Fractional Solitons in Excitonic Josephson Junctions

www.nature.com/scientificreports OPEN Fractional Solitons in Excitonic Josephson Junctions Ya-Fen Hsu & Jung-Jung Su received: 16 June 2015 accepted: 16 September 2015 Published: 29 October 2015 The Josephson effect is especially appealing to physicists because it reveals macroscopically the quantum order and phase. In excitonic bilayers the effect is even subtler due to the counterflow of supercurrent as well as the tunneling between layers (interlayer tunneling). Here we study, in a quantum Hall bilayer, the excitonic Josephson junction: a conjunct of two exciton condensates with a relative phase φ0 applied. The system is mapped into a pseudospin ferromagnet then described numerically by the Landau-Lifshitz-Gilbert equation. In the presence of interlayer tunneling, we identify a family of fractional sine-Gordon solitons which resemble the static fractional Josephson vortices in the extended superconducting Josephson junctions. Each fractional soliton carries a topological charge Q that is not necessarily a half/full integer but can vary continuously. The calculated current-phase relation (CPR) shows that solitons with Q = φ0/2π is the lowest energy state starting from zero φ0 – until φ0 > π – then the alternative group of solitons with Q = φ0/2π − 1 takes place and switches the polarity of CPR. Excitons, the electron-hole pairs bound by Coulomb interaction, can reach spontaneous phase coherence and form excitonic supercurrent1–10 when properly induced11. They are in close analogy to the Cooper pairs in s-wave superconductors in that both can be described by the SU(2) BCS-type theory12. The overall charge neutrality of excitons, however, requires that the electron- and hole-components to be spatially separated for the electrical current detection1; the separation should be sufficiently small to maintain the excitonic coherence. Thank to the advance of semiconductor manufacturing technology, excitonic superfluid in such geometry is readily realized in Quantum Hall bilayers (QHBs)9,10,13. Unique effects for excitons in bilayer include the interlayer tunneling anomaly14–21 and the current counterflow17,22–27. Both offer exotic twists to the already fascinating supercurrent phenomena – among all the Josephson effect. First proposed and demonstrated in superconductor28, Josephson effect is regarded as an unambiguous test to superfluidity or superconductivity. The dc Josephson effect, in particular, describes the zero-bias supercurrent occurring in a Josephson junction – a device consisting of two coupled condensates with a relative phase applied. The Josephson effect is usually characterized by its current-phase relation (CPR)28,29. Although best known in the standard sinusoidal form, the CPR can go beyond sinusoids to genuinely reflect the junction geometry and the composite material’s properties29. In the context of exciton condensation, the attention has been on a seemingly similar but practically different effect, the Josephson-like effect14–21,30–38. The few pioneering works39–42 that are actually on the Josephson effect (dc), however, have yet included the interlayer tunneling in dynamics; the sinusoidal Josephson relation is thus applied directly. Here we actively include the interlayer tunneling in the equation of motion to obtain the appropriate CPR for the excitonic Josephson junction. It turns out that the obtained CPRs actually go beyond the standard sinusoidal form. Moreover, the fractional solitons emerge in static. Similar solitons with exactly half quanta have been realized in only few specially designed superconducting systems43–47. Such half-integer solitons generated in a controllable fashion can be strong candidates for quantum qubits44. The fractional solitons we discuss here are even more profound: it embraces continuously varying fractions that are not limited to half or full integers46,47. Because of the continuously varying nature, abrupt occurrence of solitons with increasing relative phase φ0 is not observed – the solitons appear progressively starting from infinitesimal φ0. Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan. Correspondence and requests for materials should be addressed to J.-J.S. (email: ) Scientific Reports | 5:15796 | DOI: 10.1038/srep15796 1 www.nature.com/scientificreports/ Figure 1. Schematic illustration of the excitonic Josephson effect. In the counterflow geometry (upper half), a relative phase φ0 is generated between the two excitonic condensates, EC1 and EC2. The navy arrows indicate the current flows. Note that currents are allowed to flow between layers while the net current should conserve. Lower half of Fig. 1 demonstrates the pseudospin picture of the excitonic Josephson effect. The relative phase φ0 corresponds to an angle difference in two pseudospin Zeeman fields (red arrows). The Zeeman fields attempt to work against the pseudospin stiffness to align the local pseudospins (pink arrows) with themselves. There are two configurations, a direct solution (DS) and a complementary solution (CS) that correspond to incline angles equal φ0 and φ0 −  2π, respectively. Methods System setup. The excitonic Josephson junction we consider is illustrated in Fig. 1 (upper half). In an excitonic bilayer, the two constituent layers are separately connected to allow counterflow; blue arrows in the figure show antiparallel current flow. Interlayer tunneling is represented by electron-current flow between the two layers. This tunneling should conserve the sum of current in the two layers. A constant relative phase φ0 is then applied between the two excitonic condensates, EC1 and EC2. By applying interlayer voltage pulse on EC2, we can prepare the junction at a relative phase φ039,40,48. Such a designated φ0 is reached by controlling the magnitude or the duration of the applying voltage pulse. In the absence of lateral electric field is zero, the edge current would not play a role in the present discussion. We detail in the following sections how the system is treated by solving the Landau-Lifshitz-Gilbert (LLG) equation49 for pseudospins50. This approach is especially powerful in mapping out the pseudospin distribution and thus the phase profile. Pseudospin model and the equation of motion. We begin with the excitonic wave function in the quantum Hall bilayer6   θ (X )  †     c + sin  θ (X )  e iφ (X ) c †  0 . Ψ = ∏ cos  ↑ ↓ X X      2    2  X   (1 ) The ↑ (↓ ) denotes the which layer degree of freedom and X the guiding center quantum number of the lowest Landau level. Vacuum state 0 indicates no electron in either layer. When representing Ψ as a vector in the Bloch sphere, θ(X) and φ(X) become the associated polar and azimuthal angles. Such a m (classical) – the exciton system is ultimately mapped vector is hereafter referred to as the psuedospin → to a pseudospin ferromagnet. In the limit of smooth textures, the energy functional can be expressed by pseu (...truncated)