Topological superconductivity in planar Josephson junctions – narrowing down to the nanowire limit
We theoretically study topological planar Josephson junctions (JJs) formed from spin-orbit-coupled two-dimensional electron gases (2DEGs) proximitized by two superconductors and subjected to an in-plane magnetic field . Compared to previous studies of topological superconductivity in these junctions, here we consider the case where the superconducting leads are narrower than the superconducting coherence length. In this limit the system may be viewed as a proximitized multiband wire, with an additional knob introduced by the phase difference between the superconducting leads. A combination of mirror and time-reversal symmetry may put the system into the class BDI. Breaking this symmetry changes the symmetry class to class D. The class D phase diagram depends strongly on and the chemical potential, with a weaker dependence on . In contrast, the class BDI phase diagram depends strongly on both and . Interestingly, the BDI phase diagram has a “fan”-shaped region with phase boundaries which move away from linearly with . The number of distinct phases in the fan increases with increasing chemical potential. We study the dependence of the JJ’s critical current on , and find that minima in the critical current indicate first-order phase transitions in the junction only when the spin-orbit coupling strength is small. In contrast to the case of a JJ with wide leads, in the narrow case these transitions are not accompanied by a change in the JJ’s topological index. Our results, calculated using realistic experimental parameters, provide guidelines for present and future searches for topological superconductivity in JJs with narrow leads, and are particularly relevant to recent experiments on InAs 2DEGs proximitized by narrow Al superconducting leads (A. Fornieri et al., arXiv:1809.03037).
Majorana zero modes (MZMs) Alicea (2012); Elliott and Franz (2015); Leijnse and Flensberg (2012); Beenakker (2013); Leijnse and Flensberg (2012); Stanescu and Tewari (2013); Sarma et al. (2015); Beenakker and Kouwenhoven (2016); Aguado (2018); Lutchyn et al. (2018) are not only of fundamental interest but can also be used as the building blocks for a fault tolerant quantum computation Kitaev (2003); Nayak et al. (2008). These MZMs exist in the vortex core of two-dimensional (2D) topological superconductors (TSCs) Ivanov (2001); Read and Green (2000) or at the edge of 1D TSCs Kitaev (2001). The theoretical proposals on TSCs Kitaev (2001); Read and Green (2000); Lutchyn et al. (2010); Oreg et al. (2010); Fu and Kane (2008); Choy et al. (2011); Nadj-Perge et al. (2013); Brydon et al. (2015) have triggered a tremendous amount of experimental effort to realize TSCs in different platforms ranging from 1D nanowire Mourik et al. (2012); Rokhinson et al. (2012); Deng et al. (2012); Das et al. (2012); Churchill et al. (2013); Finck et al. (2013); Albrecht et al. (2016); Gül et al. (2018); Chen et al. (2017); Deng et al. (2016); Suominen et al. (2017); Nichele et al. (2017); Zhang et al. (2018, 2017); Sestoft et al. (2018); Deng et al. (2018); Laroche et al. (2019); van Veen et al. (2018); de Moor et al. (2018); Grivnin et al. (2018); Vaitiekėnas et al. (2018), topological insulators Xu et al. (2015); Sun et al. (2016), and ferromagnetic atomic chains Nadj-Perge et al. (2014); Feldman et al. (2017); Jeon et al. (2017); Pawlak et al. (2016). Recently, a two-dimensional electron gas (2DEG) with strong spin-orbit coupling (SOC) and proximitized by two spatially separated superconductors (SCs), thus forming a Josephson junction (JJ), was proposed as a new platform to engineer TSC Pientka et al. (2017); Hell et al. (2017). Compared to the other setups, this system has the advantage of being able to be tuned into TSC by changing not only the strength of the applied magnetic field but also the superconducting phase difference across the JJ Pientka et al. (2017); Hell et al. (2017). Recent experiments Fornieri et al. (2018); Ren et al. (2018) using this setup have observed some evidence of the Zeeman- and phase-tunable topological superconductivity in form of zero-bias conductance peaks.
In the presence of an additional symmetry which is a product of the mirror and time-reversal symmetries Pientka et al. (2017); Hell et al. (2017), the topological planar JJ belongs to the BDI symmetry class in the tenfold classification Ryu et al. (2010); Kitaev (2009), characterized by a topological invariant . This invariant corresponds to the number of MZMs at the junction’s end. Breaking this symmetry changes the symmetry class to D with a index. It was shown in Ref. Pientka et al. (2017) that for JJs with SCs whose width is much larger than the coherence length , the class BDI and D phase diagrams have weak dependence on the chemical potential but depend strongly on both the Zeeman field and . Moreover, if is not externally controlled, then as the Zeeman field is varied the system undergoes a first-order topological phase transition (TPT) where the phase of the ground state jumps from (trivial) to (topological) or vice versa. This phase jump is accompanied by a minimum in the critical current which can be used as an experimental probe for the TPT.
Motivated by recent experiments on InAs 2DEGs proximitized by narrow Al SCs Fornieri et al. (2018), in this Letter we study the topological superconductivity in planar JJs with narrow SCs (), see Fig. 1. We further examine the relation between this system and a 1D multiband nanowire TSC Lutchyn et al. (2011); Stanescu et al. (2011). We establish numerically and analytically that the class D phase diagram depends strongly on the in-plane magnetic field applied along the junction, but only weakly on the superconducting phase difference . This is due to the presence of multiple normal reflections that originate from the interfaces of the SC leads with the vacuum. At the same time, the normal reflections make the phase diagram more sensitive to the 2DEG chemical potential. In contrast, the BDI phase diagram is strongly dependent on both and . Crucially, it exhibits a “fan”-shaped region emerging from at where the BDI phase boundary lines diverge away from linearly with . The number of distinct BDI phases in the fan increases with chemical potential. In addition, the critical current through the junction has minima as a function of . These minima correspond to discontinuous transitions of the value of that minimizes the free energy. However, unlike the case of wide SC leads, here these transitions are not necessarily accompanied by a change in the topological index.
The Hamiltonian for the planar JJs [Fig. 1(a)] in the Nambu basis is
with being the annihilation operator of an electron with spin and momentum . Throughout most of this paper, we assume the JJ to be infinitely long. The Pauli matrices and act in particle-hole and spin spaces, respectively, and . Here, is the effective electron mass in the 2DEG, is the chemical potential, is the Rashba SOC strength and is the Zeeman energy due to the applied in-plane magnetic field . The proximity-induced pairing potential in the 2DEG is [see Fig. 1(a)]
The Hamiltonian in Eq. (Topological superconductivity in planar Josephson junctions – narrowing down to the nanowire limit) anticommutes with the particle-hole operator where denotes complex conjugation. When and or , the Hamiltonian commutes the standard time-reversal operator (where ) and thus it belongs to the symmetry class DIII Ryu et al. (2010); Kitaev (2009). It also commutes with the mirror operator along the - plane, i.e., . While the and symmetries are broken when and/or , the Hamiltonian remains invariant under the product Pientka et al. (2017). Since , the system belongs to the class BDI. The presence of and symmetries implies that the Hamiltonian anticommutes with the chirality operator . Breaking the symmetry reduces the symmetry class from BDI to D.
To obtain the phase diagrams, we calculate the topological invariant following Ref. Tewari and Sau (2012). Since the chirality operator obeys , it has eigenvalues . In the basis where is diagonal, the Hamiltonian is block off-diagonal (since ). The topological invariant () is calculated from the phase of the determinant of the off-diagonal part. The winding of this phase from to gives the topological invariant of the class BDI. The index of class D is simply the parity of , i.e., Tewari and Sau (2012); Kitaev (2001).
Figure 1(b) shows the class D phase diagram of a JJ with narrow leads (), calculated numerically. The phase diagram shows a sequence of TPTs from the trivial () to topological () phases. In contrast to the case of wide SC leads Pientka et al. (2017), the phase boundaries depend only moderately on .
The BDI phase diagram [Fig. 1(c)], on the other hand, depends strongly on both and . For , the BDI topological invariant is , except at where the gap closes. As increases, the gap closing point expands into a fan-shaped region containing phases with different values of .
These features of the phase diagram can be understood qualitatively as follows. Phase transitions where changes require gap closings at , while transitions with an even change in occur as a consequence of gap closings at the Fermi wavevector, . In the limit where , the system can be treated as a multiband quantum wire Lutchyn et al. (2011); Stanescu et al. (2011), with an induced gap that is smaller than the energy spacing between subbands. For generic values of the chemical potential , the spectrum at is gapped for all , and therefore the phase diagram depends only weakly on . This state of affairs changes at special values of and , where the chemical potential enters a new subband (see Sec. B of Ref. sup () for details). Independently of , a gap closing occurs at for and . This gap closing occurs as a consequence of the mirror symmetry, where the effective induced gap, which is a spatial average of the gap of two symmetric SC leads, vanishes for and .
As shown in Fig. 2, the gap closing point at and expands into a “fan”-shaped region in the phase diagram with phase boundaries which move away from with slopes which are linearly proportional with and decrease with increasing . To understand this fan, in the following we derive analytically the dependence of the superconducting gap in a given subband on the and . For simplicity, we work in the limit where . The dispersion of the 2DEG, shown in Fig. 3(a), exhibits two concentric circular Fermi surfaces. SOC locks the spin orientation to the momentum, such that the outer and inner Fermi surfaces (labeled by ) have different in-plane spin orientations. When a Zeeman field is applied, the spin tilts towards the Zeeman field direction. Moreover, to the leading order in the Zeeman field also shifts the two Fermi surfaces uniformly along in the opposite direction by [see Fig. 3(a)].
We now take into account the finite size of the system in the direction. We denote the transverse wavefunctions of the normal Hamiltonian () by , where is the band index, , and we label each subband according to the eigenvalue () of the state at in the limit [see Fig. 3(b)]. A weak Zeeman field mixes the two mirror eigenvalues and opens a gap at but does not strongly affect the wavefunctions at , such that we may keep using the labeling of the subbands. Note that the walls at mix states with different values of (See Sec. C of Ref. sup () for the explicit expression of the wavefunction).
Proximitizing the 2DEG with SCs induces intraband pairing potentials [see Fig. 3(b)]; in the limit , we may neglect the inter-band matrix elements of the pairing potential. The pairing potentials can be obtained from the first-order degenerate perturbation theory, and are given by (see Sec. D of Ref. sup ())
To the leading order in Zeeman energy, the intraband pairing potential for the -th band can be written as
where is a function of , and , while and are functions of , , , and (see Eqs. (D) and (D) in Ref. sup ()). The zeroth-order term of the gap in the Zeeman energy can be understood intuitively as follows. For JJs with narrow SCs (), electrons undergo multiple normal reflections from the edges of the SCs before they can be Andreev reflected. As a result, the gap is the average of the left and right superconducting gaps, i.e., which vanishes at . This gap closing also follows from the fact that the Hamiltonian respects the mirror and time-reversal symmetries at for which implies that (see Ref. sup () for details). Since and are even functions of while is an odd function, at and [see Eq. (4)].
Expanding Eq. (6) around , we have the gap-closing points moving away from according to
Thus, inside the fan in the BDI phase diagram, the gap closing lines of each subband move away from with slopes which are inversely proportional to . This can be seen in Fig. 2. The number of the gap closing lines increases with increasing as there are more occupied subbands for a larger . Since these are gap closings at , they are accompanied by changes in by , but do not affect .
As increases, the fan of BDI phase boundaries intersect the class D phase boundary where changes. As seen in Fig. 2, at each of these intersections, either three or four different phases meet. The four-phase intersection points signify simultaneous gap closings at both and . The three-phase intersection points happen when two gap closings at are moved by varying and , merge at , and get lifted (See Sec. E of Ref. sup () for details).
The BDI symmetry can be broken by applying a transverse in-plane magnetic field (along ), disorder that breaks the mirror symmetry, or if the two SCs have different gaps or different widths. Applying a transverse Zeeman field tilts the spectrum, which reduces the gap and results in gapless regions (see Sec. F.1 of sup ()). On the other hand, the gap-closing points at are lifted when the BDI symmetry is broken by disorder or an asymmetry of the left and right SCs Pientka et al. (2017); Hell et al. (2017); Haim and Stern (2018); sup (). Breaking the BDI symmetry also results in the hybridization of MZMs residing at the junction’s end, leaving either zero or one mode at zero energy (see Sec.F of Ref. sup ()).
where is the free energy of the system, is the temperature, and are the eigenvalues of the Hamiltonian. The critical current is
Figure 4 shows and as a function of for a JJ with narrow leads at temperature , and for two different values of . The critical current oscillates as a function of with an amplitude that decays with . For small , e.g., eV [Fig. 4(a,c)], at the critical Zeeman field where the critical current exhibits a minimum, the phase at which the free energy is minimal changes from to . Unlike the JJs with wide SCs Pientka et al. (2017), this phase jump does not necessarily imply a TPT due to the weak dependence of the class D TPT on [Fig. 2(c)]. For larger values of SOC, e.g., eV [Fig. 4(b,d)], the critical current exhibits a minimum with a shallower depth and at a larger critical Zeeman field. This minimum, however, is not accompanied by a discontinuous change of that minimizes the free energy. Some insight into these behaviors can be obtained by calculating the energy-phase relation of the junction perturbatively in , for two different limits: and (see Sec. G of Ref. sup ()).
In conclusion, we have studied topological superconductivity in planar JJs with narrow SCs. Due to multiple normal reflections from the SC edges, the topological superconductivity of JJs with narrow leads depend strongly on the chemical potential and the class D phase diagram depends only weakly on the superconducting phase difference. On the other hand, the BDI phase diagram is strongly dependent on the superconducting phase difference. Finally, we show that the minima in the critical current of JJs with narrow leads do not necessarily indicate TPTs. These results are directly relevant to recent experiments Fornieri et al. (2018), and elucidate the consequences of the BDI symmetry on the phase diagram of these systems.
Acknowledgements.This work was supported by NSF-DMR-MRSEC 1420709. A. S. and E. B. are supported by CRC 183 of the Deutsche Forschungsgemeinschaft. A.S. acknowledges support from the Israel Science Foundation, the European Research Council (Project LEGOTOP), and Microsoft Station Q.
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- (57) See Supplemental Material for (i) the details on the tight-binding Hamiltonian, (ii) dependence of class D phase diagrams on the chemical potential and superconducting phase difference, (iii) wavefunction of a Rasbha particle in a strip, (iv) detailed derivation of the dependence of the superconducting gap on the Zeeman energy and superconducting phase difference, (v) explanation of the phase boundaries in the BDI phase diagram, (vi) effects of breaking the BDI symmetry, and (vii) derivation of the Josephson current at zero and finite temperature.
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Supplemental Material for “Topological superconductivity in planar Josephson junctions – narrowing down to the nanowire limit”
Appendix A Tight Binding Hamiltonian
The Hamiltonian of Eq. (Topological superconductivity in planar Josephson junctions – narrowing down to the nanowire limit) can be written in the tight-binding form as
with () being the creation (annihilation) operator of an electron with spin on site where and . The hopping strength is denoted by (for the numerical simulation in this paper, we use meV) where with being the lattice constant and the spin-orbit coupling strength is denoted by where . The Zeeman field is along the direction and is taken to be uniform throughout the system. The proximity-induced superconductivity is nonzero only for and .
In the limit where , the Hamiltonian can be Fourier transformed using as
We use the tight-binding Hamiltonian to calculate the phase diagram, gap, energy spectrum and Josephson current. The numerical simulations on this tight-binding Hamiltonian are performed using the Kwant package Groth et al. (2014).
Appendix B Dependence of class D Phase diagram on the chemical potential and superconducting phase difference
Figure S1 shows (a) class D boundary lines as functions of Zeeman field and chemical potential for two different values of superconducting phase difference: and and (b) class D boundary lines as functions of Zeeman field and superconducting phase difference for several values of chemical potential . The class D boundary lines separate the trivial phase at low Zeeman field from the topological phase at high Zeeman field. We note that at and , the gap closing points at , which signify a BDI phase transition, can also coincide with the gap closing point at which signifies a class D phase transition. This happens when the chemical potential enters a new subband [see the blue lines in Fig. S1(a) for meV and meV]. We note that the zero critical field for at these fine-tuned chemical potentials exist only when the BDI symmetry is preserved. Breaking the BDI symmetry will move the critical field from zero to some finite values Fornieri et al. (2018). Since the standard time reversal is broken for away from , the gap opens and increases in magnitude as is tuned away from . To close the gap at away from , a finite Zeeman field is required. As a result, the class D phase boundary (as a function of and ) has a cusp at and [as shown in Fig. S1(b) for = 4.08 meV]. The dependence of the class D phase diagram on the superconducting phase difference becomes weaker when the chemical potential is tuned away from this special point, i.e., when the band bottom of the occupied subband closest to the chemical potential moves further away from the chemical potential [see Fig. S1].
In general, narrowing the SC leads makes the class D phase boundaries of a planar JJ depend more on the chemical potential and less on the superconducting phase difference due to the enhancement of multiple normal reflections from the interface of the superconductors with the vacuum. The narrower is the JJ, the greater is the amplitude of the normal reflections. In the limit where the width is sufficiently narrow (), the phase diagram of JJs with narrow leads will be similiar to that of multiband nanowires Lutchyn et al. (2011); Stanescu et al. (2011).
Appendix C Wavefunction of a Rashba particle in a strip
In order to derive the dependence of the superconducting gap on the phase difference and the Zeeman field, we first calculate the wavefunction of a particle with Rashba SOC and Zeeman field in a strip of width with infinite potential walls. Due the confinement in the direction, the energy spectrum consists of multiple bands where we label each of them by an index . In the following, we going to work in the limit where and solve for the wavefunction perturbatively in the Zeeman energy .
As we consider a system which is translationally invariant along (), we can write the wavefunction as
where are two-component spinors of the -th band. For the case of zero Zeeman field, we have the Fermi wavevector for the -th band as at satisfying
where . The Fermi surface consists of two concentric circles with radius as depicted in Fig. 3(a) of the main text. At fixed , there are four values of satisfying Eq. (S-5), which are denoted by where and
The subscripts and denote the outer and inner Fermi surfaces, respectively, while the subscript denotes the sign of . Due to the spin-orbit momentum locking, the spin rotates along the Fermi surface with the Rashba-induced spin-rotation angle given by
In the following, we are going to solve for the spinor perturbatively in in the regime near where the fan in the phase diagram emerges. To the first order in the Zeeman energy, the spinor can be written as
where is the zeroth-order and is the first-order perturbed wavefunction due to the Zeeman energy. At , the Hamiltonian commutes with the mirror operator and we can label the zeroth-order spinor as where and correspond to the even and odd eigenstates of the mirror operator , respectively:
Due to the confinement in the direction, the spinor at momentum is a superposition of four components, with amplitudes corresponding to the spins at four different Fermi momenta () satisfying Eq. (S-6), i.e.,
where are coefficients of each mode. The coefficients are determined from the boundary conditions:
where the walls are at . The number of coefficients can be reduced by using the mirror reflection symmetry . Since the spinor is either even or odd under reflection
we then have
The equation for the odd mirror eigenstate is identical to Eq. (S-14) except with cos replaced by sin:
The above derivation for (where is the SOC energy) follows the derivation for given in the Appendix A of Ref. Berg et al. (2012).
When Zeeman field is introduced, it shifts the center of momentum of the Fermi surfaces and tilts the Rashba-induced spin-rotation angle towards the magnetic field direction [see Fig. 3(a) of the main text]. To first order in , the change in the center of momentum of the inner () and outer () Fermi surfaces is given by
The Zeeman field rotates the Rashba-induced spin-rotation angle by
To get the a simple and compact analytical expression for the wavefunction, in the following we will focus on the regime where where
To the first order in Zeeman energy, the change in the spinor can be written as