Interplay of classical and “quantum” capacitance in a one dimensional array of Josephson junctions

# Interplay of classical and "quantum" capacitance in a one dimensional array of Josephson junctions

## Abstract

Even in the absence of Coulomb interactions phase fluctuations induced by quantum size effects become increasingly important in superconducting nano-structures as the mean level spacing becomes comparable with the bulk superconducting gap. Here we study the role of these fluctuations, termed “quantum capacitance”, in the phase diagram of a one-dimensional (1D) ring of ultrasmall Josephson junctions (JJ) at zero temperature by using path integral techniques. Our analysis also includes dissipation due to quasiparticle tunneling and Coulomb interactions through a finite mutual and self capacitance. The resulting phase diagram has several interesting features: A finite quantum capacitance can stabilize superconductivity even in the limit of only a finite mutual-capacitance energy which classically leads to breaking of phase coherence. In the case of vanishing charging effects, relevant in cold atom settings where Coulomb interactions are absent, we show analytically that superfluidity is robust to small quantum finite-size fluctuations and identify the minimum grain size for phase coherence to exist in the array. We have also found that the renormalization group results are in some cases very sensitive to relatively small changes of the instanton fugacity. For instance, a certain combination of capacitances could lead to a non-monotonic dependence of the superconductor-insulator transition on the Josephson coupling.

###### pacs:
74.20.Fg, 75.10.Jm, 71.10.Li, 73.21.La

The Josephson’s effect, (1); (5) reveals the central role played by the phase of the order parameter in superconductivity. It has been exploited in a broad spectrum of research problems and applications: from the study of the pseudogap phase in high materials (6) , fluctuations above (7) and cold atom physics (28) to spintronics (12) and quantum computing (13). Of special interest is the study of an array of superconducting grains separated by thin tunnel junctions, usually referred to as Josephson junctions (JJ). The physical properties of JJ arrays are very sensitive to the grain dimensionality, the presence of Coulomb interactions and dissipation (15); (17); (16); (2); (21) (see also the review (25)). Usually it is assumed that each single grain is sufficiently large so that the amplitude of the order parameter, the superconducting gap, is well described by the bulk Bardeen-Cooper-Schriffer (BCS) theory. Moreover it is also commonly assumed that a simple capacitance model is sufficient to account for Coulomb interactions. The phase of each grain is therefore the only effective degree of freedom of the JJ array.

Within this general theoretical framework a broad consensus has emerged on the main features of JJ arrays: for long 1d arrays at zero temperature with negligible dissipation, the existence of long range order depends on the nature of the capacitance interactions. For situations in which only self-capacitance is important superconductivity persists for sufficiently small charging effects (2) provided that the Josephson coupling is strong enough. Despite spatial global long-range order a state of zero resistance will strictly occur only in the case in which the super current is induced by threading a flux in a ring-shaped JJ array (23); (19). A current in a long but finite linear JJ array will eventually induce a resistance though for sufficiently strong Josephson coupling it is hard to measure it as its typical time scale can be much longer that the experimental observation time. At any finite temperature the resistivity is always finite as a consequence of the unbinding of phase anti-phase slips.

In the opposite limit in which only mutual-capacitance is considered, even small charging effects induce a superconductor insulator transition. The combined effect of the two types of charging effects, considered in (10), can also lead to global long-range order. On a single junction, dissipation by quasiparticle tunneling only renormalizes (3) the value of the capacitance. However dissipation caused by a ohmic resistance (14) induces long range correlations between phase slips and anti phase slips that restore superconductivity provided that the normal resistance is smaller than the quantum one. In order to illustrate the profound impact of dissipation it is worth noting that a state of zero resistance in a 1D JJ array can in some cases coexist (15) with an order parameter whose spatial correlation functions are short-ranged.

The closely related problem of a quantum nanowire was addressed in (19); (18) by employing instanton techniques to model phase tunneling and then mapping the resulting effective model onto a 1+1d Coulomb gas where one of the dimensions is imaginary time. For an infinite wire in the zero temperature limit a superconductor-insulator Berezinsky-Kosterlitz-Thouless (BKT) transition occurs as a function of the system parameters. The role of vortices in 1+1d is played by phase slips which correspond to configurations for which the amplitude of the order parameter vanishes and the phase receives a boost. By contrast at finite temperature – a similar argument holds for finite length – the time dimension is compactified so, in the absence of dissipation, the Coulomb gas analogy breaks down since for long separations phase and anti-phase slips become uncorrelated. As a consequence phase coherence is lost and the resistance is always finite (20); (19); (10).

As was mentioned previously all these results assume that the amplitude of the order parameter of each grain, which enters in the definition of the Josephson coupling energy, is not affected by any deviations from the bulk limit and that the phase dynamics is induced only by classical charging effects. Although these assumptions are in many cases sound there are situations in which corrections are expected.

In sufficiently small grains close to the critical temperature it is well documented that homogeneous path integral configurations different from the mean field prediction, the so called static paths, contribute significantly to the specific heat and other thermodynamical observables (26). For single nano-grains at intermediate temperatures it has been shown recently (9) that, even in the limit of vanishing Coulomb interactions, deviations from mean-field predictions occur due to the non trivial interplay of thermal and quantum fluctuations induced by finite size effects. Experimentally it is also well established (27); (8) that substantial deviations from mean-field predictions occur in isolated nano-grains. Indeed it has recently been reported (8); (29) that quantum size effects enhance the superconducting gap of single isolated Sn nanograins with respect to the bulk limit.

It is therefore of interest to understand in more detail the role of these finite size effects in arrays of ultrasmall JJ where the mean level energy spacing of single grains is smaller, but comparable, to the superconducting gap. This paper is a step in this direction. We study the stability of phase coherence in arrays of 1D JJ at zero temperature. Our formalism includes the above quantum fluctuations induced by size effects, charging effects and dissipation by quasiparticle tunneling. Starting from a microscopic Hamiltonian for a 1D JJ ring-shaped array of nanograins at zero temperature, we map the problem onto a Sine Gordon Hamiltonian where we identify the region of parameters in which long-range order persists in the presence of phase fluctuations. In the limit of vanishing charging energy, relevant for cold atom experiments, we find the minimum size for which the JJ array can be superfluid as a function of the wire resistance in the normal state. We also show that quantum fluctuations induced by finite size effects can in principle stabilize superconductivity in the limit of a negligible self-capacitance energy but a finite mutual capacitance energy. We have also identified a region parameters in which it is observed a non-monotonic dependence of the superconductor-insulator transition on the Josephson coupling.

## I The model

We consider the system sketched in Fig.(1), consisting of an array of superconducting grains with periodic boundary conditions and a total magnetic flux passing through it, that can be modeled by the Hamiltonian:

 H= L∑r=1HBCSr+HSCr+HMCr,r+1+HTr,r+1. (1)

Each isolated superconducting grain is described by the BCS term,

 HBCSr =∑α,σϵα,rc†α,σ,rcα,σ,r (2) −grδr(∑αc†α,1,rc†−α,−1,r)(∑α′c−α′,−1,rcα′,1,r),

accounting for the effective attractive electron-electron interactions in the region where the grain size is much smaller then the bulk superconducting coherence length. label single-particle states related by time reversal symmetry with energies , is the spin label and and are, respectively, the mean level spacing (inversely proportional to the grain volume) and the dimensionless coupling constant of grain . We further assume the presence of self and mutual capacitive terms of the form

 HSr =12CSr(^Nr−NSr)2, (3) HMr,r+1 =12CMr(^Nr−^Nr+1−N%Mr)2, (4)

accounting for the repulsive Coulomb interaction within each grain and between electrons in neighboring grains. is the total number of electrons, is the self-capacitance the of grain and the mutual capacitance between nearest neighbor grains and . The constants and can be adjusted by applying suitable gate voltages. Finally, the hopping of electrons between grains is captured by the term

 HTr,r+1=∑αα′σTα,α′r,r+1c†α,σ,tcα′,σ,r+1+h.c., (5)

where the hybridization matrix is proportional to the overlap of the single-particle wave functions of two neighboring grains. In the regime of interest here - small grain sizes with respect to the bulk coherence length - the simplifying assumption that the hybridization is energy independent can safely be used and thus simplifies to

 HTr,r+1=tr,r+1∑σ(∑αc†α,σ,r)(∑αcα,σ,r+1)+h.c., (6)

with the total flux passing through the ring.

## Ii Finite size corrections to the action of a Josephson Junction’s Array

### ii.1 Partition function in the path integral formalism

In this section we write the partition function in the path-integral form and identify the finite size corrections to the action. This is done by inserting complex-valued Hubbard-Stratonovich fields (HSF) to decouple the BCS term in the superconducting channel, real valued HSF , conjugate to the number of particles on each grain, and real valued HSF , conjugate to the difference of the number of particles in neighboring grains. Using the notation , the partition function reads , with the action

 S=−Ψ†G−1Ψ+∫β0dτ∑r[1grδrΔ†rΔr+CSr2(VSr)2+iNSrVSr+CMr2(VMr)2+iNMrVMr], (7)

where the full Green’s function is given by

 G−1 = ⎛⎜ ⎜ ⎜ ⎜⎝G−11T21T†21G−12⋱⋱⋱⎞⎟ ⎟ ⎟ ⎟⎠, (8)

and

 G−1r = (−∂τ−~εα,r(τ)Δr(τ)Δ†r(τ)−∂τ+~εα,r(τ)), (9)

is the inverse of the electronic propagators restricted to grain . Here we defined and the hybridization matrix .

Integrating out yields the action

 S=−Trln[−G−1]+∫β0dτ∑r[1giδrΔ†rΔr+CSr2(VSr)2+iNSrVSr+CMr2(VMr)2+iNMrVMr] (10)

solely in terms of the HSF.

We apply the unitary transformation

 U Missing or unrecognized delimiter for \right

with to the electronic propagator in order to render real its off-diagonal anomalous elements , where . Note that for odd one has that , where denotes the trace over anti-periodic functions (fermionic) and the trace over periodic functions (bosonic). For a generic we will denote for even and for odd. Whenever we have two such indices we will use for the time periodicity in indices and . Note however that this complication is only formal as we will be interested in the low temperature properties of this action where the distinction between even and odd ’s can be safely ignored (33). After this transformation we get

 ~G−1 = U†G−1U=⎛⎜ ⎜ ⎜ ⎜⎝~G−11~T21~T†21~G−12⋱⋱⋱⎞⎟ ⎟ ⎟ ⎟⎠, (11)

with

 Missing or unrecognized delimiter for \right (12)

and

 ~Tr+1,r=⎛⎝tr+1,rei12[ϕr+1(τ)−ϕr(τ)]00−¯tr+1,re−i12[ϕr+1(τ)−ϕr(τ)]⎞⎠. (13)

Moreover, assuming the hopping amplitude to be small, we may develop the term to second order in and obtain the action

 Missing or unrecognized delimiter for \left (14)

### ii.2 Leading behavior in δ

The action above Eq.(14) is suitable for a saddle-point expansion in both and fields since the action for each grain is an extensive quantity in the number of electrons within that grain . Notice however that the saddle-point equations cannot be explicitly evaluated as depends on . We proceed by noting that is small, as the phase varies smoothly as a function of for sufficiently low temperatures. Formally we set , and where the subscript denotes the static component (constant in ) of the different quantities and the fluctuation around the static value, to be considered at quadratic order, are denoted by , and . Physically, is the amplitude of the condensate on grain and the terms leads to a renormalization of the chemical potential: .

For equally spaced levels and a particle-hole-symmetric single-particle density of states the tunneling term can be simplified at low temperatures (34)

 Tr[~Gr+1~Tr+1,r~Gr~T†r,r+1]≃CJi8∫dτ{∂τ[ϕr+1(τ)−ϕr(τ)]}2−Ici2∫dτcos[ϕr+1(τ)−ϕr(τ)+ϕtr]

where is the phase of the hopping term , is the quasi-particle induced capacitance and is the junction’s critical current between grains and , given respectively by (35)

 CJi=24|tr+1,r|2δrδr+1×∫∞sr,0dν1∫∞sr+1,0dν2ν1ν2(ν2+ν1)3√(ν21−s2r,0)(ν21−s2r+1,0) (15)

and

 Icr=8|tr+1,r|2δrδr+1×∫∞sr,0dν1∫∞sr+1,0dν2sr,0sr+1,0(ν2+ν1)√(ν21−s2r,0)(ν21−s2r+1,0) (16)

Note that for these expressions simplify to and with the normal state resistance of the junction.

With these approximations the action reads

 S[s,ϕ,V]=S0+∫dτ∑r{Ωrδs2r(τ)+CSr2δVSr(τ)2+CMr2δVMr(τ)2+12Cδ,rφ2r(τ)−i⟨N0,r⟩∂τϕr(τ)+CJr8(∂τ[ϕr+1(τ)−ϕr(τ)])2−Icr2cos[ϕr+1(τ)−ϕr(τ)+ϕtr]} (17)

where

 S0=∑i{Trln[−~G−1r,0]+β[1grδrs2r,0+CSr2(VSr,0)2+iNSrVSr,0+CMr2(VMr,0)2+iNMrVMr,0]} (18)

only depend on the static saddle-point values, , , ,

 ~G−1r,0(iωn) = (iωn−~εα,rsr,0sr,0iωn+~εα,r) (19)

and where we also define

 φr(τ)=δVSr(τ)+δVMr(τ)−δVMr−1(τ)−12∂τϕr(τ) (20)

and the finite size induced self-capacitance .

Eq. (17) is now suitable to a static-path treatment (9) once the fluctuations are integrated out. Here, as we are only interested in the phase dynamics at low temperatures we set the static components to their mean-field values and integrate out the gapped fluctuations both in the and fields. In the limit the final action in terms of the phase degrees of freedom and assuming translational invariance in the couplings , , , is given by,

 S = 18∫dτ∑r,r′∂τϕr(τ)[CR]r,r′∂τϕr′(τ) +i12∑r⟨N0,r⟩∫dτ∂τϕr(τ) −Ic2∑r∫dτcos[ϕr+1(τ)−ϕr(τ)+ϕtr] (21)

where is the capacitance matrix, with the discrete derivative: , is the phase of the hopping term, is the average number of electrons in grain and

 ~CS = (1CS+δ2)−1 (22)

is the grain self-capacitance renormalized by quantum finite size effects. Note that on the lattice , with , for sake of simplicity we use the notation to denote the lattice Laplacian .

Eq.(21) is the central result of this section, it contains the effective low energy theory for a junction at , including charging effects, quasiparticle dissipation and for the first time quantum fluctuations induced by finite size effects . The Berry phase term - second term of Eq.(21) - ensures that, in the ground-state (i.e. for ), the average number of electrons on each grain is even (33). In the following we assume that this condition is fulfilled and drop this term.

Note that for a set of isolated finite-size grains with no superconducting phase ensues as the action in Eq.(21) reduces to with the phase stiffness controlling the exponential time decay of the order parameter correlation function : .

## Iii Superconducting Transition

### iii.1 Hamiltonian Formulation

In this section we analyze the action given by Eq.(21), without the Berry phase term as we assume an even number of electrons in each grain. The calculation is carried out by first mapping this equation onto an equivalent Coulomb gas model. The Coulomb gas is subsequently transformed into a Sine-Gordon action for which a perturbative RG treatment can be effectively performed.

First we provide a description of the model in terms of the effective low energy Hamiltonian for the phase degrees of freedom in order to make contact with previous works where this effective description is taken as the starting point of the calculation. The initial step is the discretization of the imaginary time in Eq.(21): (with and ) . Using the identity

 limΔτ→0∑n=n1,..,nNe−Δτ2n.A−1.n+Δτibn=⎛⎝√2πdet(A)Δτ⎞⎠Ne−Δτ2b.A.b (23)

the partition function can be rewritten as , with

 S[ϕ,n] =∑~τ,r,r′2Δτn(~τ,r)[C−1R]rr′n(~τ,r′) −∑~τ,rin(~τ,r)[ϕ(~τ+1,r)−ϕ(~τ,r)] −Ic2∑~τ,rΔτcos[ϕ(~τ,r+1)−ϕ(~τ,r)+ϕtr] (24)

In this form, Eq.(24) can readily be interpreted as the Trotter-sliced action coming from the Hamiltonian

 H = ∑rr′2^nr[C−1R]rr′^nr′ (25) −Ic2∑rcos[^ϕr+1−^ϕr+ϕtr]

where , the variable conjugated to , is the number of Cooper-pairs in grain .

### iii.2 Partition function of the Coulomb gas

We follow the procedure of (36) to re-write the action of a Josephson junction array in terms of the partition function of a classical Coulomb gas. Using the Villain decomposition of the cosine term

 ezcos(θ) ≃ I0(z)∞∑m=−∞e−12μ(z)m2eimθ (26)

with a modified Bessel function of the first kind, valid for both, large and small respectively with

 μ(z) = {−2ln(z/2)\ for z≪1z−1\ for z≫1, (27)

Eq.(24) can be written as

 Missing or unrecognized delimiter for \left (28)

where we relabel in Eq.(24) and in Eq.(26) in order to interpret as an integer field living on links of a square lattice - an integer-valued one-form on the square lattice - with corresponding to time-like and to space-like links.

Integrating out the field yields the divergence-free constraint

 ∂n ≡ Δ1n1+Δ0n0=0, (29)

where is the discrete derivative along the time direction. Locally such constrain can be satisfied by writing as the rotational of an integer valued field living on the centers of plaquettes - an integer-valued lattice 2-form - or in components: , , where the subscript of denotes that this field lives on spacial-temporal plaquettes. The operator can be seen as the lattice exterior coderivative. Globally, the most generic solution of the constraint in Eq.(29) includes a non-trivial divergence-free field that cannot be written as a rotational. On a torus, such general solution can be decomposed as . More explicitly,

 n0(~τ,r) = −Δ1a01(~τ,r)+∑α=0,1cαbα0(~τ,r) (30) n1(~τ,r) = Δ0a01(~τ,r)+∑α=0,1cαbα1(~τ,r) (31)

where and (with ) are integer-valued 1-forms on the lattice that cannot be written as a rotational. They are chosen, see Fig.(2), to have a minimum flux along time and space directions respectively: , . are integer-valued coefficients labeling different topological sectors. Note that in the infinite volume limit, i.e. zero temperature and , the terms can be dropped in the solution as the space becomes topologically trivial. Later on we will drop the contribution as we are interested in the zero temperature limit.

In terms of the field and the integers and , the partition function is given by the unconstrained sum with

 S[a,c] =∑~τ,r,r′2Δτ[Δ1a(~τ,r)−∑αcαbα0(~τ,r)]× [C−1R]rr′[Δ1a(~τ,r′)−∑αcαbα0(~τ,r′)] −ic1Φ+12∑~τ,rμ(IcΔτ2)[Δ0a(~τ,r)+cαbα1(~τ,r)]2 (32)

where the total flux .

Using the Poisson summation formula to improve the convergence of the sum over Eq.(32) (36) and integrating over yields

 Z = ∑m,cδ∑m=0e−S[c,m]eic1Φ (33)

where the sum over is restricted such that the so-called neutrality condition is fulfilled (36) and

 S[c,m] = (2π)22∑~τ~τ′rr′m(~τ,r)G(~τ−~τ′,r−r′)m(~τ′,r′) (34) −2πi∑αcα∑~τ,rm(~τ,r)(∂−1bα)(~τ,r).

with the inverse of the operator defined in Eq.(29). The last term in Eq.(34) for can be simplified to

 ∑~τ,rm(~τ,r)(∂−1b1)(~τ,r) = ∑j[(Δ20+Δ21)−1¯Δ0m](0,r).

The Green’s function is given by

 G−1=−4Δτ~C−1SΔ21⎡⎢ ⎢⎣1{1−C−1M~CSΔ21}−~C−1SCJΔ21⎤⎥ ⎥⎦−1−μ(IcΔτ2)Δ20 (35)

In summary, after integrating over the field that represents small phase fluctuations, the action in Eq.(34) is given solely in terms of topological excitations, , that can be interpreted as an instanton field representing a phase slip. The corrections due to non-vanishing values of and do not change the nature of the long-range interaction between the phase slips, as they multiply higher powers of the discrete Laplacian. Nonetheless they appear in Eq.(34) in inequivalent ways, further we will see this translates to different contribution to the monopoles energy to create monopole pairs.

### iii.3 Flux quantization

To understand how the flux piercing the ring gets quantized in the superconducting phase, where the density of instantons (phase slips) vanishes, let us examine the partition function given in Eq.(33). For simplicity let us first take the zero temperature limit in order to ignore the field. The flux is imposed to the system assuming that the magnetic field far from the ring is constant and perpendicular to the axes in Fig.(1). A complete description of the system array+field should include the dynamics of the electromagnetic field as well. However this is too involved and not really needed here, the only thing that is required is to remember that the spacial distribution of the electromagnetic field (and thus the flux piercing the ring) is itself determined by an action containing the electromagnetic contribution plus the coupling of the electromagnetic field to the instanton configurations given by the last term of Eq.(34).

Performing the summation over in Eq.(33) one observes that the partition function of a system with flux can be written as

 Z=∑m,c1δ∑m=0 δ2π(Φ−Φ1m)×e−(2π)22∑~τr,~τ′r′m(~τ,r).G(~τ−~τ′,r−r′).m(~τ′,r′) (36)

where is the -periodic delta function and . To the action of the free electromagnetic action one should thus add the monopole contribution . Directly from Eq.(36) one can observe that if the density of phase-slips vanishes (i.e. ) then and thus has to be quantized in multiples of . When phase-slips proliferate, is a fraction of , for a generic configuration of instantons , the summation over all configurations allows for a continuum value of .

### iii.4 Superconducting-Insulating Transition

Having understood how the flux gets quantized once instantons are suppressed, let us neglect the topological terms (i.e. set in Eq.(33)), in order to study the superconducting-insulating transition. A simple way of addressing this question is to map the problem to the Sine-Gordon model. The main result we report in this section is that the superconducting insulating phase transition is Kosterlitz-Thouless like, even in the presence of a finite and . This extends the results of Ref.(10), where the case , is considered. Nonetheless and renormalize the instanton-core energy in rather different ways. By studying how this energy gets renormalized we obtain the behavior of the superconducting-insulating transition line as a function of , , and . We note that also includes a term coming from quantum fluctuations induced by finite size effects that so far had not investigated in the literature.

The first step to get the Sine-Gordon action is to regularize the instanton interaction kernel at the origin