a finite offset by which topological superconductivity can be distinguished from conventional superconductivity. Our prediction can be immediately tested in nanowire systems currently used for Majorana fermion experiments and thus offers a new and realistic approach for detecting topological bound states.

pacs:
74.50.+r, 85.25.Cp, 71.10.Pm

Non-abelian anyons are the building blocks of topological quantum computers bib:Nayak2008 (). The simplest realization of a non-abelian anyon are Majorana bound states (MBSs) in topological superconductors (TSs) bib:Alicea2012 (). It has been proposed that such a TS can be induced by an -wave superconductor (SC) in systems of nanowires with spin-orbit interaction (SOI) subject to a Zeeman field bib:Lutchyn2010 (); bib:Oreg2010 (); bib:Klinovaja2012 (); bib:Kouwenhoven2012 (), in chains of magnetic atoms bib:Yazdani2013 (); bib:vOppen2013 (); bib:Perge2014 (); bib:Meyer2015 () and in topological insulators bib:Fu2008 (); bib:Hart2013 (); bib:Pribiag2015 (); bib:Wiedenmann2015 (); bib:Bocquillon2016 (); bib:Deacon2016 (). However, providing experimental evidence for the existence of this new phase of matter has remained a major challenge.

Here we present a new qualitative indicator of MBS based on Josephson junctions (JJs). In JJs the Josephson current is offset by a finite phase, , so that a finite supercurrent flows even when the phase difference between the superconducting leads and the magnetic flux enclosed by the Josephson junction (JJ) vanishes. Such JJs have been discussed in systems based on unconventional superconductors bib:Larkin1986 (); bib:Yip1995 (); bib:Sigrist1998 (); bib:Kashiwaya2000 (); bib:Asano2005 (), ferromagnets bib:Buzdin2008 (); bib:Chan2010 (); bib:Goldobin2011 (); bib:Sickinger2012 (), quantum point contacts bib:Reynoso2008 (), topological insulators bib:Dolcini2015 (), nanowires bib:Yokoyama2014 (); bib:Campagnano2015 () and diffusive systems bib:Alidoust2013 (); bib:Bergeret2015 (). Recently, the connection between JJs based on nanowires and TSs has also been discussed bib:Nesterov2016 (). Most relevant for the present work, the emergence of a JJ was theoretically predicted bib:DellAnna2007 (); bib:Egger2009 (); bib:Egger2013 () in a system of a quantum dot (QD) with SOI subject to a Zeeman field when coupled to -wave superconducting leads and observed in recent experiments bib:Kouwenhoven2016 (). Interestingly, the ingredients for observing a JJ in this type of system largely overlap with those required to generate MBSs. In this work, we focus on two models for JJs based on QDs which, compared to previous studies bib:DellAnna2007 (); bib:Egger2009 (); bib:Egger2013 (), are in the singlet-triplet anticrossing regime. In the first model, two -wave SCs are tunnel coupled via a two-orbital QD with SOI and subject to a Zeeman field, see Fig. 1(a), wherein we find a finite phase shift caused by the SOI-induced mixing of singly occupied states of the QD. In the second model, replacing the two -wave SCs by two TSs, see Fig. 1(b), we again find a finite phase shift which results from the singlet-triplet mixing of the doubly occupied QD states. When the spin-orbit vector and the magnetic field are orthogonal, the system is invariant under a composition of time reversal and mirroring in the plane perpendicular to , under which the superconducting phase goes to opposite itself; because the energy must be invariant under this symmetry, there can be no terms that are odd in the superconducting phase difference in the Hamiltonian and thus no non-trivial phase offset bib:Chan2010 (); bib:Flensberg2016 (). However, unlike the ground state of the SC leads, the ground states of the TS leads transform nontrivially under the above transformations and we thus anticipate a nonzero phase shift. Indeed, we show that the phase shift is equal to for the -wave superconducting leads, while for the TSs leads, which can, consequently, be used as a new qualitative indicator of MBSs.

Josephson junction models. Our starting point for both of the JJ models outlined above is the Hamiltonian

(1)

where corresponds to the model with -wave SC leads and TS leads, respectively. The first term in this expression is the Hamiltonian of an isolated QD. Here, describes a QD with two orbitals at energy difference with respect to a gate voltage . The particle number operator of orbital is with the electron annihilation operator with spin quantized along the -axis in orbital . The intraorbital (interorbital) Coulomb interaction strength is (). Furthermore, describes a Zeeman field along the -axis of magnitude with the electron -factor and the Bohr magneton. Lastly, describes the SOI on the QD, where , in which , is the angle of the SOI vector with respect to the Zeeman field, and is the vector of Pauli matrices.

The second term in Eq. (1) describes the isolated superconducting leads. For the first model, , where is the quasiparticle annihilation operator in SC with momentum , pseudospin , and energy with the superconducting gap and the single-electron dispersion relation in the normal metal state. The non-degenerate ground state of the -wave superconductors, , is defined so that . For the second model, we assume that the localization length of the MBS wavefunctions is much smaller than the length of TSs. We also neglect contributions of bulk quasiparticles which is valid for energies much smaller than the energy gap. Consequently the MBSs are at zero energy and . As a result, the ground state of the TS leads is four-fold degenerate which, upon choosing a fixed parity subspace, becomes two-fold degenerate. In the following, we consider the odd parity subspace, however, the results for the even parity ground state subspace are identical.

The last term in Eq. (1) describes the tunnel coupling between the superconducting leads and the QD. For the first model, it is given by

(2)

with being the annihilation operator of an electron with momentum and spin in SC . It is related to the quasiparticle operators by and with coherence factors and . The tunneling Hamiltonian also contains the superconducting phase of SC and real, spin and momentum-independent tunneling amplitudes . For the second model, the coupling of the TSs and the QD is given by

(3)

with being the MBS in TS which is spatially closest to the QD bib:Lopez2013 (). We assume that its partner at the opposite end of the TS does not couple to the QD. However, they form non-local fermionic operators and .

Figure 1: (Color online) Setups for JJs. (a) Two -wave SCs (red) are tunnel coupled via a QD (yellow) with two orbitals and . The QD is subject to an external Zeeman field at some relative angle to its SOI axis . (b) Same visual encodings. The SCs are replaced by two TSs (blue). The QD now couples to the two inner MBS (crosses) of the TSs. (c) Spectrum of the bare QD as a function of for the double occupancy sector. Red bands contribute to our effective description, green bands do not. We have chosen  meV, ,  meV and  meV,  meV, so that  mT. (d) Same as (c) but for the single occupancy sector with mT.

We now proceed with a discussion of in the regime of , which is common in typical experiments bib:Kouwenhoven2016 (). First, we address the case of a doubly occupied dot, . For , the spectrum consists of three singlet (triplet) bands which are constant (split) as a function of the Zeeman field. As experimentally observed in bib:Fasth2007 (), for finite and , the singlet and triplet bands anticross, see Fig. 1(c). In all following discussions, we operate the QD in the regime close to the anticrossing of the singlet and the triplet which occurs at the Zeeman field . Here, is the vacuum state on the dot. The effective Hamiltonian, valid to lowest order in , which acts in the two-level subspace spanned by and is . The spectrum of is given by with corresponding orthonormal eigenstates

(4)

Here, are real functions of the system parameters, see bib:supplemental ().

Second, we discuss the case of a singly occupied dot, . For , the energy levels for opposite spins split as a function of the Zeeman field. For finite and , an energy gap opens up at the crossing point of the spin-up band in orbital and the spin-down band in orbital , see Fig. 1(d). We will denote the four eigenvalues of the singly occupied sector by for . The corresponding orthonormal eigenstates are given by

(5)

Here, , are real functions of the system parameters, see bib:supplemental (). The relative imaginary unit in both Eq. (4) and Eq. (5) is due to the SOI. We adjust the filling and the gate voltage of the QD, so that its ground state is given by while its first excited states are given by and for some fixed . The seperation between to the states with is assumed to be large, . Finally, the remaining occupancy sectors of the QD, whose energies are much larger than the QD-lead coupling, are not relevant for our results and are hence omitted.

Detecting topological superconductivity. In order to calculate the superconducting current, we tune the chemical potential of the superconductors close to the level. We require for the SC JJ that with the normal-state density of states of the leads at the Fermi energy and for the TS JJ that , so that in both cases the states and on the QD serve as virtual tunneling states. Our approach is valid for angles where is a critical angle determined by the conditions above bib:supplemental (). Furthermore, we work in a temperature regime of . The effective tunneling Hamiltonian () valid up to fourth (second) order in the tunneling amplitudes acting on the ground state of the isolated dot and -wave (odd parity) ground state of the uncoupled leads is

(6)

with and , The first term in Eq. (6) arises due to Cooper pair tunneling across the SC JJ or non-local fermion tunneling across the TS JJ which splits the ground states of the TS leads. The second term is an energy offset, due to processes for which there is no such transport. At zero temperature, the Josephson current, defined by with the ground state energy of the coupled system, is given by

(7)

where the critical current is . Because in the TS case the ground state is a function of , the sign of the Josephson energy also depends on the phase difference: when is the ground state energy and otherwise. In the SC case the ground state is independent of and therefore .

Figure 2: (Color online) (a) On the left hand side of the equality: the virtual tunneling sequence which leads to in terms of the eigenstates of the effective dot Hamiltonian. Because the states and are superpositions of the doubly and singly occupied eigenstates of in the absence of SOI, respectively, can be written as a sum of the virtual tunneling processes in that basis; two examples of which, contributing to the and terms, are shown on the right side of the equality. Electron spin (quasiparticle pseudospin) is denoted by (). Notice that it is the superposition of singly occupied dot states, e.g. in the process (solid red box), that leads to a finite contribution and therefore a finite . (b) Same as (a) but for the case of . As compared with the SC case, it is the singlet-triplet mixing that induces a finite phase shift, e.g. in the contribution to . Here, or are the eigenvalues of and .

Notice that there is a finite phase shift only when . As such, we now turn to a more detailed comparison of the coefficients in Eq. (6). For the BCS JJ,

(8)

The prefactor , which is not relevant for the phase shift , includes the coherence factors and energy denominators picked up in the perturbation theory bib:supplemental (). Thus, the SC JJ exhibits in general a finite phase shift, when . For , the sign of the supercurrent is determined by and . We now explain the sequence of intermediate states which leads to the contributions in Eq. (Detecting Topological Superconductivity with Josephson Junctions). Our initial state on the QD is . To reach the first intermediate state, we remove one electron from the QD, whereupon its state changes to , and we create an excitation on SC 1 (2). Next, we use the superconducting condensate to create an electron on the QD and an excitation on SC 2 (1). This changes the QD state to bib:supplemental (). Third, we return to by absorbing one of the dot electrons and the excitation on SC 1 (2) into the condensate. Finally, we go back to the initial state by transferring the excitation on SC 2 (1) back on the QD. Because is a superposition of different singly occupied QD orbitals, in the first and third step of this sequence the electron on the QD switches orbitals while preserving spin with amplitude while it stays in the same orbital with amplitude or . Thus, the contribution originates from processes in which the electron switches orbitals exactly once, while the remaining processes yield the contribution. The mixing of singlet and triplet states in gives an overall prefactor, which due to the normalization of the states, drops out of Eq. (Detecting Topological Superconductivity with Josephson Junctions). Most interestingly, for the case when the relative angle between Zeeman field and SOI axis is the phase shift vanishes, see Fig. 3(a). On a microscopic level, this is because now the SOI only mixes opposite spins in different orbitals, for and for bib:supplemental (). This restricts the number of allowed virtual tunneling processes. In particular, processes which move the spin between the orbitals without flipping it are prohibited, and see Fig. 2(a). However, unlike the SC JJ, the TS JJ still allows for nonzero phase shift in that case, see Fig. 3(a). At , we find that the coefficients in Eq. (6) for the TS JJ when are given by

(9)

where the prefactor includes the energy denominators of the perturbation theory bib:supplemental (). In comparison to the SC JJ, the sign of the supercurrent at in the TS JJ is determined by parity . If the parity fluctuates, the supercurrent exhibits fluctuations as well. So the observation of a phase shift requires sufficiently long parity life times which can be up to minutes bib:Kouwenhoven2015 (). When we find that and . For we recover the same feature when , see Fig. 3 in bib:supplemental (). In both cases this is the special case of a JJ for TS. We now focus on the case when . Recalling that is a superposition of singlet and triplet states, we identify the processes that contribute to Eq. (9): comes from virtual tunneling sequences taking a singlet to a triplet state, with amplitude , and the corresponding sequences taking a triplet to the singlet state, with an amplitude . When the order in which the nonlocal fermion is created or destroyed is opposite between these processes, the tunneling sequences differ in phase by and acquire the same tunneling coefficients so that their sum is proportional to , see Fig. 2(b) and bib:supplemental (). Distinctly, originates from sequences that take the singlet () or triplet () to itself. In both cases there exist two sequences that, again, differ in phase by but have the same tunneling coefficients, so that their sums are .

Figure 3: (Color online) (a) Phase shift (left panel) and Josephson current at (right panel) for and with . System parameters are chosen as in Fig.1 with ,  meV,  meV,  meV and  meV. Compared to the SC JJ the phase shift (Josephson current at ) is non-zero for the TS JJ. (b) Experimental proposal. A nanowire (dark grey) is proximity coupled to an -wave SC (red). An electric field along the -direction at the SC-wire contact induces a wire SOI axis along the -direction. An external Zeeman field is applied orthogonal to . A QD (yellow) is created by depleting the electron density via gates (light grey regions). A backgate contacted to the QD (not shown) induces an electric field along the direction axis and hence a SOI axis along the direction. To measure and , is rotated in the plane orthogonal to .

Discussion. We propose an experiment based on our observation that in general . We consider a nanowire setup similar to bib:Kouwenhoven2016 (), see Fig. 3(b). The wire SOI axis , induced by an electric field along the axis at the SC-wire contact, is orthogonal to an external Zeeman field . Via gating we create a tunnel coupled QD as a short slice in the wire. Furthermore we contact the QD to a backgate generating an electric field along the axis so that the dot SOI axis is along the -axis. We adjust the size of the QD so that the singlet-triplet anticrossing occurs for Zeeman fields close to the topological phase transition, where is the chemical potential of the SCs and . Also we adjust the gate voltage and the filling of the dot so that its ground state is , while its first excited states are and . Lastly, the chemical potential of the nanowire leads is tuned to . We now position the Zeeman field orthogonal to both and . When we now tune the system across the topological phase transition by varying , we observe a change in the phase shift of the Josephson current from to some non-trivial . Moreover, we can even determine the full dependence of the phase shift and Josephson current by rotating in the plane orthogonal to . Interestingly, for typical system parameters of a nanowire QD JJs we find that, at zero phase difference between the leads, pA while nA, which corresponds to an increase by three orders of magnitude.

Conclusions. We have introduced a new qualitative indicator for the detection of topological superconductivity based on a QD JJ. We found that for this setup the trivial SCs always form a JJ while the TSs can form a JJ with . We have also seen that this change in phase shift is accompanied by a significant increase in the magnitude of the critical current. These observation can be probed by simple modifications of recent experimental setups in nanowire QD JJs bib:Kouwenhoven2016 ().

Acknowledgments. We acknowledge support from the Swiss NSF and NCCR QSIT. We are grateful to J. Klinovaja for useful comments.

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Supplemental Material to ‘Detecting Topological Superconductivity with Josephson junctions’

Constantin Schrade, Silas Hoffman, and Daniel Loss

Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland

Appendix A A Quantum dot with spin orbit interaction in a Zeeman field

This first section of the supplemental material provides a more detailed discussion of the model for an isolated QD with SOI subject to an external Zeeman field as given by in the main text. The Hilbert space of the system is spanned by the occupation number states

(10)

where is the occupation number of an electron with spin in orbital . Since the total number of electrons on the QD is conserved, we can adress each sector with fixed total occupation number separately.

a.1 Double occupancy sector

We start with an analysis of the double occupancy sector. A basis is given by the singlet states

(11)

and the triplet states

(12)

Representing in terms of these basis states we find that

(13)

Here, the top left block acts on the singlet subspace, while the bottom right block acts on the triplet subspace and the off-diagonal blocks contain the SOI which couples the singlet to the triplet subspace. The spectrum of is depicted in Fig. 1(c) of the main text. The effective Hamiltonian, valid to lowest order in , which acts in the two-level subspace spanned by and is

(14)

It contains the bare energies of the singlet and the triplet on its diagonal. The SOI interaction then couples these levels via the off-diagonal terms. The spectrum of is given by

(15)

We see that the effect of the SOI is the opening of an energy gap at the crossing point of the bare singlet and triplet energy levels. In terms of the angle between the Zeeman field and the SOI axis, the gap is maximal when and vanishes when . The eigenstates of are

(16)

where the coefficients are given by

(17)

The mixing of the singlet and the triplet is minimal when or and it is maximal when .

a.2 Single occupancy sector

We next discuss the single occupancy sector of the QD which is spanned by the basis states

(18)

The matrix representation of in terms of these basis states is given by

(19)

Here, the top left block acts on the subspace of orbital , while the bottom right block acts on the subspace of orbital . The off-diagonal blocks contain the SOI which couples the orbital to the orbital. The spectrum of is depicted in Fig. 1(d) of the main text and is given by

(20)

Here, for , is the Kronecker delta. The eigenstates of are of the form

(21)

We now determine the coefficients and for the different relative angles between Zeeman field and SOI axis.

a.2.1 Zeeman field and SOI axis are orthogonal ()

For , the SOI is proportional to so that we expect the eigenstates of to be linear combinations of opposite spins in different orbitals. Indeed, we find that the only coefficients which are non-zero are given by

(22)

The remaining coefficients are vanishing,

a.2.2 Zeeman field and SOI axis are parallel ()

In the case of , the SOI is proportional to . Consequently, we expect the eigenstates of to be mixtures of same spins in different orbitals. For , we find that the non-vanishing coefficients are given by

(23)

The remaining coefficients are all zero, . For , we find find that

(24)

As before, the remaining coefficients vanish, .

a.2.3 Zeeman field and SOI axis are non-orthogonal and non-parallel ()

We assume that ; for we note that is already diagonal. When , the SOI is proportional to both and . This means that the SOI mixes states of all spin species in all orbitals. We find that the components of the respective eigenstates are given by

(25)

where is a normalization factors which we choose so that . The normalization also ensures that when the expressions above reproduce the the corresponding limiting cases.

Appendix B An s-wave Superconductor Josephson junction

This second section of the supplemental material gives a more detailed discussion of the SC JJ described by in the main text.

b.1 Effective tunneling Hamiltonian

Figure 4: Tunneling sequences (up to hermitian conjugation) of the SC JJ for contributions . We use the basis . Filled (empty) dots are used to visually represent a filled (an empty) level.

We begin with a derivation of the effective tunneling Hamiltonian . Compared to the main text, we allow for a slightly more general tunneling Hamiltonian with spin-dependent tunneling amplitudes,

(26)

Because it is only the relative phase between the two superconductors which is a physical quantity, we assume that while . We now briefly discuss the different tunneling processes which can occur in the system. Therefore, we rewrite in terms of the quasiparticle operators,