Quantitative Magneto-Optical Analysis of Superconductors
Roberto Gerbaldo, Gianluca Ghigo,Laura Gozzelino, Francesco Laviano,
Enrica Mezzetti and Bruno Minetti
Dipartimento di Fisica
POLITECNICO di TORINOCorso Duca degli Abruzzi, 2410129 TORINO
OutlineAimed to study the local electrodynamical properties of
superconductors, we explored the powerful capabilities of the magneto-optical (MO) technique:
• evaluation of local values of the magnetic induction, BZ(x,y), from MO measurements
• reconstruction of the electrical current density distribution from magnetic induction maps(inversion method)
Finally, we present some experimental results to show ourimprovement of existing inversion methods to achieve more precise results (“in-plane field effect” correction).
Principle of the MO technique:indicator film
The magnetic sensor is a doped (Bi, Lu) ferrite film, grown on a gadolinium gallium garnet (GGG) transparent substrate.
On the other face of the ferrite film there are a mirror and a protective layer
The spontaneous magnetization of the ferrite lies in the film plane.
When the external magnetic field is not zero, the localmagnetization of the ferrite rotates out of the indicator plane
Si3N4 Al
Principle of the MO technique:Faraday effect
The local pertubation of the magnetization vectorcan be detected in the following way:
The polarization plane of the light refracted by the ferrite isrotated by an angle (αF) proportional to the lenght of the light path into the medium and to the local magnetizationcomponent along the light direction (Faraday effect).
With a second polarizer (analyzer), the Faradayrotation is visualized as light intensity variations.
Principle of the MO technique:Experimental set-up
Example of MO measurementMO imaging of magnetic bits in 3.5” floppy disk tracks.
0 50 100 150 200 250 300 350 400 450 500 550-0.6-0.4-0.20.00.20.40.60.8
i (a.
u.)
position (pixels)
Calibration: Model (1)SAA MEB /=
αsinˆ BBz =⋅= kB
( ) ( ) αcosˆˆ 22BBxy =⋅+⋅= jBiB
φα sinSF MC=
Anisotropy field*
“In-plane field”
Faraday rotation angle (C is a constantand MS the spontaneous magnetization)
(E.1)
Starting electrodynamical model of the indicator film:
)]cos(1[MB)cos1(E SA φαφ −−⋅⋅+−⋅ (E.2)
* At fixed temperature, all the parameters characterizing the indicator film are constants.
Calibration: Model (2)By minimization of (E.2), we found the expression for φ(B):
Axy
z
BBB+
= arctanφ
If the angle between the polarizer and the analyzer is θ, by the Malus law, the output light intensity is:
(E.3)
)(cos F θα ++= 2MAX0 III (E.4)
⎥⎥
⎦
⎤
⎢⎢
⎣
⎡+
+++= θ
2z
2xyA
zS2MAX0
B)B(B
BCMIII cos
From (E.1), (E.3) and (E.4), we obtain the relation between the output light intensity and the magnetic induction in the indicator plane:
(E.5)
-200 -100 0 100 200
0.3
0.4
0.5
0.6
0.7
experimental fit with (E.6)
I / I M
AX
BZ (mT)
Calibration curve: collection of light intensity values in a point far fromthe sample in function of the localmagnetic field (Hext) and the fit with(E.6). The saturation points are indicated by arrows.
(T =4.15 K, θ =45°).
For conventional MO analysis*, the term Bxy is negleted and the abovecurve is used to recurse the BZ values all over the indicator surface.
*M.R. Koblischka and R.J. Wijngaarden, Supercond. Sci. Technol. 8, 199 (1995); Ch. Jooss et al., Physica C, 266 235 (1996); A.A. Polyanskii et al., in Handbook of Superconducting Materials (App. C.3.4), IOP Publishing (1999)
Calibration: Curve
The “Inversion problem”For( ))()()( 0 rHrHrB indext += µ 0)( =⋅∇ rJ
the Biot-Savart law is valid:
[ ]
[ ]
[ ]∫∫∫
∫∫∫
∫∫∫
∫
−+−+−
−⋅−−⋅+
+−+−+−
−⋅−−⋅+
+−+−+−
−⋅−−⋅=
=−
−∧=
''')'()'()'(
)'()',','()'()',','(4
ˆ
''')'()'()'(
)'()',','()'()',','(4
ˆ
''')'()'()'(
)'()',','()'()',','(4
ˆ
''
)'()'(4
)(
3222
0
3222
0
3222
0
300
dzdydxzzyyxx
xxzyxJyyzyxJ
dzdydxzzyyxx
zzzyxJxxzyxJ
dzdydxzzyyxx
yyzyxJzzzyxJ
rd
yx
xz
zy
ind
πµ
πµ
πµ
πµµ
k
j
i
rrrrrJrH
The third contribute (along z) is of interest for MO measurements, butthe 3D general formulation for the inverse problem requiresinformation from invasive measurements (troughout the thickness).
2D ApproximationIf the sample is thin (for thickness d<λ the electrodynamic fielddistributions are really 2D), we could treat the electrical currentdensity distribution as averaged over the thickness :
[ ]∫ ∫∫−−+−+−
−⋅−−⋅=
2/
2/ 3222
00 '''
)'()'()'(
)'()','()'()','(4
),,(d
d
yxz dzdydx
zhyyxx
xxyxJyyyxJhyxH
πµ
µ
(being h the height of BZ measurement)
(E.7)
The inverse problem of (E.7) can be uniquely resolved in k-space bythe convolution theorem*. Taking bidimensional Fourier transformationsfor all the fields involved in (E.7), we obtain:
'),(~),(~2
),,(~ 22)'(2/
2/2222
00 dzekkJ
kk
kkkJ
kk
kihkkH yx kkzh
d
dyxy
yx
xyxx
yx
yyxz
+−−
−∫ ⎟⎟
⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
+−
+=
µµ (E.8)
*B. J. Roth et al., J. Appl. Phys. 65, 361 (1989); Ch. Jooss et al., Physica C 299, 215 (1998)
2D SolutionThe integration in z can be easily performed:
⎟⎠⎞
⎜⎝⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
−
kdk
ekkJkk
kkJkk
ihkkHhk
yxyx
yxxy
yxz 2sinh),(~),(~
2),,(~ 0
0µ
µ (E.9)
Coupled with continuity equation for the current density in k-space, (E.9) yields a resolvable system:
⎪⎪⎩
⎪⎪⎨
⎧
−=
⎟⎠⎞
⎜⎝⎛=
−
y
xyxyyxx
y
hk
yxxyxz
kk
kkJkkJ
kdk
ekkJihkkH
),(~),(~
2sinh),(~
2),,(~ 0
0µ
µ(E.10)
The numerical algorithm based on this formalism reconstruct the electrical current density distribution (Jx(x,y) and Jy(x,y), absolute valueand sign) from BZ distribution in a plane over the sample surface.
Bean superconductor (square) in fully-penetrated critical state*
*E.H. Brandt, Phys. Rev. B, 52 15442 (1995)
BZJxJy|J|
-600 -400 -200 0 200 400 600-101
J y (10
11 A
m-2)
pixels
Test of the method
Thin film superconductorYBa2Cu3O7-δ film grown on YSZ substrate with CeO2 buffer layer (TC ~88K, thickness of the superconductor 400 nm)*.
1 small holes (few microns of diameters)
2 deep scratch
3 garbage on not-superconducting layers
* The sample was kindly supplied by the joined team “Edison-Europa Metalli-IMEM/CNR”
MO measurements &Current density reconstruction
BZ | J|
T = 25K µ0Hext = 7.4 mTT = 25K µ0Hext = 52 mTT = 25K µ0Hext = 115.1 mT
-1000 -500 0 500 1000
0.0
0.5
1.0
1.5
2.0
[mT]
Applied fields
0 7.4 17.5 23.2 37.6 52 72 86.4 100.8 115.1
position [µm]
|J|
[1011
Am
-2]
Besides the behaviour intrinsicto the superconductor, it isremarkable the presence of a spurious current density outsidethe sample (located betweenthe peaks). Its value and spatialextension increase together the supercurrent.
This effect was first observed with MO by T.H. Johansen*, whodemonstrated that the supercurrents induce an appreciable in-planemagnetic field. High in-plane field implies underestimation of the BZ data and subsequent spurious current distribution.
*T.H. Johansen et al., Phys. Rev. B 54, 16264 (1996)
Profiles
The relation between the local magnetic induction valuesevaluated with the inverse equation of (E.5), indicated as BZ, and those from the inverse of (E.6) (conventional MO, BZ|0) is:
In-plane field effect correction (1)
0|1 ZA
xyZ B
BB
B ⎟⎟⎠
⎞⎜⎜⎝
⎛+= (E.11)
To evaluate the Bxy values, induced by the supercurrents in eachpoint, we built a numericaliterative procedure:
• the inversion of the conventionally evaluated induction data is used in direct Biot-Savart law calculations, as anapproximation of the actual current density distribution, tocalculate Bxy.
• (E.11) gives the corrected BZ values.
Bx(n)(x,y)
andBy
(n)(x,y)
Magneto-opticalmeasurement BZ
(0) (x,y)Jx
(0)(x,y)and
Jy(0)(x,y)
Jx(n)(x,y) and Jy
(n)(x,y)
BZ(n+1) =(1+ Bxy
(n)/ BA) BZ(0)
calibration
inversionI t e r a t i v e a l g o r i t h m
In-plane field effect correction (2)
The algorithm is stopped when Bz(n+1)≈ Bz
(n)
Current density correction (YBa2Cu3O7-x film with square shape)
-1000 -500 0 500 1000-2
-1
0
1
2
Position (µm)
J y (10
11 A
m-2)
µ0Ha (mT) 0 46.2 14.6 57.7 23.2 69.2 34.7
-1000 -500 0 500 1000
-1
0
1
µ0Ha (mT) 0 46.2 14.6 57.7 23.2 69.2 34.7
J y (10
11 A
m-2)
Position (µm)
Current density values along the mid-line of the sampleStandard inversion of Bz Corrected values with iterative inversion
|J(x,y)|T = 5 K; µ0Happ = 69.2 mT.
0 200 400 600 800 10000,0
0,5
1,0
1,5
2,0
iterations: 0 1 2 5 10 50
T = 5 Kµ0Ha = 69.2 mT
Position (µm)|J
| (10
11 A
m-2)
By the standard procedure, the quantitative values of the current density are affected over all the measurement surface. It doesn’t suffice to cut the edge region: also the central part of the profile is strongly affected by the in-plane field effect.
Iterations
In summary, after the correction of the measured Bz, we have estimated also the local values of Bx, By, Jx, Jy, thus obtaining the full induction distribution at the superconductor surface.
For thin superconducting sample, the critical state problem is tri-dimensional. The flux lines inside the specimen are strongly curved by the supercurrents which are almost bi-dimensional. This picture is valid if the sample thickness, d, does not exceed several λ (London) lengths. If d is lower than 2λ, the current distribution is truly bi-dimensional.
),(),(
tanarc),(yxByxB
yxZ
xy>=<θ
G P and Brandt E H, PRB 62 6800 (2000)
Magnetic field line curvature
The supercurrentflows clockwisein the xy plane(upper picture) ifthe applied fieldis directed as the z axis.
-1000 -500 0 500 1000
0
20
40
60
80
<θ >
(°)
position (µm)
-1000 -500 0 500 10000
20
40
60
80
|Bxy|Bz
(mT)
position (µm)
<θ > is the exit angle of the flux line from the sample surface. Its measure can be performed directly with our procedure.
For the test superconducting film early presented, at T= 5K and µ0Happ= 52 mT, the θdistribution confirms the previous picture of strongly curved flux lines, whose curvature changes across the sample area (and inside the sample). This can be directly deduced from the profile of Bz and Bxy across the square mid-line.
<θ > profile across the mid section of the sample. In the central part there is the Meissnerregion with Bz=0 (but, there, Bxy≠0).
Curvature measurement (1)
Curvature measurement (2)
3D
Sample*: YBCO film grown by thermal co-evaporation** with a buffer layer of CeO2 (40 nm) on yttrium stabilized zirconia. The as-grown sample (1x1 cm2) was etched in squares of about 1.2 mm side long. The thickness is 400 nm, Tc is 88 K and ∆Tc~0.4K
QMO analysis: measurement of Bz(x,y), evaluation of Jx(x,y), Jy(x,y), Bx(x,y), By(x,y), θ (x,y) in function of H and T.
Irradiation: 23+Au ions at 250 MeV with dose equivalent field BΦ=100 mT. INFN LNL
YBCO film optical photo
** Utz B., Semerad R., Bauer M., Prusseit W., Berberich P. and Kinder H., IEEE Transaction on Applied Superconductivity, 7 1272 (1997)
* The sample was kindly supplied by the joined team “Edison-Europa Metalli-IMEM/CNR”
Experiment: Vortex curvature detection in presence of anisotropic pinning
Cross-section
YBCO/CeO2interface. high resolution
In-plane viewwith a columnardefect
high resolution
TEM
Cross-section
YBCO/CeO2/substrateinterfaces
In-plane viewlowresolution
Magneto optical measurements
Experimental parameters: zero field cooling, external magnetic field directed perpendicular to the sample surface and raise up by steps of 1.5mT every 3 s.
T = 5 K
virgin irradiated
µ0Happ(mT) T = 15 K T = 25 K
virgin irradiated virgin irradiated
14.6
43.3
86.4
Two features can be addressed to defects introduced by irradiation: • less vortex penetration inside the sample• higher linearity of Bz near the Meissner area, which indicates pronounced lowering of the current density with increasing vortex curvature, signature of strong correlated pinning along the c axis (coincident with the z axis).
-750 -250 0 250 7500
50
100
150
Virgin @5 K
µ0 Ha
(mT)86.47257.743.328.914.6
position (µm)
BZ (
mT)
Irradiated @5 K
Magnetic induction along z direction (Bz) on the superconductor surface, along the mid-line of the square: in this plane the Lorenz forces generated bythe supercurrents flowingin the other two quadrantscancel each other.
QMO: Bz(x0,y)
-800 -600 -400 -200 0 200 400 600 800-2.5-2.0-1.5-1.0-0.50.00.51.01.52.0
Virgin@5 K; µ0Ha (mT) 14.6 28.9 43.3 57.7 72 86.4
Irradiated@5 K; µ0Ha (mT) 14.6 28.9 43.3 57.7 72 86.4
J x(x0,y
) (10
11 A
m-2)
position (µm)
The features seen in the Bz profile are found alsoin the current density distribution:
• Less penetration depthcorrespond to anincreased currentdensity magnitude;
• The higher linearity is traducedin a smooth decreasing of the current density value towards the center.
In any case, strong pinning by ab-planes or deep decreasing uponlocal field it is not found
Mikitik G P and Brandt E H, PRB 62 6800 (2000)
Pinning by:defects ||c axis ab planes
QMO: Jx(x0,y)
2;exp1)( 0 dJB
BqBpJBJ C
crcr
ZCZ
µ=⎥
⎦
⎤⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
0 20 40 60 80 1000.0
0.5
1.0
1.5
2.0
The parameters seems to bring no quantitative information, except for the critical current density. From the graph, we note the strong decreasing of the pinning enhancement by the nanotracks with increasing temperature.
Virgin µ0Happ= 72 mT; T= 5 K 15 K 25 K
Irradiated µ0Happ= 72 mT; T= 5 K 15 K 25 K Fit with Brandt's equation
J (10
11 A
m-2)
BZ (mT)
T (K) JC(1011 Am-2)
p q
5 1.1892.022
-1.34-0.8
5.943.55
15 1.1391.721
-0.55-0.48
3.352.98
25 1.0081.332
-0.82-0.69
3.533.26
35 0.7720.969
-1.05-2.13
2.94.87
40 0.6210.832
-4.87-3.25
5.845.87
45 0.4780.675
-1.89-1.02
0.973.71
Very general formula from*:
*Mikitik G P and Brandt E H, PRB 62 6800 (2000)
QMO: J(B)
-800 -600 -400 -200 00
20
40
60
80
0 200 400 600 8000
20
40
60
80 Virgin@5 K; µ0H
QMO: <θ>(x0,y) (°)a (mT)
14.6 28.9 43.3 57.7 72 86.4
Irradiated@5 K; µ 14.6 28.9 43.3 57.7 72 86.4
position (µm)
0Ha (mT)
The flux line curvature is determined mostly by geometry, exceptin the case of very strong correlated pinning when the defects lock the vortices if their angle is less than a critical value (10° is typical for irradiation induced nanotracks*)
*Silhanek A. V. , Civale L., Avila M. A., PRB 65 174525 (2002)
QMO: J(θ)
In the virgin sample, the current density can beconsidered constant when the curvature is lessthan 60° (pinning by weak correlated defectsand by point pins distributed homogeneously). On the contrary, after the nanotracksimplantation, the pinning appears to be more anisotropic and the current density depends in a more pronounced way from the tilt angle θ.
The comparison of J(θ) with Brandtwork confirms the strong pinning bydefects correlated wih c axis after irradiation and the less effectivecorrelated pinning in the virgin sample.
0 20 40 60 800.60.81.01.21.41.61.82.02.2
<θ > (°)
|J| (1
011 A
m-2)
Virgin@5 K; µ0Ha (mT) 14.6 28.9 43.3 57.7 72 86.4
Irradiated@5 K; µ0Ha (mT) 14.6 28.9 43.3 57.7 72 86.4
0 20 40 60 80 1000.0
0.5
1.0
1.5
2.0
for µ0Ha=72 mT
BZ (mT)
J (10
11 A
m-2)
J(B) J(BZ(x))
Because the sample displaysvery homogeneos structuralproperties, we tried to obtainthe J(B) dependence fromeach picture: for the studiedpart of the sample this is the case!
The J(BZ(x)) profile was taken on the upper mid-vertical line of the square (after irradiation@5 K). The J(B) curve correspond to a single point not far from the edge. The same result was found foreach point of the sample which have reached the critical state during the experiment.
J(B) VS J(BZ(x))
We take as the critical current density values, JC, those corresponding to the condition of maximum pinning efficiency of the defect (maximum vortex length pinned for correlated defects).
0 10 20 30 40 50 60 700.40.60.81.01.21.41.61.82.0 VIRGIN
IRRADIATED
J C (10
11 A
m-2)
T (K)
0 10 20 30 40 50
0.2
0.4
0.6
0.8
J IRR/
J Virg
in-1
T (K)Possible phasetransition? Single vortexpinning to collettive behaviour?
• The strong enhancement at low temperature may be due to a Mott insulatorphase, where single vortices are strongly pinned by the extrinsic nanotracks.
Jc(T )
0 200 400 600 8000.0
0.2
0.4
0.6
0.8
1.0
JC = 1.78 (1011 Am-2)
JC = 1.189 (1011 Am-2)
Virgin5K
1-X
P / a
µ0 Ha (mT)
The penetration depth, Xp(H), canbe measured thoughtout magneto-optics without invoking any model.
From the fit (JC the sole freeparameter), it seems that the simplestrip-Bean model* can account forthe observed behaviour. On thecontrary, the JC value found is toohigh with respect to the BZ-inversion and the correspondingBean model curve, with theinversion JC value, is plotted forcomparison.⎟⎟
⎠
⎞⎜⎜⎝
⎛−=
dJH
aaHX
C
P πcosh)(
Beam model for XP in strips:
Magnetic flux front depth
*E.H. Brandt and M. Indenbom, Phys. Rev. B, 48 12893 (199
• Magneto-optics gives an unparelleled quantity of information on the electrodynamics of thin superconductors, but it is necessary to take intoaccount the full magnetic induction distribution.
• The tri-dimensionality of flux profile is of extreme importance in determining the electrodynamic responce in dependence of the defect type (pinning anisotropies). Then, the analysis of the pinning properties must be local and should consider all the magnetic field components.
• With such analysis, we found fingerprints of defect-vortex correlation in the 3D-flux profile, resulting in particular features in J(x,y,H,B,T). In particular, The evaluation of the full magnetic induction distribution at the sample surface gives insight into the anisotropic pinning nature of the extrinsic nanotracks and of intrinsic correlated defects and also on the vortex curvature imposed by the sample geometry.
Conclusion (1)
We presented a self-consistent analysis methoddesigned for high resolution Magneto-Optical
measurements.
The method yields the quantitative local information about:
• Magnetic induction distribution BZ(x,y),
accuracy ~1%.
• Electrical current density distribution componentsJx(x,y) and Jy(x,y).
Conclusion (2)
From preliminary measurements on superconductors, it turns out that local information is needed for electrodynamical description.
