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In this paper , we discuss the full counting statistics of superconducting quantum dot contacts. We discuss the effects both of phonon and onsite electronic interaction focusing on the experimentally most relevant case of strong onsite electronic interactions. We find that in general , the Josephson effect and multiple Andreev reflections in these systems are strongly suppressed due to the onsite interaction. However, in case resonant phonons are found , the effect of the onsite interaction can be overcome.

Superconducting nanoscopic systems represent one of the most interesting classes of systems [

The properties of superconductor-quantum dot-superconductor (S-QD-S) systems are particularly relevant and rich due to the manifold energy scales being present in the system (superconductor gap Δ , phonon frequency ω D , Coulomb interaction U, tunneling rate Γ as already analyzed in previous works [

The highly nonlinear transport characteristics of the S-QD-S setup we wish to analyze have been studied in the non-interacting setup [

The quantity of our primary interest is the full counting statistics (FCS) in terms of the cumulant generating function (CGF) χ. It represents a very convenient tool for the calculation of a variety of transport properties. It is directly related to the probability distribution function P(Q) to transfer Q elementary charges during a fixed very long measurement time τ. By simple derivation with respect to some parameters (counting fields) χ gives all cumulants (irreducible moments) of P(Q). This allows to measure and understand in depth the different transport properties contributing to the CGF. The Hamiltonian for the system under consideration is given by

H = H L + H R + H d + H T L + H T R + H p h o n + H d o t . (1)

The superconducting electrodes on the left and right can be described by BCS Hamiltonians with the gap Δ of the superconducting terminal. We assume both gaps to be identical in order to simplify the analysis as no large-scale new features are to be expected.

In equilibrium the respective Hamiltonians can be written as ( α = L / R )

H α = ∑ k , σ ϵ k α k , σ + α k , σ + Δ ∑ k ( α k , ↑ + α − k , ↓ + + α − k , ↓ α k , ↑ ) . (2)

using e = ℏ = k B = 1 . The voltage is applied symmetrically so that μ L = μ R = V / 2 . The corresponding Green’s functions are given in [

For the resonant level model applied here [

U P = exp [ λ 0 ω 0 d + d ( b + − b ) ] , (3)

which leads to a Hamiltonian where the electron-phonon interaction characterized by the coupling λ 0 and resonance frequency ϖ 0 is completely absorbed in the tunnel part of the Hamiltonian. We are left with

H ˜ T α = γ α [ α + ( x = 0 ) e λ 0 ϖ 0 ( b + − b ) d + h . c . ] , (4)

H ˜ d o t = ( Λ d + λ 0 2 ϖ 0 ) d + d , (5)

replacing H T α and H d o t . We absorb the polaron shift for the dot energy by a redefinition of ϵ d → Λ d − λ 0 2 / ω 0 .

The general solution of the system described by Equation (1) seems out of question and approximations to proceed are generally necessary.

We want to first consider λ 0 = 0 (without electron phonon interaction). We want to consider the case of strong onsite interaction and an odd number of electrons on the quantum dot. In this case the most prominent effect is the emergence of a spin 1/2 Kondo resonance around the Fermi edge for temperatures below the Kondo temperature T K . In normal conducting systems the Kondo temperature is directly related to the onsite interaction via [

T K = 2 U Γ n π exp ( − π U 8 Γ n ) , (6)

where Γ n refers to the tunnel rate between the normal conductor and the quantum dot. Due to the additional energy scale Δ in the problem, two scenarios may occur: for large T K / Δ the Kondo resonance couples to the quasiparticles in the superconductor leading to a behavior similar to the one for normal conducting systems [

We evaluate the cumulant generating function χ ( λ , φ ) using the generalized Keldysh technique [

χ ( λ , φ ) = ∑ q e i λ q P q ( φ ) , (7)

where P q ( φ ) is the probability of charge q to be transferred during a given (long) measurement time τ .

Partial derivations of χ ( λ , φ ) with respect to λ give direct access to the cumulants (irreducible moments). For our system the connection to the Hamiltonian is given by

ln χ ( λ , φ ) = 〈 T C exp ( − i ∫ C T λ ( t ) d t ) 〉 , (8)

where T λ ( t ) denotes the expectation value of H T L + H T R with the substitution R σ → R σ e − i λ ( t ) / 2 . C means the Keldysh contour and T C means time ordering on it. The counting field λ ( t ) has to be both time and contour dependent. It changes sign on the different branches of the Keldysh contour to account for the charge transfer. Additionally, λ ( t ) is nonzero only during the measurement interval [ 0 , τ ] . We use the standard expression

∂ ln χ ( λ , φ ) ∂ λ = − i τ 〈 T C ∂ T λ ( t ) ∂ λ 〉 λ , (9)

to find the CGF as the counting field derivative.

Compared to the case of contacts between normal metals and superconductors the counting field derivative has normal but also anomalous contributions leading to

∂ ln χ ( λ , φ ) ∂ λ = − i γ 2 ∑ σ [ 〈 T C R σ + d σ 〉 λ e − i λ 2 + h . c . ] + [ 〈 T C R σ + d σ + 〉 λ e i λ 2 + h . c . ] , (10)

We proceed by defining the exact-in-tunneling λ-dependent dot Green’s functions and the free electrode Green’s functions in Keldysh space:

D 0 σ λ ( t , t ′ ) = − i 〈 T C d σ ( t ) d σ + ( t ′ ) 〉 , D ˜ 0 λ ( t , t ′ ) = − i 〈 T C d ↑ ( t ) d ↓ ( t ′ ) 〉 , g α σ ( t , t ′ ) = − i 〈 T C α σ ( 0 , t ) α σ + ( 0 , t ′ ) 〉 0 , f α ( t , t ′ ) = − i 〈 T C α ↑ ( 0 , t ) α ↓ ( 0 , t ′ ) 〉 0 . (11)

Equation (10) can now be integrated with respect to λ to access the CGF. The normal contributions give rise to MARs and quasi-particle tunneling whereas the anomalous part gives rise to Josephson tunneling. We discuss both parts separately.

We first evaluate the second part of the expression in Equation (10). Following [

ln χ ( λ , φ ) = 2 τ ∫ d ω 2 π ln { 1 + 4 Γ a 2 [ ( ω − Λ D ) 2 + 4 Γ a 2 ] [ ( e i λ ( cos φ + sin φ ) − 1 ) + ( e − i λ ( cos φ − sin φ ) − 1 ) ] } θ ( Δ − | ϖ | Δ ) (12)

Γ a = Γ c Δ Δ 2 − ω 2 P ( ϖ + V 2 ) . (13)

The inelastic tunnel processes associated with the emission and absorption of phonons are described by the function P ( ϵ ) being the Fourier transform of the phonon-phonon correlation function. For this correlation function we assume the phonons are thermally distributed, which may be due to coupling to a thermal environment given by the substrate of backgate [

J ( ϵ ) = γ B ω [ ω ω 0 − 1 ] 2 + [ [ γ B ω 0 ] ( 2 λ 0 ) 2 ] 2 . (14)

The phonon correlation function can now be analytically to be:

P ( ϵ ) = e − ρ γ B π R e { ∑ k , l = 0 ∞ ρ γ B , a k k ! ρ γ B , b l l ! i ω + Ω 0 k − Ω 0 * l + i Γ 0 } , (15)

where Ω 0 = ϖ 0 ξ + i γ B 2 , ξ = 1 − γ B 2 4 ϖ 0 2 . The functions ρ γ B , ρ γ B , a , ρ γ B , b are given by

ρ γ B = λ 0 2 2 ϖ 0 ϖ 0 2 − γ B 2 4 [ coth [ β Ω 0 2 ] Ω 0 2 + coth [ β Ω 0 * 2 ] ( Ω 0 * ) 2 ] , ρ γ B , a = λ 0 2 2 ϖ 0 ϖ 0 2 − γ B 2 4 coth [ β Ω 0 2 ] − 1 Ω 0 2 , ρ γ B , b = λ 0 2 2 ϖ 0 ϖ 0 2 − γ B 2 4 coth [ β Ω 0 * 2 ] − 1 ( Ω 0 * ) 2 . (16)

In the case of small Γ c treated here we find the first harmonic for V < 2 Δ :

〈 I a 〉 ( τ ) ∝ Γ c 2 sin [ φ ( τ ) ] , (17)

which corresponds to the dc and ac Josephson current depending on whether a voltage is applied or not.

Typically we would therefore expect a reduced Josephson current given the low transparency of the contact. We would not expect pronounced features of the phonons given that only the phase would depend on voltage.

We go over to the evaluation of Equation (10). The calculation can be carried forward as in [

T a , n = 2 ∏ k = 1 ⌈ n − 1 2 ⌉ 4 Γ A R k ( ϖ − ϵ D ) 2 + Γ A R k 2 θ ( Δ − | ω − k V 2 | Δ ) × ∏ k = 1 ⌊ n − 1 2 ⌋ 4 Γ A R k ( ϖ − ϵ D ) 2 + Γ A R k 2 θ ( Δ − | ω + k V 2 | Δ ) × | ω − n V 2 | | ω + n V 2 | ( ω + n V 2 ) 2 − Δ 2 ( ω − n V 2 ) 2 − Δ 2 (18)

Γ A R k = Γ c Δ ( Δ 2 − ( ω − k V 2 ) 2 ) 1 2 P ( ϖ + k V 2 + V 2 ) . (19)

The result for the CGF for V < 2 Δ is then

ln χ ( λ , τ ) = 2 τ ∫ d ω 2 π ln { 1 + ∑ n = 2 ∞ [ ( e i λ n − 1 ) n F ( ϖ − n V 2 ) ( 1 − n F ( ϖ + n V 2 ) ) + e − i λ n n F ( ϖ + n V 2 ) ( 1 − n F ( ϖ − n V 2 ) ) ] } (20)

where n F is the Fermi function.

We observe that multiple Andreev reflections are strongly suppressed as T a , n ∝ Γ c 2 n . The suppression can only be overcome in case conductance is boosted by a resonant phonon.

For voltages above the gap we recover the previously discussed situation for normal conducting systems as in [

To conclude, we have derived the cumulant generating function for a superconducting quantum dot involving both Josephson tunneling as well as multiple Andreev reflections. Whereas we discovered that the Josephson effect is typically strongly suppressed in these setups and only slightly affected by resonant phonons, multiple Andreev reflections are strongly affected and show clear signs when resonant phonons are involved.

We expect that these results pave the way towards future usage of these junctions as transistors as they show a clearly nonlinear behavior. We would also believe that the corresponding current and noise characteristics can be observed in experimental setups soon. The corresponding analysis of the correspondence between the model developed here and the experimental results would clearly mark the next step also given the approximations in the Kondo limit taken here.

The author would like to thank Ingo Deppner for numerous discussions.

The author declares no conflicts of interest regarding the publication of this paper.

Soller, H. (2019) Full Counting Statistics of Superconductor-Quantum Dot-Superconductor Contacts in the Presence of Interactions. Open Journal of Applied Sciences, 9, 386-393. https://doi.org/10.4236/ojapps.2019.95032