Quantum Phase Tunneling in 1D Superconductors

1
Quantum Phase Tunneling in 1D Superconductors
University of Jyväskylä, Department of Physics,
Jyväskylä, 40014, FINLAND

• K. Arutyunov, M. Zgirski, M. Savolainen, K.-P.
  Riikonen, V. Touboltsev

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
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 lt dTcfluct
3
Fluctuations vs system dimensionality
superconductor
normal metal
S
N
top
bottom
3D
no contribution of N inclusions normal current
is shunted by supercurrent ? abrupt bottom
S inclusions reduce the total system resistance ?
rounded top sFLUCT (T-Tc) (2-D/2) (Aslamazov
Larkin)
2D
N inclusions block the supercurrent ? rounded
bottom (Langer Ambegaokar)
1D
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 x(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)
x(T)
Infinitely long 1D wire of cross section s
vs ltlt x(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)
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 Bc2
x(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) ltlt RN the effective resistance is
proportional to the corresponding probability
R(T) exp (- DF / kBT)
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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 ? 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
ion beam sputtering enables non-destructive
reduction of a nanowire cross section
ion beam sputtering provides smoth surface
treatment removing original roughness
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R(T) transitions vs. wire diameter
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!
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Current-induced activation of phase slips

• At a given temperature T lt Tc transition to a
  resistive state can be induced by a strong
  current

I-V characteristics
Ic (T)
Sample L 10 mm vs 70 nm
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.
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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)
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 IRN(x / L), where I is the
measuring current.
Time-averaged voltage ltVgt DVQPS (t0 GQPS).
Defining the effective resistance as R(T) Reff
ltVgt / I, one gets
Reff / RN (x / L) (t0 GQPS)
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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?
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Conclusions

• ion beam sputtering method has been developed to
  reduce the cross section of lift-off
  pre-fabricated Al nanowires

• the method enables galvanomagnetic measurements
  of the same nanowire in between the sessions 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

• 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