Transcript Dimensionality of a superconductor

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.
What determines experimentally observed shape of a
superconducting transition R(T)?
homogeneity of the sample
response time of the measuring system
R
R
TC1 TC2 TC3
T
quick response measurements,
but inhomogeneous sample
thermodynamic fluctuations
R
TCMIN
TCMAX
TCMIN
T
realtively homogeneous sample,
but very slowly response
TCMAX
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
T
Fluctuations vs system dimensionality
N
normal metal
S
superconductor
top
bottom
3D
2D
S inclusions reduce the
total system resistance 
no contribution of N
inclusions: normal
current is shunted by
supercurrent 
rounded top
abrupt bottom
sFLUCT ~ (T-Tc) –(2-D/2)
(Aslamazov – Larkin)
N inclusions block the
supercurrent 
1D
rounded bottom
(Langer – Ambegaokar)
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.
7
, m
6
5
l = 30 nm
4
(Al) = 1.6 m
m
0(Sn) = 0.23
0
3
2
Aluminum
1
Tin
0
-0.080
-0.060
-0.040
-0.020
0.000
Temperature dependence of coherence length
c

where 
T-Tc, K
,
clean limit l 
l dirty limitl 
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)
Infinitely long 1D wire of cross section s
√s << (T)
Experiment:
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) s:
DF = Bc2 (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)
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)
Phase slip concept
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.
DF  Bc (T )  ,
2
F (free energy)
Let us consider macroscopically
coherent superconducting state. It can
be characterized by a wave function
   (T )  s
 DF
thermal  exp(
)
k BT
DF
 DF
quantum  exp(
)
q
0
2
phase j
4
’s denote rates for both processes.
Both processes in case of non-zero
current lead to energy dissipation 
finite resitance
Existing experiments on QPS
N. Giordano and E. R. Schuler, Phys.
Rev. Lett. 63, 2417 (1989)
A. Bezyadin, C. N. Lau and M. Tinkham, Nature
404, 971 (2000)
N. Giordano, Phys. Rev. B 41, 6350
(1990); Phys. Rev. B 43, 160 (1991);
Physica B 203, 460 (1994)
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
!
Samples: fabrication & shape control
Objective: to enable R(T) measurements of the same nanowire with progressively reduced diameter
RESEARCH FLOW-CHART
1. E-beam lithography /
development
Before sputtering
Fabrication
50 nm
2. Evaporation of Aluminium
3. Lift-off
SEM/AFM
After sputtering
Measure ments / control
image control
Ion Beam
Sputtering
SEM /AFM
Image/profile
number of scans
Before sputtering
After sputtering
wire height (nm)
control
R(T)Measurement
ion beam sputtering enables
non-destructive reduction of
a nanowire cross section
ion beam sputtering provides
’smoth’ surface treatment
removing original roughness
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!
Current-induced activation of phase slips
At a given temperature T < Tc transition to a resistive state can be induced by a strong current *
I-V characteristics
Ic (T)
Sample:
1.E-04
1.3424 K +/- 0.2 mK
1.3416 K +/- 0.1 mK
L = 10 mm
100
√σ ~ 70 nm
80
1.3342 K +/- 0.1 mK
1.3302 K +/- 0.1 mK
1.E-04
<Tc> = 1.378 K
1.3252 K +/- 0.1 mK
1.3203 K +/- 0.2 mK
1.3145 K +/- 0.1 mK
1.3090 K +/- 0.1 mK
1.3037 K +/- 0.2 mK
8.E-05
1.2990 K +/- 0.1 mK
2/3
)
1.2948 K +/- 0.1 mK
Ic 2/3 (nA
1.2900 K +/- 0.1 mK
V (V)
6.E-05
60
40
4.E-05
critical voltage
criterion
20
2.E-05
0
0.E+00
0.E+00
2.E-07
4.E-07
6.E-07
8.E-07
1.E-06
I (A)
-100
-80
-60
-40
-20
0
T - Tc (mK)
Ic ~ T3/2
single step transition
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.
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 (T), then only a single phase slip can be activated at a time: simplified model
QPS are activated at a rate: QPS = B exp (-SQPS),
where B ≈ (SQPS / t0) · (L / ), SQPS = A·(RQ / RN)·(L / ), A ~ 1, RQ = h / 4e2 = 6.47 k
RN – normal state resistance, t0 = h / D duration of each QPS.
V
t0
Each phase slip event creates instantly a voltage jump:
DVQPS = I·RN·( / L), where I is the measuring current.
QPS
Time-averaged voltage <V> = DVQPS · (t0 · QPS).
DVQPS
Defining the effective resistance as R(T) ≡ Reff = <V> / I,
one gets:
Reff / RN = ( / L) · (t0 · QPS)
t
Experimental evidence of QPS
1
original
RN = 37.1 Ohm
0.1
0.01
R / RN
etched 25 min
RN = 75.6 Ohm
L-A
Tc = 1.2760 K
ℓ = 25 nm
Bc(0) = 8 mT
√σ = 80 nm
0.001
L-A
Tc = 1.3745 K
ℓ = 8 nm
Bc(0) = 12 mT
√σ = 35 nm
0.0001
G-Z
A = 0.011
0.00001
0.000001
1.260
1.280
1.300
L = 5 mm
1.320
1.340
1.360
1.380
1.400
T (K)
After etching the wire becomes thinner
Top part (in logarithmic scale) of the R(T) transition can be nicely fitted by the LangerAmbegaokar 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?
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