Topological Superconductivity in a Planar Josephson Junction

We consider a two-dimensional electron gas with strong spin-orbit coupling contacted by two superconducting leads, forming a Josephson junction. We show that in the presence of an in-plane Zeeman field, the quasi-one-dimensional region between the two superconductors can support a topological superconducting phase hosting Majorana bound states at its ends. We study the phase diagram of the system as a function of the Zeeman field and the phase difference between the two superconductors (treated as an externally controlled parameter). Remarkably, at a phase difference of $\pi$, the topological phase is obtained for almost any value of the Zeeman field and chemical potential. In a setup where the phase is not controlled externally, we find that the system undergoes a first-order topological phase transition when the Zeeman field is varied. At the transition, the phase difference in the ground state changes abruptly from a value close to zero, at which the system is trivial, to a value close to $\pi$, at which the system is topological. The critical current through the junction exhibits a sharp minimum at the critical Zeeman field and is therefore a natural diagnostic of the transition. We point out that in the presence of a symmetry under a mirror reflection followed by time reversal, the system belongs to a higher symmetry class, and the phase diagram as a function of the phase difference and the Zeeman field becomes richer.

Popular Summary

As in all superconductors, electrons in the interior of a topological superconductor form pairs that are able to carry electric current without dissipating energy. Uniquely, however, topological superconductors also allow for single electrons—in the form of exotic particles known as Majorana fermions—to move on their surfaces. The search for topological superconductors has gained a lot of attention in the last few years, motivated by both pure scientific interest and the hope of using them as building blocks for quantum computers. At the heart of this search are materials that become topological superconductors within a range of controllable environmental parameters, such as magnetic field or chemical potential. We propose a setup that relies on a new experimental knob—the phase difference between two superconductors—and show how that knob can be tuned to create a topological superconductor.

The system we propose is a superconductor-normal-superconductor Josephson junction, a two-dimensional semiconducting strip sandwiched between two superconductors. The strip has strong spin-orbit coupling and sits in a parallel magnetic field. We show that when the phase difference between the superconductors is equal to $\pi$, the system is almost guaranteed to be in a topological phase with Majorana bound states at its ends. Under certain conditions, the system also tunes itself to the desired phase difference. Our analysis is based on a calculation of the subgap spectrum of states that are bound to live within the semiconducting strip.

Our system has already been realized in recent experiments, and part of those experimental observations may be interpreted as supporting our theoretical analysis.

Figure 1

(a) A Josephson junction is formed in a 2DEG with Rashba spin-orbit coupling by proximity coupling it to two $s$-wave superconductors with relative phase difference $\varphi$. An in-plane magnetic field is applied parallel to the interface between the normal and the superconducting regions. (b) The bound-state spectrum in a narrow junction for $k_x=0$. The spectrum in the absence of a Zeeman field is twofold degenerate and is indicated by the grey lines. In the presence of the Zeeman field, the spectrum for the two spin states (plotted in red and blue) is split, allowing for the appearance of a topological phase. (c) Phase diagram as a function of the Zeeman field in the junction, $E_{Z,J}$, given in units of the Thouless energy $E_T=(\pi /2)v_F/W$ and the phase difference $\varphi$. The solid lines are the phase boundaries in the absence of any normal backscattering at the superconducting-normal interface, while the dashed lines correspond to a junction transparency of 0.75 and a phase $k_{F}W+{\phi }_N=3\pi /8$ as defined in Sec. 3a. The arrows indicate the range of $\varphi$ values between the two zero energy crossings in (b) for which the system is topological.

Figure 2

(a) A Zeeman field along $x$ shifts the two Rashba-split Fermi surfaces of the 2DEG in opposite directions along $y$. The arrows indicate the orientation of the spin at each point on the Fermi surface. (b) The phase difference ${\varphi }_{\mathrm{GS}}$ that minimizes the ground-state energy (upper panel) and the critical current modulation (lower panel) as a function of the Zeeman field obtained numerically using a tight-binding model for the system (see Appendix pp4-s2 for details of the model). The parameters used are as follows: $W=5$, $W_{\mathrm{SC}}=10$, $t=1$, $\alpha =0.1$, $\mu =-2.4$, and $\mathrm{\Delta }=0.3$ [31]. The left (right) panel corresponds to a temperature of $T=0.05\mathrm{\Delta }$ ($T=0.3\mathrm{\Delta }$). (Note that we set $k_B=1$ throughout the paper.) The light blue color indicates the region in the parameter space for which the system is in the topological phase. As the Zeeman field is varied, the system undergoes a series of first-order topological phase transitions, in which ${\varphi }_{\mathrm{GS}}$ changes abruptly between values lying in the topological and trivial regions of the phase diagram. The critical current exhibits minima at the points of the phase transitions. As the temperature is increased, the minima become deeper.

Figure 3

(a–c) Numerical results for the phase diagram as a function of the Zeeman field and the chemical potential for different values of $\varphi$, obtained using the scattering matrix approach (see Appendix pp4-s1 for details). The width of the junction is $W=5(m\mathrm{\Delta }{)}^{-1/2}$. The phase boundaries are determined by a gap closing at $k_x=0$. At low chemical potentials, $\mu \simeq \mathrm{\Delta }$, a finite amount of normal reflection from the superconducting leads results in oscillatory modulations of the phase boundaries as a function of $\mu$. The topological phase significantly expands as the phase difference between the leads is tuned from 0 to $\pi$. (d) Same as (c) but with a rectangular potential barrier at each interface between lead and junction. The barrier has height $200\mathrm{\Delta }$ and width $0.01(m\mathrm{\Delta }{)}^{-1/2}$. This corresponds to a transparency of 0.82 at $\mu =10\mathrm{\Delta }$. The presence of a reflecting barrier leads to stronger modulations of the boundaries.

Figure 4

Phase diagram as a function of the Zeeman field in the leads $E_{Z,L}$ and the phase difference $\varphi$ in the limit of a narrow junction with $E_{Z,J}\ll E_T$. For $E_{Z,L}>\mathrm{\Delta }$, the system becomes gapless. The solid lines are the phase boundaries in the absence of any normal backscattering, while the dashed lines correspond to junction transparency of 0.75 and a phase $k_{F}W+{\phi }_N=3\pi /8$.

Figure 5

A phase diagram of the system as a function of the Zeeman field and $\varphi$ in the presence of the mirror symmetry $M_y$ defined in the text. The values of the $\mathbb{Z}$ invariant corresponding to each region are indicated on the figure. Regions with odd (even) $\mathbb{Z}$ indices, corresponding to topological (trivial) regions of the D class, are filled with shades of blue (red). Note that, by our definition of $\varphi$, the phase diagram is not invariant under $\varphi \to \varphi +2\pi$ since this operation flips the sign of the superconducting gap function. The figure was obtained using a tight-binding version of the model (see Appendix pp4-s2) with the following parameters: $W=5$, $W_{\mathrm{SC}}=5$, $t=1$, $\alpha =0.5$, $\mu =-2.75$, and $\mathrm{\Delta }=0.3$. Note that normal reflection is implicitly present in this model because of the finite width of the superconductors.

Figure 6

The bulk gap calculated along a cut in Fig. 5 with $E_{Z,J}=0.6$. When $|{\mathrm{\Delta }}_1|=|{\mathrm{\Delta }}_2|=0.3$, the system is in the BDI symmetry class. The bulk gap closes when the system undergoes topological phase transitions between regions with different (odd) $\mathbb{Z}$ indices. When $|{\mathrm{\Delta }}_1|\ne |{\mathrm{\Delta }}_2|$, the effective time-reversal symmetry is broken, and the bulk becomes gapped for all values of $\varphi$.

Figure 7

(a–c) Energy spectrum across the topological phase transition calculated using a tight-binding model for the system (see Appendix pp4-s2 for details). The tight-binding parameters used are $W=5$, $W_{\mathrm{SC}}=20$, $t=1$, $\alpha =0.5$, $\mu =-2.8$, and $\mathrm{\Delta }=0.3$. The Fermi momenta $k_{F,1/2}$ are calculated in the absence of a Zeeman field. The values of $\varphi$ and $E_{Z,J}$ for which the spectra are plotted are indicated by crosses on the phase diagram shown in (d). The phase diagram is obtained by calculating the topological invariant for class D, $Q=\mathrm{sign}[\text{Pf}(H_{k=\pi }{\tau }_x)/\text{Pf}(H_{k=0}{\tau }_x)]$ [32]. In (a), the system is in the trivial phase; in (b), the gap at $k_x=0$ closes, and the system undergoes a topological phase transition; and in (c), the system is in the topological phase.

Figure 8

Induced gap as a function of system parameters evaluated in the continuum model using the scattering matrix approach (see Appendix pp4-s1 for details) for $W=1(m\mathrm{\Delta }{)}^{-1/2}$, $m{\alpha }^2=9\mathrm{\Delta }$, and $E_{Z,L}=0$. In the left panel, $\mu /\mathrm{\Delta }=20$. The diamond-shaped gap closing lines indicate the boundary between the trivial and the topological regions. Additional regions of small gap occur in the vicinity of BDI phase transitions, where the gap closes at nonzero momenta. In the right panel, $\varphi =\pi$, and a sizable topological gap is obtained in a very broad range of Zeeman fields with hardly any dependence on the chemical potential.

Figure 9

Local density of states at the edge (left panel) and in the center (right panel) of the junction as a function of energy and phase difference. In a range around $\varphi =\pi$, a Majorana state forms at the edge. The result is obtained numerically from a tight-binding model (see Appendix pp4-s2) using the following parameters (energies and length are in units of the hopping strength and lattice spacing): $\alpha =0.5$, $E_{Z,J}=E_{Z,L}=0.1$, $\mathrm{\Delta }=0.25$, $\mu =-3.75$ (measured from the center of the tight-binding band), junction width $W=4$, width of the superconducting leads $W_{\mathrm{SC}}=8$, and length $L=200$. We plot a spatial average of the density of states over a rectangle spanning the entire width of the junction in the $y$ directions and the first 10 sites from the edge (left panel) or the most central 10 sites (right panel) in the $x$ direction. For presentation, the local density of states has been convoluted with a Gaussian with a standard deviation of $0.02\mathrm{\Delta }$.

Figure 10

The upper panel shows the bound-state energies of the two spin species (plotted in red and blue) and the energies of their particle-hole symmetric states (indicated by dashed lines) for a single momentum $k_x<k_{F,1}$ in a narrow junction, as the Zeeman field is varied. The contributions to the ground-state energy (obtained by summing over the negative energy states) and the Josephson current are plotted for each value of the Zeeman field in the lower panel in blue and green, respectively. At $E_{Z,J}=E_T/2$, the value of $\varphi$ for which the energy is minimized shifts from 0 to $\pi$. This transition is accompanied by a minimum in the critical current.

Figure 11

The ground-state energy of the junction as a function of the phase difference $\varphi$ in the limit of a narrow junction with a single bound state for all $k_x$ [see Eq. (16)]. For $k_F>k_{\mathrm{SO}}$, at $E_{Z,J}=E_T/2$, the system undergoes a first-order phase transition as the ground state shifts between the two local minima. In the left panel, $E_{Z,J}=0.45E_T$, and the system is in the trivial phase in its ground state. In the right panel, $E_{Z,J}=0.55E_T$, and the system is in the topological phase in its ground state.

Figure 12

Numerical phase diagram and analytical estimates for two limiting cases. The green lines show the solution in the limit $B\ll \mu$, $\mathrm{\Delta }$ given by Eq. (a5) expanded to linear order in $B/\mathrm{\Delta }$. The normal reflection in this limit is given by $r\simeq \mathrm{\Delta }/2\mu$ and ${\phi }_N\simeq 0$. In the opposite limit $\mu \ll B<\mathrm{\Delta }$ (red line), we use Eq. (a11). Both panels show the same data for a width of $W=14(m\mathrm{\Delta }{)}^{-1/2}$ and $\varphi =0$.

Figure 13

The critical current for the $k_x=0$ mode in a narrow junction at different temperatures. As the temperature is increased, the contrast of the modulations is increased, with the minima at $E_{Z,J}=(n+1/2)E_T$ becoming deeper.