Dichotomy in short superconducting nanowires: Thermal phase slippage vs. Coulomb blockade

– Quasi-one-dimensional superconductors or nanowires exhibit a transition into a nonsuperconducting regime, as their diameter shrinks. We present measurements on ultrashort nanowires (∼40–190 nm long) in the vicinity of this quantum transition. Properties of all wires in the superconducting phase, even those close to the transition, can be explained in terms of thermally activated phase slips. The behavior of nanowires in the nonsuperconducting phase agrees with the theories of the Coulomb blockade of coherent transport through mesoscopic normal metal conductors. Thus it is concluded that the quantum transition occurs between two phases: a “true superconducting phase” and an “insulating phase”. No intermediate, “metallic” phase was found. Under certain conditions, usually associated with a critical resistance per square [1,2], critical total resistance [3–6], or a characteristic diameter [7,8], a wire made of a superconducting metal loses its superconductivity and acquires two signatures of insulating behavior: i) The resistance increasing with cooling and ii) a zero-bias resistance peak [3, 4]. There are many models that capture certain features of the SIT in 1D wires. Some rely on the “fermionic” mechanism, in which disorder combined with electron-electron repulsion suppresses the critical temperature, Tc, to zero [9]. In other, “bosonic”, models the order parameter remains nonzero in the “insulating” phase while the coherence is destroyed by proliferating quantum phase slips (QPS) [10–15]. Existing theoretical models frequently predict a quantum superconductor-insulator transition (SIT) in thin wires [11, 13, 16, 17], driven, in many cases, by the interaction of the fluctuating phase with the Caldeira-Leggett environment [18]. Conditions that make QPS experimentally observable and the relation of the QPS to the SIT are still being actively researched [1–8,19–22]. Here we present a quantitative analysis of the transport properties of ultrashort nanowires in each of the two phases —the insulating phase and the superconducting phase. We show that the insulating phase is characterized by the normal-electron transport and governed by (∗) Present address: Brookhaven National Laboratory Upton, NY 11973, USA. (∗∗) Present address: Department of Physics, University of Utah Salt Lake City, UT 84112, USA. (∗∗∗) E-mail: [email protected] c © EDP Sciences Article published by EDP Sciences and available at http://www.edpsciences.org/epl or http://dx.doi.org/10.1209/epl/i2006-10275-5 506 EUROPHYSICS LETTERS Fig. 1 – (a) SEM micrograph of an ∼8 nm wide nanowire (light) suspended over the trench (black) in SiN. The sputtered MoGe film was 5.5 nm thick. The white regions at the ends of the wire indicate that this wire is suspended straight, without kinking and entering into the trench. The R(T ) curves for insulating and superconducting wires are shown in (b) and (c). The arrow in (b) shows RN for sample D. In (c), solid curves indicate fits to the LAMH-TAPS theory. The fitting parameters are the coherence lengths, 70.0, 19.0, 11.5, 9.4, 5.6, and 6.7 nm, and the critical temperatures, 1.72, 2.28, 3.75, 3.86, 3.80, and 4.80K, for samples 1–6, respectively. The corresponding normal resistances and the lengths, determined from the SEM images, are 5.46, 3.62, 2.78, 3.59, 4.29, 2.39 kΩ and 177, 43, 63, 93, 187, 99 nm, respectively. the Coulomb blockade physics [23, 24]. The wires in the superconducting phase exhibit good agreement with the Langer-Ambegaokar-McCumber-Halperin (LAMH) theory of thermally activated phase slips (TAPS) [25–27], without any QPS contribution. The TAPS physics is dominant, even in the vicinity of the SIT. Thus we conclude that the observed transition occurs between a truly superconducting phase (which shows no QPS and thus the resistance approaches zero resistance at T = 0) and an insulating phase in which the wire is in the normal state and the transport is controlled by weak Coulomb blockade. The nanowires, ranging in length between 43 and 187 nm, are fabricated by sputtering of amorphous Mo79Ge21 alloy on top of suspended fluorinated single-wall carbon nanotubes [3,4, 20]. The wires are homogeneous as is seen from scanning electron microscope (SEM) images (fig. 1a). The electrodes are deposited during the same sputtering run as the wire itself. Since Ar-atmosphere sputter-deposition is isotropic, and due to the small diameter of the nanotubes (∼1–2 nm), the wires, which occur on the outer surface of the nanotube, form seamless connections to the electrodes. The homogeneity of wires is confirmed also by the fact that their normal resistance is close to that estimated from the sample geometry and known bulk resistivity (∼200μΩcm) [3, 28]. Transport measurements are performed in He and He cryostats equipped with leads filtered against electromagnetic noise [29]. One sample (sample F) was measured down to ∼20mK. Resistance vs. temperature, R(T ), data for insulating and superconducting samples are shown in figs. 1b and c, respectively, where R(T ) ≡ dV/dI at V → 0. The resistive transiA. T. Bollinger et al.: Dichotomy in short superconducting nanowires 507 tion observed in all samples at higher temperatures is that of the film electrodes, which are connected in series with the wire. As the electrodes go superconducting, the total sample resistance equals the wire resistance. The only changing parameter amongst all samples is the nominal thickness of MoGe (4.0–8.5 nm). Consistently, the critical temperature of the electrodes decreases gradually as we proceed from the superconducting sample corresponding to the largest amount of MoGe sputtered (sample 6) to the insulating sample with the smallest amount of MoGe sputtered (sample B). The resistance measured immediately below the film transition is taken as the normal (or high-temperature) resistance of the wire, RN . The R(T ) curves of superconducting samples are shown in fig. 1c. The fits are made using the LAMH-TAPS formulas [20]. As the resistance starts to sharply drop with the cooling, the LAMH model becomes valid, and the data exhibit an excellent agreement with the fits. For the two thinnest samples (1 and 2), which have the lowest Tc’s, the thermodynamic critical field Hc(T ) in the free energy barrier for phase slips had to be modified: We used the empirical expression [30] Hc(T ) ∝ 1− (T/Tc), instead of the usual Hc(T ) ∝ 1− T/Tc. After such modification the fits matched the data. A striking result is that our set of ultrashort wires exhibits an excellent agreement with the TAPS model, without requiring any QPS contribution [7, 10]. This is even true for wires in the vicinity of the SIT. Good agreement with the LAMH model indicates that the observed superconducting regime is a “true” superconducting phase, i.e. the wire resistance is expected to rapidly approach zero as T → 0. In this regime, the wires R(T ) exhibits a negative curvature on ln(R) vs. T plots, which is an indication that the contribution of QPS [7, 10, 19, 21, 31] is negligible. Such QPS-free regime is new and was not seen on longer wires [7, 8, 19,22]. The R(T ) curves of nonsuperconducting wires show a qualitatively different behavior (fig. 1b). We term them “insulating” because they reproducibly show i) an upturn at the lowest temperatures, i.e. dR(T )/dT < 0 (down to ∼20mK, as was tested for sample F), and ii) a zero-bias resistance peak, i.e. dV (I)/dI < 0 for V → 0 (I is the bias current and V is the bias voltage). The observed abrupt change from TAPS to the insulating behavior strongly suggests that a quantum phase transition does occur in ultrashort nanowires. This is in contrast to longer wires, which exhibit a crossover [7, 8, 19, 22] from a quasi-superconducting to a quasi-normal regime. We speculate that this SIT takes place due to coupling of QPS to gapless excitations in the environment [11, 13–16], similar to the Chakravarty-Schmid transition [31–34]. The QPS-free regime can be understood assuming that the QPS are completely suppressed by a Caldeira-Leggett environment (e.g. produced collectively by the QPS cores). In the insulating phase the QPS proliferate and completely suppress superconductivity, again due to normal cores associated with each QPS. To understand dR(T )/dT < 0 seen in our insulating samples, we consider the theories of the Coulomb blockade (CB) in diffusive normal wires [23, 24, 35, 36]. Nazarov showed that the CB can survive in a setting in which two plates of a capacitor C are connected by a homogeneous normal wire (which now plays the role of a tunnel barrier), even if its resistance is much lower than the von Klitzing constant, RK = h/e, provided that the wire acts as a coherent scatter [23]. Golubev and Zaikin (GZ) [24] derived useful I(V ) formulas, enabling direct comparisons with experiments. At high temperatures (kBT > EC , where EC = e/2C is the charging energy) the zero-bias conductance, G(T ) = 1/R(T ) is G(T ) G0 1− β [ EC 3kBT − ( 3ζ(3) 2π4 g + 1 15 )( EC kBT )2] , (1) 508 EUROPHYSICS LETTERS Fig. 2 – Pekola et al. [38] conductance function γ(T ) ≡ G0[G0−G(T )]−1 plotted vs. temperature for all insulating samples. The solid lines are fits to the GZ theory (eq. (2)). Values of the fitting parameter G0 for samples A–H are G −1 0 = 6.14, 7.14, 7.93, 7.76, 9.78, 9.97, 17.33, and 26.10 kΩ, respectively. As expected, they are close to the corresponding normal resistances of the wires: RN = 6.43, 7.54, 8.25, 8.35, 10.33, 10.50, 18.05, and 32.46 kΩ. The predicted range of the offsets is shown by arrows. The lengths of these wires are 46, 45, 140, 105, 140, 49, 120, and 86 nm. where β = 1/3 for diffusive wires. Also, ζ(3) ∼= 1.202, g ≡ G0RK , and G0 is the conductance in the absence of the CB. Apart from the second-order term (EC/kBT ), this result is the same as obtained by Kaupinnen and Pekola for a single tunnel junction [37]. Originally such result was derived for a primary thermometer [38]. The same expression was derived by Joyez and Esteve [39] for a single tunnel junction (i.e. for the dynamic Coulomb blockade), though in their case the value of parameter g is defined differently: GJE ≡ RK/Renv (Renv is the impedance of the environment). If the ratio EC/kBT is a small parameter, the diverging terms of eq. (1) can be removed by rewriting it as γ(T ) ≡ G0 G0 −G(T ) ∼= 3kBT βEC + 9 β ( 3ζ(3) 2π4 g + 1 15 )