1. Now I want to tell you one (in my opinion) interesting system. I will be based on this article. Always, when I
prepare presentation for journal club, I learn something new. Before this I didn’t work with such materials.
This article consist of a lot of formulas and a little figure, but I will use only some of them. And I try to explain
the main idea and results.
2. Exciton condensate ate ordered states of matter in which macroscopic phase coherence is established
through the condensation of electron-hole pairs. In this bound state electron and hole are attracted to each
other by the electrostatic Coulomb force. And exciton is regarded as an elementary excitation of condensed
matter that can transport energy without transporting net electric charge. Exciton atom looks like a hydrogen
atom, but the binding energy much smaller and the particle site much larger than a hydrogen atom. Signatures
of exciton condensation have been reported in quantum Hall bilayers, in optically excited exciton and excitonpolariton cold gases.
3. In investigation of exciton condensate, very important is works about exciton condensate in two separating
layers. In such system there is possibility of the charge superfluidity, i.e. e-h-superconductivity. Undumped
current fluids into different layers in opposite directions.
4. They consider structure with exciton consensate based on SNS junction. Especially on Andreev reflection and
Andreev bound state.
5. They consider two coupled weak links between four superconducting electrode, linked by a bilayer of length
L. They assume that the top layer is electron doped (negatively charged), and the bottom layer is hole doped
(positively charged). Two layers are coupled only through the Coulomb interaction. And use only one
dimension.
6. For such structure, electron field operator have such view.
7. The exciton and superconducting condensate are described by two order parameters. The Josephson current
in this four-terminal system has two contributions. The first one is related to direct quasiparticle tunneling
through the barrier caused by the gap in the bilayer quasiparticle spectrum. This process is present in ordinary
superconductor/insulator/superconductor two-terminal setups. The second contribution, which is the main
subject of interest in this work , is linked to the possibility of direct dissipationless conversion of supercurrents
into EC counterflow currents and can be realized only in a four-terminal setup like the one shown in Fig. 1.
Importantly, in the long-junction limit, the first contribution is exponentially suppressed and only the second
one survives.
8. The conversion of supercurrent into counterflow (neutral) current hidden in the solution of the Bogolubov-de
Gennes equations takes place through a coherent Andreev-type process that involves two Cooper pairs in the
two different layers. Consider Superconductor/exciton-condensate interface, and imagine a thin normal
region between superconducting and exciton-condensate layers. Energy is smaller than the EC and
Superconducting gaps. Electron incidents from normal layer to superconductor and reflects as hole (Andreev
reflection) hole reflects as electron. (see arrow C and A). Relations between up- and down-spin amplitudes:
formula 1 An analogous process takes place at the Normal/Exciton-Condensate interface: Particle (for instance
electron (blue point)) impinging on the EC from the top layer is reflected into the bottom layer upon emission
of an electron-hole pair. (see arrow B for electron and D for hole.) when incident particle is hole, then in EC
electron-hole pair decay. Relations between top- and bottom-layer amplitudes: formula 2 By combining the
four amplitude relations: formula 3. In order to find energy we must solve this equation. Energy will depends
on both superconducting and excitonic gap. They called this state as Excitonic Andreev bound state. As usual
Andreev Bound State is responsible for the flow of Josephson current, Excitonic Andreev Bound State is also
crucial for supercurrent conversion from S to EC.
9. Consider this system in long-junction limit. In tis limit the superconducting gap is largest energy scale. The
boundary conditions at the two interfaces: see formula 3 (they used boundary conditions which were derived
by Maslov and other) The Josephson current through the alpha layer can be evaluated as: formula 4 The
supercurrent can be expressed as the sum of ground-state contribution and thermal fluctuation term. (formula
5)
10.At zero temperature Josephson current depends on the difference between the two phase differences. This
give very interesting implications: When the top and bottom junctions are polarized with the same phase bias,
supercurrent cannot flow through the EC. In this case, The Josephson current experience an exciton blockade.
In the opposite case, when phase differences have opposite sign, the Josephson current flowing through the
EC is maximal, with the critical value equal to half of the critical current of ballistic one-channel S/N/S junction.
When current flows in one layer due to a phase bias in the same layer, a current equal in magnitude but
opposite in direction flows in the other layer. When a phase bias is applied to the top layer only, supercurrent
flows and in bottom layer.
11.At finite temperature. After same simplifications with respect to these approximations they obtained such
expression for current. First term is the zero temperature contribution, while the second one encodes the
effect of thermal fluctuations. This means that, as long as thermal fluctuations are dominated by the exciton
gap, the ground state current is essentially unaffected by finite temperature. The EC counterflow channel is
responsible for an exponential enhancement of the critical current.
12.The method which is based on boundary conditions is valid only in limit in which superconducting gap is the
largest energy scale. In order to investigate short-junction they use the scattering approach.Josephson current
is equilibrium current and can be computed from this relation: formula 1 Where F is phase-dependent term of
the free energy. Energy can be written such: formula 2 where ro is density of states and epsilom is energy
measured from the Fermi energy. The density of states depends on each of the four superconducting phases. I
don’t show all way of calculation and don’t want to discuss method which they used. I want to focus on results
of this calculation. You can see expression for energy. First term describes an excitonic supercurrent
contribution, second term can be understood as the contribution to the free energy due to bound states.
13.In long junction limit quadratic term in q can be neglected. Minimization of second term with respect to qL
yields. And they obtained expression for existence of solutions of the Hamiltonian. They use this condition and
got free energy and current In short-junction limit the dominant term is first term. Again they minimize energy
with respect to qL and got condition for existence of solutions of Hamiltonian. Insert this condition they gained
energy and current.
14.In the intermediate regime, the Josephson current can be understood as interpolating between long- and
short-junction limits. They present numerically plotted picture for current. We can see that when current is
normalized with respect to the critical current, the various curves differ by a small amount when
superconducting gap changes by three orders of magnitude.They repeat that all currents in S/EC/S junction
depend on phase difference of superconducting layers.
15.Again I will not go into all stages of the calculations. Only first step and results In tunneling limit they employ
a tunneling Hamiltonian. They obtain free energy and current. On this step I finish with formula.
16.A very important consequence of the fact that the Josephson energy depends only on the difference ϕT − ϕB
is the appearance of an almost exact double periodicity in the energy of the circuit illustrated in this figure. The
cross connections in Fig ensure that fluctuations that change φT −φB are suppressed, leaving φ = ϕT = −ϕB as
the only degree of freedom. An energy that is a function of the phase difference ϕT − ϕB may be expressed as
F(ϕT − ϕB) = F(2φ), which is doubly periodic compared to a usual Josephson energy. Exact double periodicity
of the free energy is quite generally promising for decoherence-free quantum computation. It implies that the
energy of the resulting circuit has two distinguishable minima located at φ = 0 and π that are separated by a
maximum at φ = ±π/2. The two quantum states |0 and |π, corresponding to phase differences φ = 0 and π, can
be used as the logical states of a quantum computation. The fact that their energies are equal implies absence
of the dephasing processes. The large barrier between them implies that there is no tunneling and thus no
decay.