1. Quantum Phase Tunneling in 1D Superconductors K. Arutyunov, M. Zgirski, M. Savolainen, K.-P. Riikonen, V. Touboltsev University of Jyväskylä, Department of Physics, Jyväskylä, 40014, FINLAND SUMMARY 1. Introduction: 1.1 What determines shape of a superconducting transition R(T)? 1.2 Fluctuations vs. system dimensionality. 2. Thermal PS activation in 1D channels. 3. Quantum PS activation in 1D channels.

2. What determines experimentally observed shape of a superconducting transition R(T)? homogeneity of the sample response time of the measuring system thermodynamic fluctuations quick response measurements, but inhomogeneous sample realtively homogeneous sample, but very slowly response inhomogeneous sample and unrealistically fast response  T c measure,  T c sample <  T c fluct  T c exp = MAX (  T c sample,  T c measure,  T c fluct ) Hereafter we assume: homogeneous sample, the measuring system is fast enough to follow accordingly the temperature sweeps, but ’integrates’ contributions of instant thermodynamics fluctuations

3. Fluctuations vs system dimensionality SN superconductornormal metal top bottom S inclusions reduce the total system resistance  rounded top  FLUCT ~ (T-T c ) –(2-D/2) (Aslamazov – Larkin) 3D 2D 1D no contribution of N inclusions: normal current is shunted by supercurrent  abrupt bottom N inclusions block the supercurrent  rounded bottom (Langer – Ambegaokar)

4. Dimensionality of a superconductor Dimensionality of a system is set by the relation of characteristic physical scale to corresponding sample dimension L. For a superconductor this scale is set by the temperature - dependent superconducting coherence length  (T). Coherence length tends to infinity at critical temperature.

5. Thermal fluctuations J.S. Langer, V. Ambegaokar, Phys. Rev. 164, 498 (1967), D.E. McCumber, B.I. Halperin, Phys. Rev. B 1, 1054 (1970)  (T ) √  <<  (T) Infinitely long 1D wire of cross section  If the wire is infinitely long, there is always a finite probability that some fragment(s) will instantly become normal The minimum length on which superconductivity can be destroyed is the coherence length  (T). The minimum energy corresponds to destruction of superconductivity in a volume ~  (T)  :  F = B c 2  (T) , where B c (T) is the critical field. If the thermal energy k B T is the only source of destruction of superconductivity, then in the limit R(T) << R N the effective resistance is proportional to the corresponding probability: R(T) ~ exp (-  F / k B T) Experiment: J. E. Lukens, R.J. Warburton, W. W. Webb, Phys. Rev. Lett. 25, 1180 (1970) R. S. Newbower, M.R. Beasley, M. Tinkham, Phys. Rev. B 5, 864, (1972)

6. Phase slip concept Let us consider macroscopically coherent superconducting state. It can be characterized by a wave function  = |  | e i  Dependence of the free energy F vs. superconducting phase  of a 1D current-carrying superconductor can be represented by a tilted ‘wash board’ potential with the barrier height  F. The system can change its quantum state in two ways: 1. via thermally activated phase slips 2. via quantum tunneling. FF  ’s denote rates for both processes. Both processes in case of non-zero current lead to energy dissipation  finite resitance

7. Existing experiments on QPS N. Giordano and E. R. Schuler, Phys. Rev. Lett. 63, 2417 (1989) N. Giordano, Phys. Rev. B 41, 6350 (1990); Phys. Rev. B 43, 160 (1991); Physica B 203, 460 (1994) A. Bezyadin, C. N. Lau and M. Tinkham, Nature 404, 971 (2000) C. N. Lau, N. Markovic, M. Bockrath, A. Bezyadin, and M. Tinkham, Phys. Rev. Lett. 87, 217003 (2001) ’Unique’ nanowires of classical superconductors MoGe film on top of a carbon nanotube more systematic study is required !

8. Samples: fabrication & shape control Objective: to enable R(T) measurements of the same nanowire with progressively reduced diameter 50 nm Before sputtering After sputtering ion beam sputtering enables non-destructive reduction of a nanowire cross section ion beam sputtering provides ’smoth’ surface treatment removing original roughness wire height (nm) number of scans Before sputtering After sputtering

9. R(T) transitions vs. wire diameter Solid lines are fits using PS thermal activation model Langer-Ambegaokar / McCumber-Halperin Effect of sputtering The shape of the bottom part of the R(T) dependencies of not too narrow Al wires can be nicely described by the model of thermal activation of phase slips Wires are sufficiently homogeneous!

10. Current-induced activation of phase slips At a given temperature T < T c transition to a resistive state can be induced by a strong current * I-V characteristics single step transition single phase slip center activation ’short’ wire limit I c (T) I c ~ T 3/2 ’true’ 1D limit * R. Tidecks ”Current-Induced Nonequilibrium Phenomena in Quasi-One-Dimensional Superconductors”, Springer, NY, 1990. Sample: L = 10  m √σ ~ 70 nm

11. Quantum Phase Tunneling in case of a short wire (simplified model) A. Zaikin, D. Golubev, A. van Otterlo, and G. T. Zimanyi, PRL 78, 1552 (1997) A. Zaikin, D. Golubev, A. van Otterlo, and G. T. Zimanyi, Uspexi Fiz. Nauk 168, 244 (1998) D. Golubev and A. Zaikin, Phys. Rev. B 64, 014504 (2001) Full model (G-Z) only a single phase slip can be activated at a time If the wire length L is not much larger than the temperature dependent superconducting coherence length  (T), then only a single phase slip can be activated at a time: simplified model QPS are activated at a rate:  QPS = B exp (-S QPS ), where B ≈ (S QPS /  0 ) · (L /  ), S QPS = A · (R Q / R N ) · (L /  ), A ~ 1, R Q = h / 4e 2 = 6.47 k  R N – normal state resistance,  0 = h /  duration of each QPS. V t 00  QPS  V QPS Each phase slip event creates instantly a voltage jump:  V QPS = I ·R N ·(  / L), where I is the measuring current. Defining the effective resistance as R(T) ≡ R eff = / I, one gets: Time-averaged voltage =  V QPS · (  0 ·  QPS ). R eff / R N = (  / L) · (  0 ·  QPS )

12. Experimental evidence of QPS After etching the wire becomes thinner Top part (in logarithmic scale) of the R(T) transition can be nicely fitted by the Langer- Ambegaokar model of thermal phase slip activation For the thinner wire a ’foot’ develops at the very bottom part, which cannot be fitted by L-A model at any reasonable parameters of the sample Quantum phase slip mechanism?

13. Conclusions  the method enables galvanomagnetic measurements of the same nanowire in between the sessions of sputtering  a ’foot’ develops at the low temperature part of the R(T) dependencies of the very thin Al wires, which can be assosiated with quantum phase slip phenomena  the shape of the bottom part of the R(T) dependencies of not too narrow Al wires can be nicely described by the model of thermal activation of phase slips  ion beam sputtering method has been developed to reduce the cross section of lift-off pre-fabricated Al nanowires