Duality picture of Superconductor-insulator transitions on Superconducting nanowire

In this study, we investigated the electrical transport properties of niobium titanium nitride (NbTiN) nanowire with four-terminal geometries to clarify the superconducting phase slip phenomena and superconducting-insulator transitions (SIT) for one-dimensional superconductors. We fabricated various nanowires with different widths and lengths from epitaxial NbTiN films using the electron beam lithography method. The temperature dependence of resistance R(T) below the superconducting transition temperature Tc was analyzed using thermal activation phase slip (TAPS) and quantum phase slip (QPS) theories. Although the accuracy of experimental data at low temperatures can deviate when using the TAPS model, the QPS model thoroughly represents the R(T) characteristic with resistive tail at low temperatures. From the analyses of data on Tc, we found that NbTiN nanowires exhibit SIT because of the change in the ratio of kinetic inductance energy and QPS amplitude energy with respect to the flux-charge duality theory.

Scientific RepoRts | 6:27001 | DOI: 10.1038/srep27001 suggestion is conformed for short wires by phenomenological model 23 and experimentally confirmed for specimens with a length of L < 200 nm 14 . However, the data for longer specimens are inconsistent in that they exhibit R N c > R Q and show superconductivity 15,24 . Thus, there is no consensus between experiments and theories on the exact role of SIT of 1D-SNWs.
Recently, Mooij et al. proposed an idea that the concept of flux-charge duality can relate the QPS with Josephson tunneling if the roles of phase and charge are interchanged 1 . They discussed the crossover value between superconducting and insulating states at low temperatures as a function of ratio α = E s /E L , where E s and E L are the QPS energy and inductive energy, respectively. Increasing E s leads to a transition from inductively superconducting regime where E L ≫ E s to a capacitive insulating regime where E s ≫ E L . Further, they succeeded in showing the phase boundary between the superconducting and insulating state of data for MoGe SNWs 14 by assuming α = 0.3.
To observe the quantum phase slip in SNWs, specimens are required to be homogeneous and satisfy the condition d,w ≤ ξ, where ξ is superconducting coherence length. Further, poor links due to inhomogeneities in the superconducting wires can cause residual resistance at low temperatures as pointed by Altomare et al. 15 In the present report on homogenous nanowires, the R(T) characteristics of NbTiN SNWs in a broad range of the R N /L were investigated from the viewpoint of QPS mechanisms. We analyzed the data from the superconducting and insulating phase diagrams based on the flux-charge duality model using the relation (R N /R Q )/(L/ξ) versus L/ξ with a suitable parameter α and other parameters in theories 1 .

Experimental Procedure
Superconducting NbTiN films were firstly prepared by deposition at ambient temperatures on (100)-MgO substrates by DC reactive magnetron sputtering. The background pressure of the chamber was maintained below 2.0 × 10 −5 Pa. The relative amounts of argon and nitrogen were controlled by mass flow controller during sputtering. The total pressure was maintained at 2 mm torr and the substrate was not heated intentionally during deposition. Details of preparation procedures and films quality of NbTiN thin films are previously reported 25 . The NbTiN SNWs were fabricated from 2D films with d = 5 nm by a conventional e-beam lithography method and a reactive ion etching method with CF 4 plasma. The ranges of L and w of nanowires are 250 ≤ L ≤ 1000 nm and 10 ≤ w ≤ 30 nm, respectively. To eliminate the influence of the contact resistance, measurements of transport properties were performed by four-probe method. The normal state resistance, R N , is defined as the sample resistance at 20 K. The T C and H C2 were defined as the point at which the R N reached half its value.

Results and Discussion
Figure 1(a) shows the scanning electron microscopy image of typical NbTiN SNWs. Figure 1(b) presents the characteristics of R(T) for various SNWs with different values of w and L. Superconducting SNWs that have dR/dT > 0 under low temperatures and low R(Ω) characteristics, experience the initial drop of R(Ω) almost at the same temperature owing to the superconducting transition. An increase in R N causes T c and the residual resistance to monotonically decrease and increase, respectively. Prior to the detailed discussions on the SIT of 1D specimen from a viewpoint of quantum phase transition, we will present some transport properties of the present SNWs from the characteristics of low dimensional superconductors. Figure 2(a) shows a typical R(T) of the NbTiN SNWs with L = 500 nm and w = 20 nm at various external magnetic fields. With an increase in H, T c monotonically decreases without field-tuned SIT even at H = 9 T. On the other hand, as shown in the inset, the 2D NbTiN with almost the same thickness (≈ 5 nm) shows that the field-tuned SIT occurs around 5-6 T where many vortexes in the film appear to transition into super-fluid states of vortexes in the dirty boson scenario [26][27][28] . The present result for NbTiN SNWs suggests that this nucleation of the vortex in the superconducting state is inhabited because of the 1D-restricted geometry of the nanowire. We consider the dimensionality and size effects of the nanowire on the upper critical magnetic field H c2 . The suppression of superconductivity by perturbations is given by the relation, , where Ψ (x) is the digamma function and δ is the pair-breaking strength which depends on the dimensionality of the specimen and the direction of the external magnetic field 29 . Expanding the function the temperature range near T c (0), where δ is given by δ = DeH/c and δ = DeH 2 d 2 /6ħc 2 for fields perpendicular and parallel to the surface of 2D specimen with d < ξ(0), respectively, and D is the diffusion constant. From the above relation, H c2 (T) near T c (0) ≡ T c0 is given by where the index n is 1 and 1/2 for magnetic fields perpendicular and parallel to the surface, respectively. When the expression for the parallel case is approximately applied to SNWs, it is expected that n approaches 1/2 with w ≈ ξ(0). By using the Eq. (1), we obtained the index n for each SNW. Figure 2(b) shows the R/L dependence of n for NbTiN SNWs, where the dotted line is the reference point. It can be seen that the index n approaches 0.5 with an increase in the R/L ratio. The inset shows the typical data of H c2 (T) for the SNW with w = 10 nm and L = 500 nm. The solid line shows Eq. (1) with n = 0.56. These results indicate the 1D transport property of the present SNWs.
To clarify the mechanism of the resistive tail for NbTiN SNWs at low temperatures shown in Fig. 1(b), we analyzed the R(T) and the voltage-current characteristics in a broad temperature range. The fluctuation of the superconducting order parameter ψ(r) plays an important role in the transport properties of the 1D superconductor. The magnitude of ψ(r) vanishes at some points in the SNWs owing to the fluctuation, and it recovers the phase slip by 2π . There are two mechanisms for the phase slip, TAPS and QPS. According to the TAPS model, dV/dI and R TAPS (T) are expressed by 9,10 Scientific RepoRts | 6:27001 | DOI: 10.1038/srep27001 is the normalized unit measured by R Q and ξ(T), and β and η are fitting parameters held constant on the order of unity 30,31 . For ξ(T), the expression ξ t (0) 1/2 2 1/2 is adopted 32 . The dV/dI is also given by QPS 0 ,QPS where I 0,QPS is expected to have a different temperature dependence from I 0,TAPS = (4ek B /h)T. Figure 3 shows the superconducting transport properties for the NbTiN SNW with d = 5 nm, w = 10 nm, L = 500 nm, T c = 10.0 K, and R N = 5.0 kΩ. Figure 3(a) shows R(T) from the measurements of current-bias (○ ), and the dV/dI ( ) at I ≈ 0 shown in Fig. 3(b). The calculation made using the TAPS model shown by the dashed line (---) cannot explain R(T) characteristic except for temperatures close T c and unfortunately, the theory strongly deviates from the data below 8 K. This discrepancy suggests that transport properties of the present NbTiN SNWs are incompatible with TAPS theory at low temperatures. The solid line is calculated using Eq. (4) with parameters of ξ(0) = 8 nm, β = 0.0013, and η = 0.024 in order to fit theory with R(T) data under a broad temperature range. The calculation using the QPS model agrees accurately with the resistive tail in the range of 5 magnitude orders. Figure 3(b) represents the I-V characteristics at temperatures of 2 K, 5 K, 6 K, 7 K, 8 K, and 9 K. The I-V curves have shown nonlinear characteristic in the superconducting region below the 9 K. This I dependence of dV/dI at each temperature agrees well with the term of cosh(I/I 0,QPS ) in the Eq. (5) as shown by the solid lines 33 . From the fitting procedure, we obtained the temperature dependences of R and I 0,QPS . The inset in Fig. 3(b) shows the I 0,QPS (T), which is almost independent of temperature as shown by the dotted line. This discrepancy with the TAPS model shown by the solid line is consistent with the experimental result that R(T) cannot be explained by the TAPS model. Before investigating the R N /L ratio dependence on T c in order to clarify the SIT of NbTiN SNWs, we will analyze the data using the theory based on the dynamically enhanced Coulomb repulsion competing the attractive interactions between electrons 7 . The theoretical expression for T c is given by a simple formula as a function of sheet resistance R sq (resistance for unit area) with the parameter γ τ = .
, where τ is the electron elastic scattering time. Figure 4 represents the R sq dependence of T c for both nanowire specimens and 2D specimens. T co values are expected to be independent of w and L as shown in Fig. 5, because the T c on the vertical axis is normalized by T c0 = 11.0 K of pure 2D films. Although R sq of 2D specimens was controlled by changing the thickness, R sq of SNW specimens with different w was controlled by changing the length L and by keeping the thickness constant at ≈ 5 nm for all SNWs. The dotted line is calculated by using the theory for impure 2D system 5 in order to fit the data (× ) with a parameter γ τ . The good agreement between the theory and data suggests that the T c depression of 2D NbTiN films is determined by the decrease in the amplitude of the superconducting order parameter that belongs to the system confirming the fermionic scenario. As for 2D NbN and NbTiN films, we have already investigated transport properties on the fluctuations and SIT 5,28 . We reported that the critical sheet resistance R c is approximately 2.2 kΩ and superconducting suppression mechanism can be explained by the localization theory. On the other hand, data for SNW specimens in the range of 10 nm ≤ w ≤ 30 nm do not collapse on the unique line calculated by the theory 7 . In addition, the depression of T c cannot be explained only by enhanced Coulomb interaction in impure superconductors.   dependence of T c for the 1D system and for 2D system in the case R N is replaced with R sq. The T c slowly decreases below the value R N /L ≈ 50Ω/nm when R/L increases, and rapidly decreases in the range above 50-60 Ω/nm. For a relation between T c and R N /L in Fig. 5, Marković et al. 19 proposed a simple criterion for the crossover value of (R N /L) c.o separating an insulating state from superconducting state. If the wire resistance at T = 0 due to the quantum phase slip is comparable to R N , the resistance drop does not appear to sustain the value of R N even at very low temperatures. Using Eq.(4) at T = 0, they obtained the normalized resistance due to quantum phase slips 24 as , where T c0 = 11.0 K and u = 8.75 nm 2 are the transition temperature of 2D film and the parameter determined from the fitting procedure, respectively. 1/A dependency of T c has also been reported in Mo 78 Ge 22 and Mo 50 Ge 50 SNWs 18 that is having greater α than that of the present NbTiN. If this relation for NbTiN SNW is valid for board range 1/A, the D c ∝ u 1/2 value is expected to be smaller than ≈ 2 nm. On the other hand, the T c (1/A) characteristic drastically decreases around 1/A ≈ 0.06 nm −2 , giving D c ≈ (4A/π) 1/2 ≈ 4.6 nm denoted by the arrow ↓ , which is approximately half of the 2D NbTiN coherence length ξ(0) ≈ 9 nm determined from the relation ξ = ≈ ħc eT dT dH . This estimation suggests that the restricted geometry of SNW allows smaller critical diameter than ξ(0) for 2D specimens.
Now, we will discuss of size dependence of T c and SI phase diagram for the present SNWs. To clarify the 1D SIT mechanisms of NbTiN SNWs, we illustrated the SI phase diagram in Fig. 6 according to the Chakravarty-Schmid-Bulgadaev theory based on the interaction of QPS and dissipative environment [20][21][22] . Such a relation between L and L /R N has been reported for MoGe SNWs 14 . The author claimed that the SIT boundary is given by a condition R N = R Q = 6.45 kΩ. However, the present NbTiN SNWs specimens with L > 500 nm do not satisfy this condition as shown by the dashed line, that is, specimens with L > 500 nm show superconductivity though R N is larger than R Q . Although the R(T) characteristic of NbTiN SNWs can be explained by the QPS theory as discussed in Figs 4 and 5, the phase diagram shown in Fig. 6 suggests that the SI boundary depends on the length of the nanowire.
QPS and the Josephson effects in SNWs are related to each other by a concept of duality transformation. According to this concept 1 , the SIT is determined by the ratio between the strength of QPS amplitude energy E S and SNW inductive energy E Li . Both energies are given by E S = eV 0 /π = a(L/ξ)k B T c (R Q /R ξ )exp(− bR Q /R ξ ) andE Li = φ 0 /2L i = 17.4k B T c (R Q /R N ), where the R ξ = R N ξ/L is the resistance of the SNW over an appropriate length, L i = 0.18ħR N /k B T c is the kinetic inductance of the wire, Φ 0 = h/2e is the flux quantum, and a and b are constants of order one. According to Mooij et al. 1 , it is expected that SIT occurs at condition E s /E Li = (aλ 2 c 2 2 Figure 7 shows the r ξ (λ) for all same NbTiN SNWs shown in Fig. 6, where ξ = 8 nm is used. To show a boundary separating the superconducting phase from the insulator phases, we calculate r ξ (λ) from Eq. (7) with input parameters b and c. Although data are not so large to make clear the boundary, we attempt to find reasonable values for b and c assuming that the theoretical line must go through a reliable point of r ξ (λ) ≈ 0.6 at λ(= L/ξ) ≈ 62 for analysis. The red-broken, red-solid, and black-solid lines are typically calculated from Eq. (7) to divide the data into superconducting and insulator phases with the use of parameters (b, c) = (0.14, 0.05), (b, c) = (0.23, 0.2) and (b, c) = (0.28, 0.5), respectively. When we take into account the theoretical suggestions that the strength p of three parameters a, b and α c is given as 0.1 < p < 1. Further, the first combination of (b, c) = (0.14, 0.05) corresponds to (α c , a, b) = (0.025, 0.5, 0.14) due to the definition c = a/α c in Eq.