Vortex matter in type II superconductors; insights from ... - iitk.ac.in

Vortex matter in type II superconductors;
insights from local magnetic imaging
A.N. Grigorenko † & S.J. Bending
Department of Physics, University of Bath, UK
†School of Physics & Astronomy, University of Manchester, UK
A.E. Koshelev
Argonne National Laboratory, Argonne, Illinois 60439, USA
Simon J. Bending, University of Bath, UK
John R. Clem
Iowa State University, Ames, IA 50011-3160, USA
T. Tamegai
Department of Applied Physics, Univ. of Tokyo, Japan
Vortex
Physics
Experiments
Vortex
Matter
Theory
BSCCO
Single
Crystals

Introduction
Type II superconductors & vortex matter
The vortex matter phase diagram in Bi 2 Sr 2 CaCu 2 O 8+δ
‘Crossing’ vortex lattices in layered superconductors
Scanning Hall probe magnetic imaging
Experimental Results; Vortex matter in Bi 2 Sr 2 CaCu 2 O 8+δ
First order melting of the vortex lattice, H = H z
– The influence of quenched disorder
‘Decoration’ of Josephson vortices by pancake vortices
Vortex lattice melting in the presence of an in-plane magnetic field
Indirect pinning and vortex stack fragmentation
Conclusions and Prospects
Simon J. Bending, University of Bath, UK
OUTLINE

Introduction to Type II Superconductors
In 1954 Abrikosov solved the Ginzburg-Landau equations in an applied magnetic
λ 1
field for κ = > . He found his famous vortex solution whereby ψ(x,y) contains a
ξ 2
periodic lattice of zeroes.
Abrikosov – 2003
Physics Nobel Prize
Meissner
Simon J. Bending, University of Bath, UK
Φ ο
Mixed phase
B
Each ‘normal’ pole is associated with a
circulating supercurrent that generates
a single quantum of magnetic flux, φ 0.
Leads to a new mixed Abrikosov
vortex phase between the Meissner
and normal states.

Bi 2 Sr 2 CaCu 2 O 8+δ
Introduction to Vortex Matter
U(r i j)
i j
Vortices are mutually repulsive - leads to
formation of an ordered triangular lattice.
VORTEX MATTER: Elastic moduli of lattice are small and the mixed state represents a form of
soft line-like matter that exhibits phases and phase transitions (e.g., melting) in which thermal
fluctuations and disorder play key roles.
- can tune vortex density and interaction strengths by many orders of magnitude by varying H z
A model system for investigating structural phase transitions.
Discovery of high T c cuprates led to renewed interest in vortex matter:-
– high operation temperatures lead to much higher thermal energies (kT)
1
– high crystalline anisotropy moves phase transition lines further from H B c2 m ~ 2
γ
Simon J. Bending, University of Bath, UK
r i j
2
φ ⎛ r
0
ij ⎞
U( rij
) ~ exp ⎜
⎜−
⎟
2
2πµ
0λ
⎝ λ ⎠
Vortices readily interact with quenched
disorder, becoming pinned on e.g.,
defects and impurities in the sample.
γ = anisotropy parameter

Pancake Vortices in Bi 2Sr 2CaCu 2O 8+δ - H=H z
Bi Bi2Sr 2Sr2CaCu 2CaCu2O 2O8+δ 8+δ
The layered structure and very strong crystalline anisotropy in the
cuprate superconductor Bi 2Sr 2CaCu 2O 8+δ (BSCCO) gives rise to
pancake vortices with the applied field along the high symmetry c-axis.
- supercurrents flow predominantly in the CuO 2 layers which are only
weakly coupled along the c-axis direction.
Simon J. Bending, University of Bath, UK
(a)
CuO 2 plane
c^
H
^
b
a^
pancake
vortex

Pancake Vortex Phase Diagram – Bi 2Sr 2CaCu 2O 8+δ
H (Oe)
10 5
10 4
10 3
10 2
10 1
Highly
disordered
glass
Bragg glass
Simon J. Bending, University of Bath, UK
Liquid / Gas
Thermally driven
H c2
0 20 40
T (K)
60 80 100
Meissner state, H c1
Thermally driven
Scanning Tunnelling Microscopy
Disorder driven
Vortex
crystal (?)
Scanning Hall Probe Microscopy
Re-entrant liquid
Adapted from a phase diagram due to Eli Zeldov.

Phase Transition Lines – Bi 2Sr 2CaCu 2O 8+δ
Khaykovich et al., Phys.
Rev. Lett 76, 2555
(1996)
Second
magnetisation
peak
E couple ~ E pin
Bragg glass
Point-like disorder
too weak to create
dislocations but
destroys true long
range order.
Giamarchi et al.,
PRB 52, 1242 (1995)
B (G)
10 5
10 5
10 5
10 4
10 4
10 4
10 3
10 3
10 3
10 2
10 2
10 2
10 1
10 1
10 1
Highly
disordered
glass
Simon J. Bending, University of Bath, UK
Disorder Disorder driven driven
Bragg glass
Depinning transition -
Glass transition line?
Thermally Thermally driven driven
E pin ~ E th
Liquid / Gas
Thermally Thermally driven driven
Vortex
crystal (?)
H c2
0 20 40 60 80 100
Meissner state, H c1
T (K) Re-entrant liquid
Energy scales:
Ecouple – J & EM coupling
Eel – Elastic energy
Epin – Pinning potential
Eth – Thermal energy
Zeldov et al.,Nature
375, 373 (1995)
First order
melting
E el ~E th

B
Sample
Scan
Voltages
Scanner
Tube
Tilt
Stage
Electronics
STM Tip
Bias
1-2 o
Hall
Probe
Simon J. Bending, University of Bath, UK
Scanning Hall Probe Imaging
Z(x,y)
B z(x,y)
PC
Hall
Voltage
I Hall
V + Hall
I -
V - Hall
I +
Hall
sensor
2µm
STM
tip
(I tip )
+
+
+
+
E c
10nm GaAs
80nm n-AlGaAs
electrons
1000nm GaAs
~100nm Hall probe wet etched in
a GaAs/AlGaAs heterostructure.
-20 -10 0 10 20
H (Oe)
Sensor can be moved to any location and
a ‘local’ magnetisation curve generated.
M l (G)
2
0
-2

First Order Vortex Lattice Melting - H = H z
Zeldov et al. (Nature 375, 373 (1995)) first unambiguously identified vortex lattice
melting in Bi 2Sr 2CaCu 2O 8+δ single crystals using local Hall probe measurements.
- vortex density (≡ B) increases slightly when system melts (like water).
M z (a.u.)
10
5
0
-5
T= 85K
-10
-15 -10 -5 0
Hz (Oe)
5 10 15
There is no proper theory of vortex melting and
nearly all theoretical works are based on the
Lindemann criterion, i.e., melting occurs when
1/2 = C L ×a,
where C L ~ 0.2 is the Lindemann constant.
Simon J. Bending, University of Bath, UK
H m
∆M~0.5G
H m H m
B m (G)
50
40
30
20
10
Bragg
Glass
Cut along a
line at T=81K
0
76 78 80 82 84 86 88
T (K)
Liquid
1/2
a

First Order Vortex Lattice Melting - H = H z
Image 1 Image 2 Image 3
The vortex system must be left to
equilibrate for several tens of minutes.
- vacancies and shear waves
frequently observed in rapid imaging.
Simon J. Bending, University of Bath, UK
Oral et al., Phys. Rev. Lett. 80, 3610 (1998)
H z = 10Oe
T = 81K

First Order Vortex Lattice Melting - H = H z
10 Oe 12 Oe 14 Oe 16 Oe 18 Oe 20 Oe 22 Oe
Linescans are generated across chains of
vortices at each magnetic field and fitted to
a pancake model due to John Clem.
Allows extraction of vortex fluctuation
amplitude quantified by the parameter σ.
B ( r, h)
=
z
0
2 2
// a
σ = standard deviation of lateral fluctuations.
J. R. Clem, Physica C 235-240, 2607 (1994) & (unpublished).
Simon J. Bending, University of Bath, UK
2
2
j
σ G
Q j z −
− ∆ i 2
− G j h iG j . r
∞ 2sφ
e e e e
3λ
Q = G + 1 λ
2 2
j j //
∑∑
( Q + G )
i= 0 j j j
G j = reciprocal vortex lattice vector.
∆∆∆∆z i = i.s - summed over consecutive CuO 2 planes
h = height of sensor above surface
∆B (G)
0.6
0.4
0.2
0.0
-0.2
H z = 14Oe
T=81K
H m
23 Oe
-0.4
-1 0 1 2
x (µm)
3 4 5
c.f., Oral et al., Phys. Rev. Lett. 80, 3610 (1998)

∆B
First Order Vortex Lattice Melting H = H z
1/2 /a grows again
at low fields
– possible signature
of re-entrant melting?
a
∆∆∆∆B (G)
0.8
0.6
0.4
0.2
Simon J. Bending, University of Bath, UK
13.355
0
T=81K
Estimated fluctuation
amplitude exhibits abrupt
jump at H m.
1/2 /a(H m) = C L ~ 0.26.
C L ~ 0.26
H m
0.4
0.3
0.2
0.1
0.0
0.0
0 5 10 15
Heff (Oe)
20 25
A single sharp
transition is observed
- probable that vortex
lines simultaneously
melt and decouple.
1/2 /a
c.f., Oral et al., Phys. Rev. Lett. 80, 3610 (1998)
1/2
a

Influence of Disorder on First Order Melting
Local magnetisation traces at 3 different
locations on a BSCCO crystal at 82K in the
locations indicated on the image (H z =38Oe).
Oral et al., Phys. Rev. B 56, 14295 (1997)
Simon J. Bending, University of Bath, UK
Point disorder and anisotropy are known to shift H m.
Local variations in these parameters, quenched
during crystal growth, lead to roughening of the
H m (x,y) landscape.
H z =-12.8Oe H z =-11.2Oe
=
–
Difference image
T=85K
Liquid
SHPM
Solid
MOI
MO difference image; H a =83Oe,
dH a =0.5Oe, T=70K.
(0.26mm x 0.34mm)
Zeldov et al., Nature 406, 282 (2000)
Placing upper bounds on the superheating field
yields estimate of S/L surface tension which is
more than 100x lower than theoretical estimates.
Melting completely dominated by disorder!

Anisotropic Vortex Structures in Bi 2Sr 2CaCu 2O 8+δ – Arbitrary H
Bi Bi2Sr 2Sr2CaCu 2CaCu2O 2O 8+δ
Simon J. Bending, University of Bath, UK
(a) (b)
CuO 2 plane
c^
H
^
b
a^
pancake
vortex
Very strong crystalline anisotropy in the cuprate
superconductor Bi 2Sr 2CaCu 2O 8+δ (BSCCO) is also
reflected in the vortex (and vortex lattice)
structure as a function of direction of applied field.
H
Josephson
vortex
c^
^
b
^a
rhombic unit cell
Realistic side view of
Josephson vortex lattice.

Tilted Vortex Instability in Bi 2Sr 2CaCu 2O 8+δ Single Crystals
Josephson
vortex
segment
pancake
vortex
Tilted Vortex
H
d = 0.866 γ. Φo
/B//
d = Φo
/ 0.866.Bz
JVs from side
A homogeneous tilt of the PV stacks costs
magnetic energy and for a wide range of applied
field angles the ground state consists of coexisting,
perpendicular JV and PV ‘crossing lattices’.
L.N.Bulaevskii et al., PRB 46, 366 (1992)
Simon J. Bending, University of Bath, UK
PVs from
top
H
Crossing Lattices
Interactions between JV and PV Lattices
CuO 2
planes
JV
j n
PV
PVs lying on stacks of JVs become
displaced (u n) due to interactions
with the JV supercurrents.
2
E = AC . u − B. j u
n 44 n n n
Although this costs the stack tilt
energy it results in a net reduction
of energy, and leads to an attractive
interaction between JVs and PVs.
j -n
A.E.Koshelev, PRL 83, 187 (1999).

‘Decoration’ of Josephson Vortices by Pancake Vortices
magnetic
particles
Bitter decoration
Simon J. Bending, University of Bath, UK
‘Decoration’ with pancake vortices
pancake
vortices
Only pancake vortices at the surface
are visible in our experiments.

‘Decoration’ of Josephson Vortices by Pancake Vortices
12.5µm
θ=0 o θ=66 o
H //
Simon J. Bending, University of Bath, UK
also seen in Bitter decoration †
H z=12Oe, H //=0Oe T=81K H z=14Oe, H //=32Oe, T=81K H z=10Oe, H //=32Oe, T=77K
25µm
H //
θ=73 o
θ=86 o θ=89 o θ=89 o
H // H // H //
H z=2Oe, H //=27Oe,T=81K H z=0.5Oe, H //=38Oe, T=81K H z89 o
1D chains of PV stacks
(Grayscale spans ~ 4G)
† C.A.Bolle et al., PRL 66, 112 (1991)
I.V.Grigorieva et al., PRB 51, 3765 (1995).
c
θ
H //
H
H z
a,b
Segments of fragmented
PV stacks? (Sublimation?
Re-entrant tilted state?)
(Grayscale spans ~ 0.4G)
H

Schematic H z – H x Vortex Matter Phase Diagram
Proposed experimental phase
diagram for the different
observed states of vortex matter
in the H
- H // domain for the
c
temperature range where this
study was performed (77-88K).
Simon J. Bending, University of Bath, UK
10
1
0.1
0.01
Log(H (Oe))
c
H c1
II. Tilted
Lattice
I. Meissner
state
VII. 2D Vortex
Liquid
III. Crossing
Lattices
IV. Composite
Lattice
VI. VI. 'Sublimed' Re-entrant 1D
Tilted Chain Lattice? State
0.01 0.1 1 10 100 1000
V. 1D Chains
H c2
VII. 2D Vortex
Liquid
Log(H (Oe))
//

Direct Measurement of Pancake Vortex Displacements
∆B (G)
(a) H // =11Oe (b) H // =16.5Oe
3.0
2.5
2.0
1.5
1.0
0.5
H //=16.5Oe
H //=11Oe
H z =1.8Oe, T=83K
0.0
2 4 6 8 10 12
x (µm)
A.N.Grigorenko et al., Phys. Rev. Lett. 89, 217003 (2002)
Simon J. Bending, University of Bath, UK
Between (a) H // =11Oe and (b) H // =16.5Oe a JV
stack has been manipulated until it lies along
the indicated pinned row of PV stacks.
Broadening of the PV stacks along the row in
the presence of the JV stack is clearly resolved in
the linescans.
Incorporating the calculated pancake vortex
displacements into a model for PV fields due to
Clem we obtain excellent agreement.
∞
2 2 −2
sΦ
exp( − G
0
i + q y + λ .n.s)
B z(x,h) = 2 ∑∑ ×
2πλ a ∫ 2 2 −2
2 2
n G G + q + λ + G + q
ch i −∞ i y i y
where u
× exp( − G + q .h).cos(G .(x − u )).dq
n
2 2
i y i n y
2
2Cnλ
≈
(n − 1 2) γsln λ r
( )
λ=450nm, r w =270nm, a ch =2.35µm, γ=640 and h=650nm
J.R.Clem, Physica C 235-240, 2607 (1994)
A.E.Koshelev, PRL 83, 187 (1999).
w

Comparison of Chain Spacings with Theoretical Predictions
∆B (G)
H z =0.7Oe, H // =49.5Oe, T=81K
4
3
2
1
line (a)
H //
line (b)
0
0 5 10 15 20 25 30
Within anisotropic London theory
the inter-chain spacing is given by:-
Simon J. Bending, University of Bath, UK
c y
c y (H // ) = 3 γ. Φo
/ 2.H//
c y (µm)
y-axis
offset?
24
22
20
18
16
14
12
10
8
6
4
2
0
T=77K
T=81K
c y (µm)=-1.3 + 107 * H // (Oe) -0.5
T=83K
T=85K
High field limit
Layer corrections
Cy (µm)
25
20
15
10
-2
0.00 0.05 0.10 0.15 0.20 0.25
5
(H //) -1/2 (Oe -1/2 )
H //=38Oe
0
76 78 80 82 84 86
T (K)
γ= 640
A.N.Grigorenko et al., Phys. Rev. Lett. 89, 217003 (2002)

Deviations from Anisotropic London Theory
At intermediate fields first
order corrections to the
vortex interactions due to the
layered structure yield
⎛ 2
9. 3. γ . s . H ⎞
//
cy ≈ cy
0.
⎜1− ⎜ ⎟
Φ ⎟
⎝ 32. 0 ⎠
A.E.Koshelev (private communication)
Decoration of the Josephson
vortex stacks does seem to
weakly perturb them!
Simon J. Bending, University of Bath, UK
c y (µm)
24
22
20
18
16
14
12
10
8
6
4
2
0
T=77K
T=81K
c y (µm)=-1.3 + 107 * H // (Oe) -0.5
T=83K
T=85K
High field limit
Layer corrections
Cy (µm)
25
20
15
10
-2
0.00 0.05 0.10 0.15 0.20 0.25
5
(H //) -1/2 (Oe -1/2 )
H //=38Oe
0
76 78 80 82 84 86
T (K)
A.N.Grigorenko et al., Phys. Rev. Lett. 89, 217003 (2002)

Vortex Melting in the Presence of an in-plane Magnetic Field
M local (G)
The melting field is a very strong function of applied field angle.
Koshelev has shown that for relatively low in-plane field components the crossing
lattice interaction leads to a linear suppression of H z at melting as a function of H //.
85K
0.5 B // = 0
0.0
-0.5
-1.0
B // = 28Oe
6 7 8 9 10 11 12 13 14
Hz (Oe)
Suppression of the c-axis melting field
with an applied in-plane magnetic field.
Linear regime of our experiments
Simon J. Bending, University of Bath, UK
Koshelev, Phys. Rev. Lett. 83, 187 (1999)
Vortex-lattice melting transition in the
H z-H // plane of BSCCO at T=85.2 K.
Mirković et al., Phys. Rev. Lett. 86, 886 (2001)

‘Decoration’ as a Probe of Josephson Vortex Fluctuations & Melting
Pancake vortex and Josephson vortex lattices expected to melt simultaneously.
‘Decoration’ of Josephson vortices is a sensitive probe of vortex fluctuations near melting.
H z =-8.0Oe H z =-8.8Oe H z =-9.6Oe H z =-10.4Oe H z =-11.2Oe H z =-12.8Oe H z =-13.6Oe
– – – – – – –
= = = = = = =
Simon J. Bending, University of Bath, UK
T=85K
Melting
H // =0
H // =36Oe

Fluctuations of Josephson Vortex Lattice near Melting
∆B (G)
1.75
1.50
1.25
1.00
0.75
0.50
0.25
T=85K
H z =8.0Oe
H z =8.8Oe
H z =9.6Oe
H z =10.4Oe
H z =11.2Oe
0.00
0 1 2 3 4 5 6 7 8 9 10
x (µm)
Linescans across ‘decorated’ Josephson
vortices at different values of Hz

Indirect Pinning of Josephson Vortices; Experiment
a-axis
H //=0
H z=8Oe, T=85K, 28µm×28µm
Quenched disorder in the crystal has a pronounced
anisotropy, and leads to vortex clustering in rows/stripes
directed along the a-axis.
Segments of a Josephson vortex stack are frequently
found to be indirectly pinned along these disordered
regions via the pancake vortices lying there.
At high T and large H z, when the line tension is small,
trapped segments can be long. ( e.g. w~9µm)
Simon J. Bending, University of Bath, UK
w
H //=36Oe
∆B (G)
12.4
12.2
12.0
11.8
11.6
~0.2G
Linescan
0 5 10
x (µm)
15 20
Higher density of PVs
in ‘pinned’ region.

Indirect Pinning of Josephson Vortices; Theory
Φ B
K =
4πγ
λ
E
×
Planar
defect
u(z)
θ
0
2 2
w
θ acc
Vortex
Φ
σ ≅
(4 π) γλ
−2.1Φ =
π γ γ λ
2
0
2 2
4
2
2
0
2
sln(3.5 s / )
Simon J. Bending, University of Bath, UK
H
Healing
region
Trapped
region
Healing
region
Problem can be solved by analogy with that of a vortex trapped by
twin boundaries (c.f. Paulius et al. PRB 56, 913 (1997)) :-
Minimisation with respect to w yields:-
8.Up
σ
w = −
2 2
K.sin θ.cos θ K.cos θ
– Assuming
E
2 ⎡σ ⎛ du ⎞ K ⎤ 2
tot ⎢ + u ⎥.dz
− Up
= ∫ 2
⎜
dz
⎟
⎢⎣ ⎝ ⎠ 2
⎦⎥
line
tension
U = E . ∆(1/
a )
cage
potential
p × ch
– Ex is crossing energy for a single PV stack crossing one JV.
∆(1/ a ch)
– is difference in PV chain density between the strongly
pinned region and elsewhere.
Assuming yields:w=9µm
(c.f., ~9µm) and θacc=62o (c.f., ~65o ∆ (1/ a ch)
~ 1PV / µ m
)
2
sin θ 2U
acc =
cosθ
σ
Downwards renormalisation of JV line tension (σ) due to repulsive
PVs attached to it would lead to a smaller pinning potential and
modelling suggests this estimate is probably 5 -10 too large.
w
pinning
energy
A.N.Grigorenko et al., Phys. Rev. Lett. 89, 217003 (2002)
acc
p

Fragmentation of Pancake Vortex Stacks
(i) (ii)
∆B (G)
(i) (ii)
H H z ~0, H // =0 Hz ~0, H // =11Oe
6
5
4
3
2
1
0
(i)
(ii)
T=85K
0 2 4 6 8 10 12 14 16
x (µm)
Simon J. Bending, University of Bath, UK
480nm
2.3µm
Fragmentation of PV stacks can occur when
‘decorated’ stacks of JVs are forced abruptly
through a region of disorder (rather rare event).
Image shows pinned chain of PV stacks
after the in-plane field was suddenly reduced
from H // =36Oe to zero.
‘Fractional’ pancake vortex stack ‘heals’
back to a connected stack after the in-plane
field was cycled back up to H // =11Oe
PV model fits data well if assume that PV stack
splits cleanly ~480nm below the surface, into
two segments a lateral distance 2.3µm apart.

Fragmentation of Josephson Vortex Stacks
Fragmentation of JV stacks occurs much more
commonly when ‘decorated’ stacks of JVs are
forced abruptly through a region of disorder.
(a) Splitting of JV stack after H // was abruptly
reversed from -33Oe to +33Oe at 77K (H z=1Oe).
(b) ‘Healing’ of the fork after the in-plane field was
reduced to H // =+22Oe (H z=1Oe).
We presume PV and JV stack fragmentation
arises when decorated “1D chains” are
driven through regions of quenched disorder
which are inhomogeneous along the c-axis.
Simon J. Bending, University of Bath, UK
H //
(a)
H //
H z =1Oe T=77K (a) H // =33Oe, (b) H // =22Oe
(b)

Scanning Hall Probe imaging has been used to investigate vortex matter in
single crystals of highly anisotropic cuprate superconductor, Bi 2 Sr 2 CaCu 2 O 8+δ .
Pancake vortex melting transition directly imaged as a function of applied field.
– fits to melting behaviour yield Lindemann constant, C L = 1/2 /a ~ 0.26.
– Increasing fluctuations at low field may be signature of re-entrant melting.
Josephson vortex fluctuations near melting line indirectly measured by
‘decoration’.
– fluctuating PVs strongly influence JV lattice due to renormalization of the phase
stiffness and Josephson energy – simultaneous melting.
Interacting ‘crossing’ pancake vortex and Josephson vortex lattices observed over
a broad range of magnetic field tilt angles.
– several new order/disorder phase transitions observed as a function of tilt angle.
– Indirect pinning of JVs via attraction to weakly pinned PVs
– Fragmentation of both PV and JV stacks observed when decorated JV stacks are
driven through a region of disorder.
Interacting ‘crossing’ vortex lattices can be exploited to achieve new functionality.
Trapped PV stacks can be manipulated by deforming the JV lattice and vice versa.
Simon J. Bending, University of Bath, UK
Conclusions & Prospects