Bardeen-Stephen flux flow law
disobeyed in Bi2Sr2CaCu2O8+δ
G. Kriza,1,2 A. Pallinger1, B. Sas1, I. Pethes1, K. Vad3, F. I. B.Williams1,4
1Research
Institute for Solid State Physics and Optics,
Budapest, Hungary
2Institute of Physics, Budapest University of Technology and Economics,
Budapest, Hungary
3Institute of Nuclear Research, Debrecen, Hungary
4Service de Physique de l’Etat Condensée,
Direction Sciences de la Matière, Comissariat à l’Energie Atomique,
Gif-sur-Yvette, France and
Aim: Measure Free Flux Flow (FFF) resitivity in the
high-Tc superconductor Bi2Sr2CaCu2O8+δ (BSCCO)
 B  ab
Transport
current JT
Abrikosov vortex
FL vs. J T and E vs. v L are independen t of material properties
v L vortex line velocity
FL  J T  zˆ 0 Lorentz force
Material properties enter via the velocity – force relation v L vs. FL ( 0  h / 2e is the flux quantum)
In convention al supercondu ctors :
 FFF   n
B
Bc 2
 n : normal state resistivit y
Bc 2 : upper critical field
Bardeen–Stephen law (BS)
Induced electric field :
E  B  vL
JT
 dissipatio n
What is known of ρFFF in high-Tc superconductors?
• No clear experimental evidence for BS law in any high-Tc SC
(nor in any unconventional superconductor)
• No theory takes into account all the essential ingredients
1
This experiment
 B 

 FFF   n 
 Bc 2 
3/ 4
FFF /n
We find in BSCCO :
0
BS law
0
B/Bc2
1
Complication: How to account for the pinning force?
Pinning force  uncertainty in the total force F  uncertainty in velocity – force relation
Solution
Model pinning
(e.g., to interpret
surface impedance)
Unpinned
vortex liquid
state near Tc
Create conditions
where pinning is
irrelevant
(our approach)
Apply high current
so that the pinning
force is negligible
in front of the
Lorentz force
Experiment
c
B  ab
BSCCO single crystal:
a
b
Current excitation:
Voltage
response
Voltage – current
(V – I)
characteristics
Typical voltage – current characteristics
Voltage (mV)
8
V/I varies with I
up to the highest current
 pinning is not negligible
T = 25 K, B = 2 T
6
4
2
0
0
200
400
600
800
Current (mA)
20
Rab= dV/dI saturates (becomes current independent)
at the high current.
If Rab = dV/dI = const, then
for I →  , V/I → Rab
T = 25 K, B = 2 T
dV/dI (m)
15
Rab
10
5
0
Ith
0
200
400
600
800
Current (mA)
The differential resistance Rab measures the high-current limit of the resistance V/I
 We assume that Rab reflects the free flux flow limit
Temperature and field dependence of the high-current
differential resistance Rab
Differential resistanceRab (m)
300
250
High temperature:
sublinear B-dependdence
200
150
100
5K
40 K
75 K
50
0
0
1
10 K
60 K
2
3
4
Magnetic field (tesla)
30 K
70 K
5
Low temperature:
T and B independent
resistance
Empirical form for the high-current differential resistance Rab
300
Interpolating function:
Bc2(T) = Bc2(0)[1– (T/Tc)2]
with Bc2(0) = 120 tesla to give
Rab (m)
250
200
Rab (m)
0
2
4
0
(T)
150
100
50
0
0.001
0.01
dBc 2
dT
5K
10K
30K
40K
60K
70K
75K
0.1
 2.7 T/K
Tc
as in Qiang Li et al.,
Phys. Rev. B 48, 9877 (1993).
1
B/Bc2
Empirical form for the B and T dependence of the resistance:
n
1   log( B / Bc 2 (T ))
Rab ( B, T )  Rab
n
Rab
is the normal resistance at Tc
  0.2 (  0.16, 0.19, and 0.21 in three different samples)
Dependence of the resistance Rab on the local resistivities c and ab
Anisotropic quasi-2d sample:
I
V
t
c
ab
l
Strong anisotropy c >> ab
 shallow current penetration
 influence on the current density
 Rab depends on both c and ab
For thick samples (t / l  c /  ab ) :
Rab  A  ab c
Valid for linear response and for asymptotically
linear resistivities in the high-current limit
geometrical factor
Our samples are well in the thick sample limit
How to disentangle c and ab from Rab?
Rab   ab c
Multicontact method:
Vtop
I
c
ab
Vtop
ab
Vbottom
c
Vbottom
This experiment was done by
R. Busch, G. Ries, H. Werthner, G. Kreiselmeyer,
and G. Saemann-Ischenko, Phys. Rev. Lett. 69, 522 (1992)
Problem: Busch et al. measured in the I → 0 limit
whereas we measured in the high-current limit
How to compare high-field and low-field resistances?
Go to the unpinned liquid phase!
40
300
78 K
B=3T
30
250
58 K
25
dV/dI (m)
Voltage (mV)
35
0
50 K
20
40 K
15
200
58 K
150
100
50 K
50
40 K
30 K
10
20 K
5
0
78 K
0
50
100
Current (mA)
150
200
0
0
20 K
50
100
Current (mA)
With increasing current, the V-I curves are less and less nonlinear
For T > Tlin linear response  "unpinned liquid phase”
(smooth crossover, no sharp change)
150
200
Magnetic field–temperature phase diagram
Magnetic field (T)
100
Bc2
Tlin
10
Bc2 upper critical field
TFOT first order transition line
1
Tirr magnetic irreversibility line
Tirr
0.1
T2nd second magnetization peak
T2nd
TFOT
0.01
0
20
40
60
Temperature (K)
80
Magnetic field–temperature phase diagram
Magnetic field (T)
100
Bc2
Glass?
10
Tlin
Vortex liquid
1
Bc2 upper critical field
TFOT first order transition line
Tirr magnetic irreversibility line
Tirr
0.1
T2nd second magnetization peak
T2nd
TFOT
0.01
0
20
40
60
Temperature (K)
80
Magnetic field–temperature phase diagram
Magnetic field (T)
100
Bc2
Glass?
10
Tlin
Pinned liquid
1
Unp. L.
Bc2 upper critical field
TFOT first order transition line
Tirr magnetic irreversibility line
Tirr
0.1
T2nd second magnetization peak
T2nd
TFOT
0.01
0
20
40
60
Temperature (K)
80
Unpinned liquid phase
For T > Tlin the V-I curves
are linear
Analysis of the multicontact data of Busch et al.
Phys. Rev. Lett. 69, 522 (1992)
FIG. 3
• Digitize isothermal sections
• Sort out data for which
T > Tlin(B) (unpinned liquid)
Single crystal resistance
(same quantity as in our experiments)
Busch et al., Phys. Rev. Lett. 69, 522 (1992)
70 K
75 K
80 K
Rab (m)
100
50
Unpinned
liquid
0
0.001
0.01
• Reproduces B/Bc2 scaling
• Reproduces logarithmic field dependence:
0.1
B/Bc2
n
1   log( B / Bc2 )
Rab  Rab
• The slope  = 0.23 is in good agreement with our results
1
In-plane (ab-plane) resistivity
Busch et al., Phys. Rev. Lett. 69, 522 (1992)
-4
10
70 K
75 K
80 K
ab ( cm)
-5
10
Unpinned
liquid
3/4
B
-6
10
-7
10
10
-3
10
-2
B/Bc2
10
-1
10
0
• ab also exhibits B/Bc2 scaling
• Exponent of best power law fit: 0.75  0.01 (too good to be true?)
 ab
n  B 


  FFF   ab
 Bc 2 
3/ 4
In-plane (ab-plane) resistivity
Busch et al., Phys. Rev. Lett. 69, 522 (1992)
-4
10
70 K
75 K
80 K
ab ( cm)
-5
10
Unpinned
liquid
3/4
B
-6
10
-7
10
10
-3
10
-2
B/Bc2
10
-1
10
0
• ab also exhibits B/Bc2 scaling
• Exponent of best power law fit: 0.75  0.01 (too good to be true?)
 ab
n  B 


  FFF   ab
 Bc 2 
3/ 4
Out-of-plane (c-axis) resistivity
Busch et al., Phys. Rev. Lett. 69, 522 (1992)
c ( cm)
8
6
4
2
0
-3
10
70 K
75 K
80 K
Unpinned
liquid
10
-2
-1
B/Bc2
10
0
10
• c also exhibits B/Bc2 scaling
• Given experimental forms for Rab and ab, we can write an experimental
form for c using Rab  A  ab c :
 B 

c  cn 
B
 c2 
3 / 4
1   log B / Bc2 2
• Reproduces the maximum below Bc2 seen earlier:
G. Briceño, M. F. Crommie, and A. Zettl, Phys. Rev. Lett. 66, 2164 (1991); K. E.
Gray and D. H. Kim, Phys. Rev. Lett. 70, 1693 (1993); N. Morozov et al., Phys. Rev.
Lett. 84, 1784 (2000).
Comparison with thin film ab data
Z. L. Xiao, P. Voss-de Haan, G. Jakob, and H. Adrian, Phys. Rev. B 57, R736 (1998)
ab ( cm)
Xiao et al., T = 77 K
Unpinned liquid
-5
10
3/4
B
Busch et al.
-6
10
-3
10
10
-2
B/Bc2
10
-1
0
10
Reasonable agreement but systematic deviation from power law
(weaker than linear on log-log plot)
Comparison with thin film ab data
H. Raffy, S. Labdi, O. Laborde, and P. Monceau, Phys. Rev. Lett. 66, 2515 (1991)
P. Wagner, F. Hillmer, U. Frey, and H. Adrian, Phys. Rev. B 49, 13184 (1994)
M. Giura, S. Sarti, E. Silva,R. Fastampa, F. Murtas, R. Marcon, H. Adrian, and P. Wagner,
Phys. Rev. B 50, 12920 (1994)
Z. L. Xiao, P. Voss-de Haan, G. Jakob, and H. Adrian, Phys. Rev. B 57, R736 (1998)
• Different thin film results do not agree with each other
• Common feature: weaker than power law, stronger than logarithmic B-dependence
 Thin film resistance is in-between single crystal Rab and ab
Macroscopic defects may force c-acis currents
Thin film resistance may be a sample-dependent mixture of ab and c
(2d topology amplifies the effect of macroscopic defects)
Comparison with c-axis-configuration single crystal c data
c-0 (k cm)-1
N. Morozov, L. Krusin-Elbaum, T. Shibauchi, L. N. Bulaevskii, M. P. Maley, Yu. I. Latyshev,
and T. Yamashita, Phys. Rev. Lett. 84, 1784 (2000)
4
Morozov et al.
T = 70 K
2
Empirical
form
0
0
20
Magnetic field (tesla)
40
• Excellent agreement with our empirical form (blue line):
 B 

c  cn 
 Bc 2 
3 / 4
1   log B / Bc2 2
•  = 0.20 from the fit is in excellent agreement with our result
• Discrepancy: 0 = 3.6 (kcm)−1 from fit is different from
0 ≈ 8 (kcm)−1 inferred by Morozov et al. (Resistance decreases
more slowly above the maximum than the fitting function.)
Quick summary of empirical forms
In - plane resistivit y :
 ab   FFF  B 3 / 4
c  axis resistivit y :
 c  B 3 / 4 log 2 B
Single crystal resistance : Rab   ab  c  log B
Sharp crossover at Tco from Rab = const (low T) to Rab  log B (high T)
Magnetic field (T)
100
Rab =  log B
Rab = const
Tlin
Tco
10
Hc2
Pinned
liquid
1
Unp. L.
Tirr
0.1
T2nd
TFOT
0.01
0
20
40
60
Temperature (K)
80
For T < Tco : Rab = const
For T > Tco : Rab  log B
Probable difference:
intervortex correlations (interlayer?)
Mismatch with thermodynamic
vortex phases:
• No change in Rab when Tirr and Tlin
crossed
• No anomaly in the
low-current resistance at Tco
Tco reflects transition in the dynamic vortex system?
Unpinned liquid dynamically restored for Tco < T < Tlin?
Dynamic ordering of vortices below Tco?
Other high-Tc materials: YBCO
1.3
Power law exponent
ab-plane resistivity
from microwave surface impedance (arb. units)
Y. Tsuchiya et al., PRB 63, 184517 (2001): Microwave surface impedance in YBCO
1.2
1.1
BS law
1.0
0.9
0.8
Exponent 3/4
0.7
0.6
0.5
0
20
40
60
80
Temperature (K)
Power law exponent agrees well with 3/4
in the high temperature limit.
No agreement at low temperature, but
this is not dynamic vortex system!
Cancellation of power law exponents in ab and c
Rab   ab  c  log B, purely logarithmi c,
although  ab is power law and a power law factor is expected in  c
 cancellation of power law exponents
In conventional superconductors:
ab  DOS(B)
 abc = const
c  DOS(B)
In a superconductor with line nodes DOS(B)  B1/2
 sublinear B-dependence is not surprising,
but the origin of exponent 3/4 is not clear.
The response of the extended line nodes is not taken into account
in existing theories.
The cancellation of power law exponents may indicate a common origin of
ab plane and c axis dissipation.
Simultaneous in-plane and interplane phase slips?
Theoretical calculations of this mechanism are in disagreement with our results.
Conclusion
• Empirical forms for the magnetic field dependence of
resistivities of BSCCO in the high-current limit:
In - plane resistivit y :
 ab   FFF  B 3 / 4
c  axis resistivit y :
 c  B 3 / 4 log 2 B
Single crystal resistance : Rab   ab  c  log B
• Some evidence that: FFF  B3/4 holds in YBCO as well
• We speculated about a dynamic transition in the vortex system
See also: Á. Pallinger, B. Sas, I. Pethes, K. Vad, F. I. B.Williams, and G. Kriza,
Phys. Rev. B (in press).
• Validity in other high-Tc?
• Theoretical underpinnings?