Quantum Interference by Vortex Supercurrents

The origin of the parabolic background of magnetoresistance oscillations measured in finite‐width superconducting mesoscopic rings with input and output stubs and in patterned films is analyzed. The transmission model explaining the sinusoidal oscillation of magnetoresistance is extended to address the parabolic background as a function of the magnetic field. Apart from the interference mechanism activated by the ring, pinned superconducting vortices as topological defects introduce a further interference‐based distribution of supercurrents that affects, in turn, the voltmeter‐sensed quasiparticles. The onset of vortices changes the topology of the superconducting state in a mesoscopic ring in such a way that the full magnetoresistance dynamics can be interpreted due to the interference of the constituents of the order parameter induced by both the ring with its doubly connected topology and the vortex lattice in it.

interference to the onset of the sinusoidal constituent in MROs.Despite previous considerations [40], the interference mechanism affects the dependence of the wave functions of the transmitted particles on spatial coordinates, rather than their density.Both the densities   of Cooper pairs (CPs) and thermally activated [46] [47] quasiparticles (QPs), realize the mixture of transmitted particles and depend on  only, while the interference-induced sinusoidal voltage Δ  (, ) = ()Δ  (, ) (see Supporting Information) relies on the variation of the sensed QP charge.Here, the measured zero-bias resistance at  = 0 is ().We will apply the same model to address the PB by the onset of vortices affecting the topology of the sample as further ring-like objects.
A sketch of a typical mesoscopic ring is represented in Fig. 1, indicating the input and output stubs, through which the voltage across the ring is measured.A magnetoresistance (MR) is acquired in proximity of the critical temperature   , when the current−voltage characteristic () displays a zero-bias resistance due to the thermal activation of phase slips (TAPS) [48] (see Fig. 2(d) in [45]) .Usually, to acquire a MR, an AC probe current   is injected and a lock-in amplifier measures the voltage across the ring as a function of .
The sinusoidal oscillation in MR has been addressed in terms of the order parameter Ψ ̃ of a finite-size ring defined by the inner/outer radius ( / ).Ψ ̃ depends on the density of the dia-and paramagnetic supercurrents, distribution of which over the ring's surface oscillates to fulfill the fluxoid quantization (FQ) condition.Once the order parameter of the ring is defined, we can evaluate the change in geometrical phase according to the transmission equation [45] Ψ ̃, = (Φ  ) Ψ ̃, , where (Φ  ) = (−1)  cos ( ),  is the winding number and Φ  =   2 with   the average radius of the ring.Information on the change of the transmitted supercurrent density is obtained using the equation where  is the electronic charge,  * is the effective mass of a single electron and  =  ×  is the vector potential of the magnetic field piercing the sample.Substitution of eq. ( 1) in eq. ( 2) yields The change of the transmitted current density must comply with the conservation of the injected probe current   =   , so that eq. ( 3) refers to a change in the spatial density of CPs.
Thermally activated QPs propagate through the domains of the sample where   = 0 [45] separating the regions of dia-and paramagnetic supercurrents.Thus, supercurrent dynamics also affect the QP spatial density that results in the component of current sensed by voltmeter revealing the interference mechanism.
Geometry, including the sizes, of the sample strongly affects the vortex supercurrents, especially because of the crowding effects [51] stimulating vortices to nucleate from interconnections between stubs and each of the voltage leads as well as the ring.Yet, confinement and vortex stiffness determine a nonlocal [7] [52] dynamics because local distribution of vortices relies on their motion all over the mesoscopic sample.The onset of dia-and paramagnetic supercurrents [53] along with antivortices [54] make the vortex matter and, consequently, the read out of QP density in each voltage lead very complicated (Fig. 2(a)).
We tackle this problem by defining an average order parameter of the sample realized through the factorization of the swirling supercurrents in the ring Ψ ̃ and the vortex supercurrents Ψ ̃ (Fig. 2(b)) as shown in the equation where the first factor is the phase contribution owing to the averaged dia-and paramagnetic supercurrents in the ring, the second factor refers to the phase contribution of the vortex supercurrents, whereas the third factor depends on the total flux Φ = ∮  ⋅ = Φ  + Φ  .The average supercurrent of the ring    is assumed to flow along the circle   = 2  .with the average radius [45]   .The same approach is applied to the vortex supercurrents swirling along the circular paths   = 2  ; Λ =  0  2 where  is the penetration depth and Φ 0 is the flux quantum.
Since the vortex nucleation in mesoscopic structures occurs through the onset of vortex rows [9] [12] [55], the term } in eq. ( 4) accounts for the onset of the first vortex row event justifying the PB in MRs.Transport measurements sense the collective coherent dynamics of all vortex supercurrents as if they were generated by a single effective vortex (see Fig. 2(c)), so that where   = 2  ; if  denotes the sample width, then the condition 2  <  must be fulfilled.The total phase of Ψ ̃ satisfies the FQ where  is either zero or an integer.According to eqs. ( 4) and ( 5), the total transmission equation is so that the total transmission function is and Φ  =   2 .By substituting eq. ( 7) in eq. ( 2) and with the support of eq. ( 8), the modulation of transmitted supercurrent density is obtained in the form where, as shown in Fig. 2(c),  , () and  , () refer to the input supercurrent densities corresponding to the ring with its doubly-connected topology and the effective vortex, respectively.The aforementioned supercurrent densities differ by their spatial distributions as a consequence of the different spatial position of the ring and the effective vortex.
In order to measure the voltage signal Δ  related to the quantum interference, the current conservation [45] as sensed by the ammeter (the corresponding quantities are labelled by the index A) can be set and it can be represented as follows where the subscripts qp, s refer to the temperature-based partition (see Supporting Information in [45]) into QPs and CPs within the injected current.The growth of  induces vortex supercurrents to swirl either in the input or in the output part of the sample.These contributions  /,  () are not sensed by the ammeter, but can be accounted for within the voltage sensitive areas [45], where the total charge is conserved  , () +  , () +  ,  () =  , () +  , () +  ,  () .( 12) Magnetic field breaks the symmetry:  , () ≠  , () for  = , ,   , and eq.( 12) can be written accounting for all increments: −  , () =  , () −  , () +  ,  () −  ,  () .
According to eqs. ( 29) and (30) of ref. [45], eq. ( 13) turns to where Δ 0 () is the measured voltage for  = 0 and () ≡ Δ 0 ()/  is the value of the zero-bias resistance at the temperature of the MR acquisition.Hence, the MR trend accounting for the PB is given by where (, ) = Δ  /  .It is important, that  ,  refers to vortex supercurrents, which, according to the B-T phase diagram, are placed as a function of , and hence of   , higher than those related to the full supercurrent phase bounded by  1 ().Thus, the inequality   <  , () is physically possible.
The model is validated employing MROs acquired on the "wide" mesoscopic ring reported in [56].The sample is realized patterning a 30 nm-thickYBa2Cu3O7- (YBCO) film.The ring has the inner and outer radii   ≅ 70 nm and   ≅ 190 nm, respectively.In Fig. 3 15).
The proposed model is further tested on MROs of superconducting rings with different sizes [45] [57] and nanopatterned films [39] as well (see Supporting Information).The effectiveness of the presented model and its direct dependence on the geometry of multiply connected samples can provide a relevant contribution for solving issues on asymmetric rings presenting oscillation of critical current depending on the direction of injected current [58] [59].
A consistent demonstration of the change in distribution of transmitted QPs must involve a measurement realized through Hall probes at the input and output stubs.Hints on the valuable use of the Hall probes are indirectly provided by the sign-reversal effect [60] [61] measured in planar films.In this regard, CPs, QPs and vortices are considered mutually connected by the interactions, which can provide an edge imbalance charge measurable through the Hall resistance.
In conclusion, we have extended the transmission model developed for the sinusoidal constituent in MROs of superconducting rings to the onset of the PB.We have argued that the onset of vortices and related vortex supercurrents affects the topology of the current flow.Consequently, the full MR dynamics can be addressed through the interference of the constituents of the order parameter induced by both the ring with its doublyconnected topology and the vortex lattice.The PB is inherent to the development of vortex supercurrents, which affect the path of QPs, introducing a further voltage variation that follows a sinusoidal law similarly to the arguments presented for the supercurrents in a ring [45].Since the vortex size is much smaller than the ring's size, typical narrow-range acquisitions prevent from observation of the background oscillation.
The validation of the transmission model can be extended to other multiply-connected superconducting structures.Here, we focus on three distinct publications presenting MRO on a patterned film [2], on a holed stripe [3] and on a film with an array of pinning centers [4].

Case 1
The patterned film is 26 nm thick and made of La1.84Sr0.16CuO4(LSCO).The film is patterned as a grid of noninteracting square loops.Each loop has an area of 100 × 100  2 and the wire realizing the web is about  = 25 wide.The sample presents a full transition temperature at about  0 ≅ 26 (see Fig. 2 of [2]).The MRO employed to validate the model has been acquired at  = 29.5(see Fig. 6 of [2]).In Fig. 2S(a), a comparison between the experimental (black dots) and the model (red curve) is represented.The experiment shows damped oscillations with a flux periodicity consistent with  2 ~2 extracted by eye from Fig. 3 of [2].The parabolic background is driven by the first vortex nucleation event that scales like  1 ~Φ0  2 = 3.2  [5], because the magnetic penetration depth is comparable with wires width.Hence, wires are so narrow to make  1 comparable to  2 .As a matter of the fact, the experimental trend is similar to the MRs of aluminum rings [6].The presence of a damping factor modifies eq. ( 16) in the following form: where   =  − 2 /  2 defines the damping of sinusoidal oscillations as a function of   ~1 [7].The fitting parameters are () = 24.7 Ω,  , ()   ⁄ = 0.15,  ,  ()   ⁄ = 1.25, (Φ  ) = 53 , (Φ  ) = 12.5  and   = 1.25 .

Case 2
The system investigated in [3] is a Nb slab 100nm thick and 385 nm wide patterned with a series of periodic square holes 120nm in size.The MR curves we are interested in, present a hole period of 385nm and have been extracted from the Fig. 5(a) of [3] (curve made of violet triangles).In Fig. 2S(b), the experimental curves are compared with the model results obtained by us.As distinct from the Case 1, the MR presents the transition up to the normal state reached at about   = 27.9Ω.Therefore, the vortex supercurrent contribution also presents a damping factor, and the MR curve follows the trend where   =  − 2 /  2 and   =  − 2 /  2 .The matching of the model with the experimental curves is obtained using () = 12 Ω,  , ()   ⁄ = 0.5,  ,  ()   ⁄ = 2.3, (Φ  ) = 151 , (Φ  ) = 48 ,   = 0.05  and   = 1.5.

Case 3
Large superconducting bridges functionalized with an array of pinning centers display peculiar MRO presenting deep and sharp minima (vortex matching effect), which are placed according to the fields   = Φ 0  2 , where  is the average distance between the pinning centers.These latter can be obtained through two main routes: patterning of antidots [8] or ion irradiation [4].Pinning centers govern the nucleation of vortices at some specific locations, but the unavoidable presence of interstitial ones promotes the onset of pinned vortices presenting slightly different radii.Hence, the sinusoidal in MR is a function of a series of fluxes Φ , =   2 with   ∈ [  ,   ] ranging around .
Here, we refer to the measurements conducted over a bridge of YBa2Cu3O7−δ (YBCO) 50 wide and 50nm thick, in which a square array of pinning centers was imprinted through the deposition of a PMMA mask and O + ion irradiation.The sample is realized with defects having a diameter of 40nm and period of 120nm.Measurement, reported as black dots in Fig. 2S(c), has been extracted from the Fig. 4(c) of [4] referring to the one acquired with the lowest excitation current.The curve has been traced out with the function where the sum runs over pinned vortices ranging in radius between   and   ,   = exp − 2 /  2 is the damping factor relative to the sinusoidal, and the last term on the right is the interstitial vortices

Discussion
Damping factors are phenomenological parameters governing the onset of decoherence driven by the magnetic field.If the acquisition range is large, MRO are traced out using damping parameters in both the sinusoidal and background contributions, but the zero-field resistance () is enough to describe the evolution of MRs even if they present a three-fold variation of resistance as shown in the above cases.Do damping factors account for the increase of QP density that grows up to   ?No they do not.In fact, damping factors are different for the geometrical and vortex contribution highlighting that 's are not related to the growth of QP density.Damping factors explain the reduction of asymmetry in the distribution of transmitted QPs.The degree of asymmetry, regarding the ring driven interference, relies on the oscillation amplitude of the effective radius that being a function of   /  [9] lowers if  − and consequently   − approaches a critical value.Hence, factors  deliver the lowering of Δ  in place of defining the transformation of CPs in QPs.As a matter of fact, () can be assumed constant in eqs.(S3), (S4) and (S5).We argue, in agreement with the phenomenology of Case 2, that the growth of MRs up to   can happen keeping the QP density constant.
To discern on vortex or QP constituents of the normal state, more investigations are needed.MRs at high fields must be acquired to realize if the normal state for  >   is the same recorded for  >   .MRs [10] [2] [11] show that the resistive state for ~ 2 −when the resistance is flat and basically independent on  − is still dependent on  despite the state for  >   where the resistance in not dependent on  at all.We speculate that the normal state for  <   can be effectively reached only injecting high currents [12] [13] guaranteeing the appropriate heating for the transition to the normal state.

Enhancement of superconductivity in nanowires
Our model is useful to address the phenomenon of the enhancement of superconductivity observed through the   of superconducting nanowires [14] [15] [16] [17].We argue that the mechanism governing the   enhancement is based on the remarkable difference between the nanowire and stub sizes.Fig. S3(a) provides a pictorial representation of the sample under investigation.A nanowire of width  bridges two large stubs, whose transversal dimensions   ,   fulfil  ≪   ,   .Consequently, the nanowire first critical field    ~Φ0 / 2 is quite larger than that for both the left () and the right () stubs, which behave as plane films.
We focus on the range  <    , when the stubs are experiencing the mixed state, whereas the nanowire is fully superconducting.The nanowire decouples the interference phenomena occurring in the two stubs inducing the value of each voltage lead   and   to depend exclusively on the vortex dynamics of the corresponding stub.This fact is a clear consequence of the standard application of the transmission model that, as usual, starts from the current conservation.The current conservation also implies that the output current from the left stub is equal to the input one in the right stub, so that where  = Δ 0  − Δ 0  .Eq (S14) drives the variation of the voltage range, in which the   is extracted.
As discussed in the article, the lowering of Δ  versus  implies the enhancement of the critical current that is the amount of injected current   () relative to a voltage drop  ≤  ℎ .In Fig. S3(b), a simulation of Δ  is reported according to a slight change in the effective vortex flux area between the two stubs.Setting a mismatch between Φ  and Φ  of about a few percent, we obtain a result similar to what was measured in [14], in which   grew up to about 2T.Bibliography

Figure 1 :
Figure 1: Sketch of a mesoscopic ring.Input/output voltage and current leads are indicated.The shadowed areas near the voltage leads are the voltage sensitive areas.

Figure 2 :
Figure 2: (a) Scheme of supercurrents flowing in the sample in the presence of vortices.(b) A simplified vortex supercurrent pattern realized by sampling the real distribution by a series of swirling supercurrents at the nanoscale.(c) Reduction of the vortex lattice to a single vortex because of averaging of the vortex supercurrents.Dashed arrows sketch the paths of injected and vortex supercurrents involved in the interference mechanism.
, black dots and red curves represent the experimental and theoretical data, correspondingly.For three temperatures of acquisition  = 84.25K,84.50K, 84.75K related to the zero-bias resistances () = 0.40Ω, 0.93Ω, 1.73Ω, the best matching parameters are 37, 1.17, 0.43 with an uncertainty of about 5%.The average radii of the vortex and the ring are   = 36 nm and   = 127 nm, respectively.

Figure 3 :
Figure 3: Comparison between the experimental MRs (black circles) of an YBCO mesoscopic ring (after [Ref.[56]]) and the theoretical behavior (red curves) provided by the present model according to eq. (15).
Figure S3: (a) Pictorial representation of a nanowire sample realized between large stubs.(b) Simulations of   as a function of the change in the vortex flux area between the two stubs represents the voltage of each stub for  = 0 with respect to the ground.Eq. (S12) means that the nanowire decouples the two vortex dynamics so that the following equations hold true Eqs.(S13) enable one to evaluate the measured voltage drop across the device Δ  = Δ   − Δ   to be Δ  (, ) = () ( ,  sin 2