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:
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K. Arutyunov, M. Zgirski, M. Savolainen, K.-P. Riikonen, V. Touboltsev
University of Jyväskylä, Department of Physics, Jyväskylä, 40014, FINLAND
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.
homogeneity of the sample
response time of the measuring system
quick response measurements, but inhomogeneous sample
realtively homogeneous sample, but very slowly response
inhomogeneous sample and unrealistically fast response
dTcexp = MAX (dTcsample, dTcmeasure, dTcfluct)
Hereafter we assume:
homogeneous sample, the measuring system is fast enough to follow accordingly the temperature sweeps, but ’integrates’ contributions of instant thermodynamics fluctuations
dTcmeasure, dTcsample < dTcfluct
no contribution of N inclusions: normal current is shunted by supercurrent
S inclusions reduce the total system resistance
sFLUCT ~ (T-Tc) –(2-D/2)
(Aslamazov – Larkin)
N inclusions block the supercurrent
(Langer – Ambegaokar)
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 x(T). Coherence length tends to infinity at critical temperature.
J.S. Langer, V . Ambegaokar, Phys. Rev. 164, 498 (1967), D.E. McCumber, B.I. Halperin, Phys. Rev. B 1, 1054 (1970)
Infinitely long 1D wire of cross section s
√s << x(T)
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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 x(T).
The minimum energy corresponds to destruction of superconductivity in a volume ~ x(T) s:
DF = Bc2x(T) s, where Bc(T) is the critical field.
If the thermal energy kBT is the only source of destruction of superconductivity, then in the limit R(T) << RN the effective resistance is proportional to the corresponding probability:
R(T) ~ exp (- DF / kBT)
’s denote rates for both processes.Phase slip concept
Let us consider macroscopically coherent superconducting state. It can be characterized by a wave function
Y = |Y| eij.
Dependence of the free energy F vs. superconducting phase j of a 1D current-carrying superconductor can be represented by a tilted ‘wash board’ potential with the barrier height DF.
The system can change its quantum state in two ways:
1. via thermally activated phase slips
2. via quantum tunneling.
Both processes in case of non-zero current lead to energy dissipation
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’Unique’ nanowires of classical superconductors
MoGe film on top of a carbon nanotube
more systematic study is required
number of scans
wire height (nm)Samples: fabrication & shape control
Objective: to enable R(T) measurements of the same nanowire with progressively reduced diameter
ion beam sputtering enables non-destructive reduction of a nanowire cross section
ion beam sputtering provides ’smoth’ surface treatment removing original roughness
Effect of sputtering
Solid lines are fits using PS thermal activation model
Langer-Ambegaokar / McCumber-Halperin
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!
L = 10 mm
√σ ~ 70 nm
At a given temperature T < Tc transition to a resistive state can be induced by a strong current *
single step transition
Ic ~ T3/2
single phase slip center activation
’true’ 1D limit
’short’ wire limit
* R. Tidecks ”Current-Induced Nonequilibrium Phenomena in Quasi-One-Dimensional Superconductors”, Springer, NY, 1990.
1 / GQPS
tQuantum Phase Tunneling in case of a short wire (simplified model)
Full model (G-Z)
A. Zaikin, D. Golubev, A. van Otterlo, and G. T. Zimanyi, PRL 78, 1552 (1997)
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If the wire length L is not much larger than the temperature dependent superconducting coherence length x(T), then only a single phase slip can be activated at a time: simplified model
QPS are activated at a rate: GQPS = B exp (-SQPS),
where B ≈ (SQPS / t0) · (L / x), SQPS = A·(RQ / RN)·(L / x), A ~ 1, RQ = h / 4e2 = 6.47 kW,
RN – normal state resistance, t0 = h / D - duration of each QPS.
Each phase slip event creates instantly a voltage jump: DVQPS = I·RN·(x / L), where I is the measuring current.
Time-averaged voltage <V> = DVQPS· (t0 · GQPS).
Defining the effective resistance as R(T) ≡ Reff = <V> / I,
Reff / RN = (x / L) · (t0 · GQPS)
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?