L12 Superconductivity I - Triumf

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15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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Jeff Sonier
Simon Fraser University
Superconductivity I
(Conventional Superconductors)
What is superconductivity?
Superconductivity was discovered in 1911by H. Kamerlingh Onnes at the
University of Leiden, 3 years after he first liquefied helium.
This year marks the 100th
anniversary of superconductivity!
TheNobel Prizein Physics 1913
Properties of a Superconductor
A superconductor is a material that exhibits both perfect conductivityand
perfect diamagnetism.
Electrical Resistance in a Metal
Resistance to the flow of electrical current is caused by scatteringof electrons.

scattering from latticevibrations(phonons)

scattering from defects and impurities

scattering from electrons
Resistance causes lossesin the
transmission of electric power and
heatingthat limits the amount of
electric power that can be
transmitted.
0 K
Temperature (K)
Resistance
TC
Mercury
(TC = 4.15 K)
Normal
Metal
Zero Electrical Resistance Onnes found that the electrical resistance of various metals (e.g. Hg, Pb, Sn)
vanished below a critical temperatureTc.
Superconducting Power Cable
Bi-2223 cable -Albany New York –commissioned fall 2006
February 2008 updated with YBCO section
Magnetic Field
Normal Metal
Cool
Magnetic Field
Superconductor
“Meissner Effect”
Perfect Diamagnetism
In 1933, Meissner and Ochsenfeld discovered that magnetic field in a
superconductor is expelledas it is cooled below Tc.
However, there is a limit as to how much field a superconductor can take!
Superconductivity is destroyed above a critical magnetic fieldHc(T),
separating the “normal” and superconducting states.
MRI machine
27 km Large Hadron Collider (LHC)
HELIOS
Superconducting Magnets
The absence of zero electrical resistance means that persistent currentsflow in a
superconducting ring. A major application of this property is superconducting
magnets. With no energy dissipated as heat in the coil windings, these magnets
are cheaper to operate and can sustain larger electric currents (and hence
produce greater magnet fields) than electromagnets.
4
M
H
H
c
Meissner
B
= 0
Normal
B
=
H
Type I
4M
Magnetic Response of Type-I and Type-II Superconductors
4
M
H
H
c1
H
c2
Normal
B
=
H
Meissner
B
= 0
Vortex State
B
<
H
Type II
STM image of vortex lattice
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“Cooper pairs”:pairs of electronscaused by electron-phonon interaction
The way to superconductivity…
1972
J. Bardeen L.N. Cooper J.R. Schrieffer
Nobel Prize in Physics
BCS Theory of Superconductivity
General idea -Electrons pair up (“Cooper
pairs”) and form a coherent quantum state,
making it impossible to deflect the motion of
one pair without involving all the others.
Zero resistance and the Meissner effect require that the Cooper pairs share
the same phase “quantum phase coherence”
The BCS state is characterized by a complexmacroscopicwave function:
)(
0
)(
ri
er




Amplitude
Phase
The pair binding manifests itself as an energy gapat the Fermi surface.
Energy Gap
kx
ky
Fermi
surface
E
Density of States
EF
Normal Metal
kx
ky
Fermi
surface
Energy gap

E
Density of States

EF
Superconductor
(T)
T
T
c
In conventional
superconductors
(0) ~ kBTc..
SC gapappears at Tc
where resistance also
vanishes.
History of Superconductivity
Liquid
nitrogen
Liquid
helium
Some Applications of SR to conventionallow-Tc
superconductors

Intermediate state of a type-I superconductor

Magnetic penetration depth

in a polycrystalline sample

Vortex core size from single crystal measurements
Conventionalsuperconductors conform to BCS or Bogoliubov theories.
The attractive interaction is usually mediated by phonons, and the pairing
symmetry of the superconducting state is s-wave.
Type-I Superconductivity
Hc
Tc
Normal
Superconducting


Magnetic Field
Temperature
Type-I
A type-Isuperconductor is one in which < , where is the magnetic
penetration depthand is the coherence length (more about these later).
In these materials there is a critical fieldHc(T) below which the material is
in the Meissner state, and above which superconductivity ceases.
Intermediate State of a Type-I Superconductor
The magnetic response of a type-I superconductor depends on the shapeof the
sample.
The field at A and C reaches
the critical field Bc
at B= Bc/2.
At Bc/2 < B< Bc, the cylinder splits up into
small normal and superconducting regions.
i.e.an intermediate state
Intermediate state in
an Al plate
Transverse-Field SR
01234567
-1.0
-0.5
0.0
0.5
1.0

P(t)
Time (
s)
Envelope
)cos()()(





tBtGtP
The time evolution of the muon spin
polarization is described by:
where G(t) is a relaxation function describing
the envelopeof the TF-SR signal.
Positron
detecto
r
Electronic clock
Muon
detector
Sample
z
x
y
P(= 0)
t
H


e
Positron
de
t
ec
t
o
r
Intermediate State of Single Crystal Sn Measured by TF-µSR
Muons stopping in Superconducting domains
only experience the randomly oriented nuclear
dipole fields, and hence do not coherently
precess. Consequently, the superconducting
volume fraction is a “missing asymmetry”.
0)cos()(
2/
22




BteatA
t
N
Muons stopping in the Normal domains
experience a local magnetic field Bthat is
dominated by the contribution of the external
field, and hence coherently precess.
V.S. Egorov et al.Phys. Rev. B 64, 024524 (2001)
Mixture of Normal
& Superconducting
Superconducting
Hc2
Hc1
Tc
Normal
Superconducting


Magnetic Field
Temperature
Type-II
Type-II Superconductivity
A type-IIsuperconductor is one in which > .
In these materials there is a phase that exists between lowerand upper
critical fields Hc1(T)and Hc2(T)where flux penetrates the material as a
regular array of magnetic flux quanta, i.e.an “Abrikosov vortex lattice”.
At magnetic fields below Hc1(T) the material is in the Meissner phase.
2
2
2
*
4
1
cm
en
s



TF-SR measurements in the vortex state
deduce an effectivemagnetic penetration depth

and superconducting
coherence lengthfrom the internal magnetic field distribution
ns
≡ density of superconducting
carriers
~ vortex core size
The length scale for spatial
variations of the superconducting
order parameter(r), which
decreases to zero at the vortex core
centre, and rises to its maximum
value outside the vortex core.
T.J. Jackson et al. Physical Review Letters 84, 4958 (2000)
)](1)[/exp()0()(tAtNtN




The time histogramof decay positrons is given by:
where
)cos()()(
0



ttGAtA







22
2
1
exp)(ttG

In polycrystallinesamples the measured internal magnetic field distribution n(B) of the
vortex lattice is nearly symmetric due to the random orientation of the crystallites with
respect to the applied field. In this case it is generally sufficient to approximate the
relaxation function G(t) by a simple Gaussiandepolarization function:
which implies a Gaussian distribution of internal fields whose second moment
2
)(B


22
)()(BBB
is related to via:
The second moment is a function of the magnetic penetration depth

and coherence
length

(i.e.two fundamental length scales). When

>>

and H >> Hc1
the following
relation is generally obeyed:
*2
2
1
)(
m
n
B
s



Density of
superconducting
carriers
K. Ohishiet al.J. Phys. Soc. Jpn. 72, 29 (2003)
Polycrystalline MgB2
(Tc
= 38.5 K)
FFTs of TF-SR signal at H= 0.5 T
)cos(
)cos()(
2/
2/
2
2
22
b
t
b
s
t
s
HteA
BteAtA
b










The TF-SR signal in the time domainis fit to the
following relation:
where the depolarization rate is related to the
second momentof the internal field distribution n(B)
as follows:
2
)(B


Vortex Lattice of a Type-II Superconductor
TF-SR is ideally suited to measure the internal magnetic field distribution
B
n(B)
e.g. field distribution of a square vortex lattice (from Mitrovic et al.)
01234
-0.2
-0.1
0.0
0.1
0.2
0.3


Asymmetry
Time (s)
16182022242628
0.000
0.005
0.010
0.015
0.020
0.025
0.030
Vanadium
H = 1.5 kOe
T = 2.5 K
Real Amplitude
Frequency (MHz)
Fast Fourier
Transform
0
400
800
1200
0
400
800
1200
x(Å)
y(Å)
0
400
800
1200
0
400
800
1200
x(Å)
y(Å)
0
400
800
1200
0
400
800
1200
x(Å)
y(Å)
0
400
800
1200
0
400
800
1200
x(Å)
y(Å)
4.84.95.05.15.2


6.95


6.75
B (kG)
n(B)
JES et al.,Phys. Rev. B 76, 134518 (2007)



G
ab
rGi
G
euuK
bBrB




2
2
1
4
0
)(
)1()(

2c
B
B
b
])1(21)[1(2
2422
bbbGu
ab


YBa2Cu3Oy
Fit of Single Crystal TF-SR Spectrato Phenomenological Model
GL-model of spatial field profile:

ab
effectivepenetration depth

ab
effectivevortex core size
-400-2000200400600800
T = 0.02 K
1.6 kOe
2.0 kOe
2.4 kOe
2.9 kOe


B - B (G)
Real Amplitude (a.u.)
1.01.52.02.53.0
120
130
140
150
160
170



(Å)
H (kOe)
V
T = 0.02 K
01234
100
200
300
400
500



(Å)
Temperature (K)
V
H = 1.6 kOe
M. Laulajainenet al.Phys. Rev. B 74, 054511 (2006)
Pure Vanadium
404406408410412
Real Amplitude
Frequency (MHz)
H = 30 kOe
676678680682
Real Amplitude

H = 50 kOe
Frequency (MHz)
H (kOe)
d (10
-3
)

(degrees)

(Å)

(Å)
(c)
(b)
(a)
0
10
20
30
40
50


900
1050
1200
1350
1500


110100
60
70
80
90
0.0
0.5
1.0


JES et al.,Phys. Rev. Lett. 93, 017002 (2004)
Vortex lattice of V
3Si (Tc
= 17 K)
TF-SR line shapes at T= 3.8 K
E
9/2
E
7/2
E
5/2
E
3/2
E
1/2
EF + 0
EF
Rapid spatial variation of the pair
potential (r) at the vortex site
Low-lying quasiparticle excitations are
boundto the vortex core.
Hess et al.PRL 62, 214 (1989)
core center
outside core
First observation of bound core states
STSonNbSe
2
Vortex Electronic Structure
Conduction by Electrons in Solids
Intervortex Quasiparticle Transfer in a Superconductor
Atoms far apart
Atoms within bonding distance
~ 10 nm
Vortices far apartVortices strongly overlap
~ 1000 nm
Shrinkingvortex cores are attributed to the delocalizationof bound
quasiparticle core states.
The magnetic field dependence of the vortex core sizer0
in V
3Si measured by
SRcan be compared to electronic thermal conductivity measurements,
which are directly sensitive to delocalized quasiparticles.
0.00.20.40.60.81.0
0.00
0.02
0.04
0.06
V3Si

0
20
40
60

e
/

N
r
0
(Å)
1/H1/2 (kOe-1/2)



Square VL
Hexagonal VL
10
0.0
0.4
0.8
1.2
1.6
NbSe2
V3
Si


(

0
2
-

2
)H (a.u.)
[Hc2(0)/H]
1/2
0.0
0.4
0.8
1.2
1.6

e
(a.u.)
e
~ N(0)deloc
= N(0)tot
-N(0)loc

[(Hc1)
2
-(H)2]H
SR & electronic thermal conductivity
The vortex cores contain localizedor bound quasiparticle states, with a density proportional
to the product of the vortex core area (~ 2) and the applied magnetic field H. Since the
electronic thermal conductivity is a measure of delocalizedquasiparticles only, the following
relation is expected to hold:
JES, Rep. Prog. Phys. 70, 1717-1755 (2007)
~
0
~1

(r)
r
1

0
T/Tc
L. Kramer & W. Pesch, Z. Phys. 269, 59 (1974)
1
is slope of (r) at
vortex core center
(r)
Kramer-Pesch Effect
Thermal population of higher energy core
states leads to an expansionof the core
radiuswith increasing T, because the higher
energy bound states extend outwards to larger
radii.
012345
50
75
100
125
150
175
H=1.9 kOe
H=5 kOe
NbSe2


r
0
(
Å
)
Temperature (K)
Hayashi et al., PRL 80, 2921 (1998)
J.E. Sonieret al., PRL 79, 1742 (1997)
R.I. Miller et al., PRL 85, 1540 (2000)
Kramer-Pesch effect observed by SR
0510152025
0
20
40
60
80
100
0.0
0.1
0.2
0.3
0.4
0.5
0.6



e
/

n

ab
(Å)

H (kOe)

0.00.10.20.30.4
50
100
150
200

ab
(Å)
NbSe
2


H/Hc2
T = 0.02 K
T = 2.5 K
T = 4.2 K
Freeze out thermal excitations of
quasiparticle core states to reveal
multibandvortices.
F.D. Callaghan et al., Rep. Phys. Rev. Lett. 95, 197001 (2005)
Vortex core size in the multiband superconductor NbSe
2
NbSe2
has twodistinct superconducting energy gaps (largeand small) on different Fermi
sheets.
At low fields the loosely boundquasiparticle states associated with the smallerenergy gap
dominate the vortex structure, but delocalize at higher magnetic field where these states
strongly overlap.