Quantum Tunneling of Normal-Superconductor Interfaces in a Type-I Superconductor

QUANTUM TUNNELING OF NORMAL-
SUPERCONDUCTOR INTERFACES IN A TYPE-I
SUPERCONDUCTOR

J. Tejada, S. Vélez, A. García-Santiago, R. Zarzuela, J.M. Hernández
Grup de Magnetisme, Dept. de Física Fonamental, Universitat de Barcelona

E. M. Chudnovsky
Lehman college, City University of New York, New York.

Outline

1. Introduction:
1.1 Type-I superconductivity.
1.2 Intermediate state and flux structures.
1.3 Magnetic irreversibility.

2. Experimental Results:
2.1 Topological hysteresis and pinning.
2.2 Thermal and non-thermal behaviors in the magnetic irreversibility.
2.3 Quantum tunneling of Normal-Superconductor interfaces.
2.4 Phase diagram of flux motion in a type-I superconductor.

3. Model: FlatteningBumping of NSI at the defects

4. Conclusion: A new physical phenomena discovered: QTI

1.1 Type-I superconductivity. Basics

A superconductor, aside to has
zero resistance, it is
characterized by the Meissner
state: In the superconducting
state any flux line cannot
penetrate inside the sample.
Perfect screening of the
external applied magnetic field

The superconducting state only can exist below a
certain critical temperature T<Tc if any.

However, for strong enough applied magnetic fields
H>Hc (or even for strong applied currents), the
superconducting state is suppressed.

H c (T ) = H c 0 [1 − (T / Tc ) 2 ]
Phase Diagram of a
NormalSuperconductor

1.1 Type-I superconductivity. Basics

Magnetic properties At different temperatures

H Hc
M H Hc(T3) Hc(T2) Hc(T1)
M

M = −H for H < H c (T )

M =0 for H > H c (T )

1.2 Intermediate state. Geometric effects
Geometry of the sample
⇓
Non uniformity of external magnetic field over the space (around the sample)
⇓
Free energy reach SC-N transition at certain H’c<Hc
⇓
SC-N transition becomes gradual between H’c<H<Hc
⇓
Coexistence of SC and N regions: Intermediate state

M Hc’ Hc

H

Formation of N-SC strips in a plane. First idea
introduced by Landau (1938)

Hc’ = (1 – N) Hc N : Demagnetizing factor

1.2 Intermediate state. Magnetic properties
B
H
Hc
M

-Hc

N ~1 N =1/2 N =1/3 N ~0 H
N ~1 N =1/2 N =1/3 N ~0
N ~ 1 infinite slab with H applied perpendicular to the surface
N = 1/2 infinite cylinder with H applied perpendicular to the revolution axis
N = 1/3 sphere
N~0 infinite cylinder with H applied parallel to the revolution axis or
infinite slab with H applied parallel to the surface

H Important points:
M =− for ′
H < H c (T )
1− N 1) All “new” properties can be described through N
H −H 2) Reversible system is expected. Penetration an
M =− c for ′
H c (T ) < H < H c (T ) expulsion of magnetic flux in the intermediate state
N
should follow same states

1.2 Intermediate state. Flux structures

Magneto-optical imaging of the flux structures formed in the intermediate state: Thin Slabs
Laminar patterns  Landau description of the IS

V. Sharvin, Zh. Eksp. Teor. Fiz. 33, 1341 (1957).
T. E. Faber, Proc. Roy. Soc. (London) A248, 460 (1958).

Applied magnetic field in plane Applied magnetic field is perpendicular to the plane
Regular strip patterns! Laberinthic patters: Random growth/movement

1.3 Magnetic irreversibility. Historical point of
view

J. Provost, E. Paumier, and A. Fortini, J. Phys. F: Met. Phys. 4, 439 (1974)

However, irreversibility was mostly observed in slab-shaped samples when the applied magnetic
field was perpendicular to the plane of the surface… WHY?
Would be a correlation between different flux structures observed, magnetization dynamics and the
shape of the sample against the direction of the applied magnetic field?

1.3 Magnetic irreversibility. Prozorov’s point of
view:
Topological hysteresis
Flux penetration: bubbles

Flux expulsion: lamellae Interpretation: there is a GEOMETRICAL BARRIER
which controls both the penetration and the
expulsion of the magnetic flux in the intermediate
state and is the responsible of both the intrinsic
irreversibility of a pure defect-free samples and
the formation of different flux patterns.
This irreversibility is called TOPOLOGICAL
HYSTESRESIS and vanishes when H tends to 0

R. Prozorov, Phys. Rev. Lett. 98, 257001 (2007).

1.3 Magnetic irreversibility. Prozorov’s point of
view:
Topological hysteresis
Different thin slab samples of Pb where studied. H was always applied perpendicular to the surface.

Different flux structures appear in the Intermediate
state depending on the history: Tubs/bubbles are
formed during magnetic field penetration and
Upper panel, sample with stress defects laberinthic patterns appear upon expulsion
Lower panel, defect-free sample

R. Prozorov et al., Phys. Rev. B 72, 212508 (2005)

1.3 Topological hysteresis. Suprafroth state
The growth of flux bubbles in the IS in a defect-free sample resembles the behavior of a froth.

Minimation of the free energy when the
Suprafroth grows:
tends to 6 interfaces for each bubble:
hexagonal lattice!

2. Experimental Results. Set-up

All magnetic measurements were performed in a commercial
superconducting quantum interference device (SQUID)
magnetometer (MPMS system) which allows to work at
temperatures down to T = 1.80 K and it is equipped with a
continuous low temperature control (CLTC) and enhanced
thermometry control (ETC) and showed thermal stability
better than 0.01 K.

In all measurements the applied magnetic field does not
exceed H = 1 kOe strength.

The samples studied were: a sphere (Ø = 3 mm), a cylinder

(L = 3 mm, Ø = 3 mm) and several disks (L ~ 0.2 mm, Ø = 6
mm) of lead prepared using different protocols (cold rolling,
cold rolling+annealing, melting and fast re-crystallization) and
MPMS SQUID Magnetometer were studied over different orientations.

2.1 Topologycal hysteresis and pinning
A h = Ø = 3 mm
t = 0.2 mm, S = 40 mm2
B
H C same as B, annealed
H (glycerol, 290ºC, 1h, N2)
D Ø = 3 mm

Sample A
1.0
Sample B
Sample C
Sample D
0.5
H
No zero M when H0

M/Mmax
0.0

-0.5

-1.0
The effect of the geometrical barrier strongly depend
on the orientation of the applied magnetic field with -750 -500 -250 0 250 500 750
H (Oe)
respect to the sample geometry.

Defects are the reason for the observation of a remnant
Topological irreversibility appears in disk (cylinder)- magnetization at zero field. Enhancement of the
shaped samples only when the applied magnetic irreversibility of the system.
field is parallel to the revolution axis.
S Vélez et al., Phys. Rev. B 78, 134501 (2008).

2.1 Topological hysteresis and pinning
1.0 Sample A A h = Ø = 3 mm
Sample B
Sample C
0.5 Sample D B t = 0.2 mm, S = 40 mm2
H C same as B, annealed
M/Mmax

0.0 (glycerol, 290ºC, 1h, N2)
D Ø = 3 mm
-0.5

Non zero M when H0
-1.0 0.15

-750 -500 -250 0 250 500 750 0.10
H
H (Oe) 0.05

M (emu)
0.00
Non-annealed sample exhibit higher remnant flux. Defects
enhance the capability of the system to trap magnetic flux -0.05

-0.10
In conclusion, defects act as a pinning centers that avoids the Sample B
-0.15 T = 3 K
complete expulsion of the magnetic flux as is expected in the -800 -600 -400 -200 0 200 400 600 800
defect-free case. H (Oe)

⇓ Parallel configuration. No irreversibility
The resemblances of domain walls in ferromagnets and the
⇓
movement on Normal-Superconductor Interfaces in type-I Absence of vortices: No type-II
superconductors points to pinning of NS interfaces at the
superconductivity
defects et al., Phys. Rev. B 78, 134501 (2008).
S Vélez


Schematic view of the texture of a NS Interface in presence
of structural defects. Around the defects, a bump in the
NS Interface could be generated. For a strong enough
pinning potential, the interface must not move freely through
the sample when the external magnetic field, H, is swept:
Enhancement of the magnetic irreversibility of the system.

E. M. Chudnovsky et al., Phys. Rev. B 83, 064507 (2011).



M(H) data of a disk with stress defects.
Effect of pinning potentials upon expulsion? REMEMBER

0.2
Topological equilibrium Points correspond to
0.0 the FC data,
m (arb. units)

whereas line is the
-0.2 whole M(H) cycle
obtained
-0.4
Solid simbols: M(H) loop
-0.6 Open simbols: FC data
0.0 0.1 0.2 0.3 0.4 0.5 0.6 R. Prozorov et al., Phys. Rev. B 72, 212508 (2005)
h



0.2
M(H) data of a disk with stress defects.
Effect of pinning potentials upon expulsion? 0.0

m (arb. units)
-0.2
0.2
Topological equilibrium?
0.0 -0.4
m (arb. units)

-0.2 -0.6 Open simbols: FC data
0.0 0.1 0.2 0.3
-0.4 h
Open simbols: FC data
-0.6 Enhanced irreversibility along the descending branch
0.0 0.1 0.2 0.3 0.4 0.5 0.6
due to the existence of defects.
h
Pinning of Normal-Superconductor Interfaces!

2.2 Magnetic irreversibility at different temperatures
The magnetic properties of any reversible type-I superconductor scale with the thermodinamical crytical
field Hc. Using the so-called reduced magnitudes:

m = M / Hc h = H / Hc

all the M(H) curves measured at different temperatures collapses in a single m(h) curve.

Actually, the magnetic properties of an irreversible sample are also related to Hc. Therefore, any deviation
between the different (and hysteretic) m(h,T) curves should be related to other thermal effects than those
related to Hc.

1.0
Cylinder Pb A defect-free sample does not exhibit thermal
H effects in the magnetic irreversibility.
0.5
⇓
The topological hysteresis is thermally
m

0.0
independent!
2.00 K
-0.5 3.00 K ⇓
4.00 K
5.00 K Equivalent flux structures should be
6.00 K
-1.0
-1.0 -0.5 0.0 0.5 1.0
formed for a given h
h

S. Vélez et al., arxiv:cond-mat.suprcon/1105.6218

2.2 Magnetic irreversibility at different temperatures.
1.0
(a) Annealead disk 0.6 2.0 K
H Thermaly dependent
2.5 K 4.5 K
upon flux expulsion
0.5 3.0 K 5.0 K
3.5 K 5.5 K
4.0 K 6.0 K
m

0.0
0.0

m
2.0 K
-0.5 2.5 K 4.5 K
3.0 K 5.0 K FC data
3.5 K 5.5 K
-1.0 4.0 K 6.0 K -0.6
1.0
(b) Cold rolled disk All curves stick togheter
H during penetration
0.5
0.0 0.2 0.4 0.6
h
m

0.0

-0.5 2.0 K 1) Thermal dependencies appear only during flux expulsion.
2.5 K 4.5 K
3.0 K 5.0 K 2) Flux penetration is quite similar for all samples and resembles how
3.5 K 5.5 K
-1.0 4.0 K 6.0 K must be the defect free one
1.0 (c)
Recristalized disk 3) At fixed T, irreversibility increases from (a) to (c) in accordance with the
H expected strength of the pinning potentials
0.5
4) For a given sample, as higher T is, smaller the irreversibility becomes.
m

0.0
⇓
-0.5 2.0 K
2.5 K 4.5 K
Thermal effects should be related to the thermal activation of the
3.0 K
3.5 K
5.0 K
5.5 K
NSI when they are pinned by the defects.
-1.0 4.0 K 6.0 K
-1.0 -0.5 0.0 0.5 1.0
The pinning potentials should follow an inverse functionality with h
h

2.3 Magnetic relaxation experiments. Basics.
One simple experiment to test the metastability of a system: Magnetic relaxation

0.2 1
2
0.0
m (arb. units)

2
-0.2
1
-0.4
0.0 0.1 0.2 0.3 0.4 0.5 0.6
h

1) We keep T fixed and we apply H>Hc. Then the magnetic field is reduced to a desired H and
subsequently, the time evolution of the magnetic moment, M(t), is recorded .
2) Keeping constant h, the temperature is increased above Tc and then reduced to a desired T.
When it is reached, time evolution of the magnetic moment, M(t) is recorded

2.3 Magnetic relaxation experiments. Basics.
One simple experiment to test the metaestability of a system: Magnetic relaxation

0.2
1 Solid simbols: M(H) loop
Open simbols: FC data
m (arb. units)

0.0 0.2
1 Metastable
2

m (arb. units)
-0.2
2
states
-0.4
0.0
0.0 0.1 0.2 0.3
h
0.4 0.5 0.6
1
-0.2
2 Stable states
0.0 0.1 0.2 0.3
h

1 (Metastable state) relax towards 2 (stable state).
We can study the magnetic viscosity at several points
1
along Mdes (h) and repeat the process at different T

2.3 Magnetic relaxation experiments: Magnetic viscosity
S.
1.00

0.96
2.0 K

Mirr (t)/Mirr (0)
2.5 K
3.0 K
0.92 3.5 K
4.0 K
4.5 K
0.88 5.0 K
5.5 K
6.0 K
6.5 K
0.84 7.0 K
5 6 7 8
2
ln [t (s)]

1
1 From the slope  S

This law is followed for any system which has a broad distribution of energy barriers
as the sources of both the metastability and any normalized magnetic relaxation rate
obtained, S.

2.3 Magnetic relaxation experiments: Quantum tunneling.
0.005
1.00

0.004
0.96
2.0 K
Mirr (t)/Mirr (0)

2.5 K
0.003
3.0 K
0.92 3.5 K

S
4.0 K
0.002
4.5 K
0.88 5.0 K
5.5 K
0.001 2
6.0 K
6.5 K
0.84 7.0 K
0.000
5 6 7 8 2 3 4 5 6 7
ln [t (s)] T (K) 1
M. Chudnovsky, S. Velez, A. Garcia-Santiago, J.M. Hernandez and J. Tejada, Phys. Rev. B 83, 064507 (2011).

S tends a finite non-zero value when T0
Quantum tunneling of the Normal-
Superconducting interfaces!

2.3 Magnetic relaxation experiments. Quantum tunneling.

1.000 1.000

0.996
0.996
h = 0.323
h = 0.296
0.992 T = 6.60 K

mirr(t)/mirr(0)
mirr(t)/mirr(0)

h = 0.269
h = 0.242 0.992 T = 6.30 K
h = 0.215 T = 6.00 K
0.988 h = 0.188 T = 5.70 K T = 3.60 K
h = 0.161 T = 5.40 K T = 3.30 K
0.988 T = 5.10 K
h = 0.134 T = 3.00 K
0.984 h = 0.108 T = 4.80 K T = 2.70 K
h = 0.081 T = 4.50 K T = 2.40 K
h = 0.054 0.984 T = 4.20 K T = 2.10 K
0.980 h = 0.027 T = 3.90 K T = 1.80 K

5 6 7 8 5 6 7 8
ln[t(s)] ln[t(s)]

Perfect logarithmic time dependence of M(t) for several T and h


2.3 Magnetic relaxation experiments. Quantum tunneling.

h = 0.00 (a) T = 2.00 K
0.012 h = 0.10 0.004 T = 4.00 K
h = 0.15 T = 5.00 K
0.009 h = 0.20 0.003
h = 0.25

0.006 0.002
S

S
0.003 0.001
(b)
0.000 0.000
2 3 4 5 6 7 0.00 0.05 0.10 0.15 0.20 0.25
T (K) h

We can identify the transition between two different regimes:
Quantum and thermal regimes

S Vélez et al., arxiv:cond-mat.suprcon/1105.6222

2.3 Phase diagram of flux motion.

0.6
0.0
h*(T) Magnetic irreversibility develops
Quasi-Free Flux Motion TQ(h), hQ(T) in this region --> h*
0.5

0.4 -0.2

m
Thermal Activation
0.3
h

2.0 K
2.5 K 4.5 K
-0.4 3.0 K 5.0 K
0.2 3.5 K 5.5 K
4.0 K 6.0 K

0.1 Quantum Tunneling 0.4 0.6
h
0.0
2 3 4 5 6 7 T increase  h* decrease
T (K)
h*  onset of the irreversibility) along Mdes

Experimental data suggest that the strength of the pinning potential barriers should follow a
decreasing magnetic field dependence.

The phase diagram of the dynamics of NSC Interfaces is developed. It shows the Quasi-free Flux
motion, Thermal Activation over the pinning potential as well as the quantum depinning regime


4. Model: FlatteningBumping of NSI at the deffects

Theoretically Experimentally Matching

TQ ~ 5 K L ~ 90 nm ~ ξ

U B (0) ~ 100 K a ~ 1 nm

E. M. Chudnovsky et al., Phys. Rev. B 83, 064507 (2011).

4. Model: quantum tunneling of interfaces

What happen with an applied magnetic field to the bumps?

Experimentally Theoretically
0.6
h*(T)
Quasi-Free Flux Motion TQ(h), hQ(T)
0.5
Decreasing
0.4
function
Thermal Activation
0.3
h

0.2
Also a decreasing
0.1 Quantum Tunneling
function
0.0
2 3 4 5 6 7
T (K)

L increase and a decrease as h increases

Bumps become flatter!!

5. Conclusion

The intermediate state of type-I superconductors
has been reactivated as an appealing
field of experimental and theoretical research
one century after the discovery of the superconductivity

THANK YOU VERY MUCH FOR
YOUR KIND ATTENTION!!

Quantum Tunneling of Normal-Superconductor Interfaces in a Type-I Superconductor

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