Beyond Zero Resistance Phenomenology of Superconductivity

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Nov 15, 2013 (3 years and 7 months ago)

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Beyond Zero Resistance


Phenomenology of Superconductivity

Nicholas P. Breznay

SASS Seminar


Happy 50
th
!

SLAC

April 29, 2009

Preview


Motivation / Paradigm Shift


Normal State behavior


Hallmarks of Superconductivity


Zero resistance


Perfect diamagnetism


Magnetic flux quantization


Phenomenology of SC


London Theory, Ginzburg
-
Landau Theory


Length scales:
l

and
x


Type I and II SC’s

Physics of Metals
-

Introduction


Atoms form a periodic lattice



Know (!) electronic states key for
the behavior we are interested in



Solve the Schro …






… in a periodic potential





E
H

)
(
2
)
(
2
2
r
V
m
r
H







)
(
)
(
K
r
V
r
V





K is a Bravais lattice vector

K

Wikipedia

Physics of Metals


Bloch’s Theorem




Bloch’s theorem tells us that
eigenstates have the form …



… where u(r) is a function with the
periodicity of the lattice …






E
r
V
m




)
(
2
2
2





E
m
H




2
2
2

)
(
)
(
r
u
e
r
r
k
i







)
(
)
(
K
r
u
r
u





r
k
i
Ae
r





)
(

Free particle Schro

Wikipedia

Physics of Metals


Drude Model


Model for electrons in a metal


Noninteracting, inertial gas


Scattering time
t





Apply Fermi
-
Dirac statistics






t
)
(
)
(
t
p
E
q
t
p
dt
d





damping term

H

E
k


E
k

E
f
E
f
m
k
E
2
2
2


http://www.doitpoms.ac.uk/tlplib/semiconductors/images/fermiDirac.jpg

Physics of Metals


Magnetic Response


Magnetism in media


Larmor/Landau diamagnetism


Weak anti
-
// response


Pauli paramagnetism


Moderate // response


Typical
c

values




c
Cu
~
-
1 x 10
-
5



c
Al
~ +2 x 10
-
5




minimal response to
B

fields



m
r

~ 1


B =
m
0
H

)
(
0
M
H
B


m
in SI

linear response

familiarly

H

E
k


E
k

E
f
E
f
H
M
c

H
H
H
r
m
m
m
c
m




0
0
)
1
(
B
Physics of Metals


Drude Model Comments


Wrong!


Lattice, e
-
e, e
-
p, defects,



t

~ 10
-
14

seconds


MFP ~ 1 nm



Useful!


DC, AC electrical conductivity



Thermal transport


Lorenz number
k/s
T



Heat capacity of solids


Wikipedia

E
m
ne
J









t
2
m
ne
p
p
0
2
2
2
2
,
1
)
(
m








t
)
(
)
(
t
p
E
q
t
p
dt
d





3
AT
T
C
v



Electronic
contribution

Lattice

1
~
'
s
fe
meas


2
8
2
2
2
10
44
.
2
3
K
W
e
k
T
L
B







s
k
8
10
6
.
2
1
.
2




meas
L
Preview


Motivation / Paradigm Shift


Normal State behavior


Hallmarks of Superconductivity


Zero resistance


Perfect diamagnetism


Magnetic flux quantization


Phenomenology of SC


London Theory, Ginzburg
-
Landau Theory


Length scales:
l

and
x


Type I and II SC’s

Hallmark 1


Zero Resistance


Metallic R vs T


e
-
p scattering (lattice interactions) at high temperature


Impurities at low temperatures




R

Temperature

Residual

Resistance

(impurities)

Electrical resistance

R
0

Lattice (phonon)

interactions

T
D
/3

Hallmark 1


Zero Resistance


Superconducting R vs T




R

Temperature

R
0

T
c

“Transition temperature”

Hallmark 1


Zero Resistance


Hard to measure “zero” directly


Can try to look at an effect of the
zero resistance


Current flowing in a SC ring


Not thought experiment


standard configuration for high
-
field laboratory magnets (10
-
20T)


Nonzero resistance


changing
current


changing magnetic
field


One such measurement


Superconductor

Circulating

supercurrent

Magnetic (dipole) field

From Ustinov “Superconductivity” Lectures (WS 2008
-
2009)

I

18
10


Cu
SC


Hallmark 1


Zero Resistance Notes


R = 0 only for DC


AC response arises from kinetic
inductance of superconducting
electrons


Changing current


electric field


Model: perfect resistor (normal
electrons), inductor (SC electrons) in
parallel



Magnitude of “kinetic inductance”:





At 1 kHz,


Normal
R
L
12
10
~


V
ac
L
R
http://www.apph.tohoku.ac.jp/low
-
temp
-
lab/photo/FUJYO1.png

Hallmark 2


Conductors in a Magnetic Field

Normal metal

Field off

Apply

field

t
E
J
B
t
B
E
B
E




























0
0
m


j
E




)
1
(
~
)
(
/
0
t
t
e
B
t
B


R
L
/

t
Hallmark 2


Conductors in a Magnetic Field

Apply

field

Perfect (metallic) conductor

Superconductor

Normal metal

Cool

Cool

Field off

Apply

field

Apply

field

Hallmark 2


Meissner
-
Oschenfeld Effect

Superconductor

Cool

Apply

field


B = 0


p
erfect diamagnetism:
c
M

=
-
1





Field expulsion unexpected; not discovered for
20 years.






H
H
M
M
H
B






c
m
0
)
(
0
B/
m
0

H

-
M

H

H
c

H
c

Hallmark 3


Flux Quantization

2
7
15
0
10
2
~
2
10
2
~
2
cm
G
e
hc
s
V
e
h









Earth’s magnetic field ~ 500 mG, so in
1 cm
2

of
B
Earth

there are ~ 2 million

0
’s.

first appearance of h in
our description; quantum
phenomenon

0

n
A
d
B







Total flux (field*area) is integer
multiple of

0


Hallmark 3


Flux Quantization

Apply uniform field

Measure flux

Aside


Cooper Pairing


In the presence of a weak
attractive interaction, the filled
Fermi sphere is unstable to the
formation of bound pairs electrons



Can excite two electrons
d

above
E
f
, obtain bound
-
state energy <
2E
f

due to attraction



New minimum
-
energy state
allows attractive interaction (e
-
p
scattering) by smearing the FS

The physics of superconductors Shmidt, Müller, Ustinov

Preview


Motivation / Paradigm Shift


Normal State behavior


Hallmarks of Superconductivity


Zero resistance


Perfect diamagnetism


Magnetic flux quantization


Phenomenology of SC


London Theory, Ginzburg
-
Landau Theory


Length scales:
l

and
x


Type I and II SC’s

SC Parameter Review

g(H)

H

H
c

g
normal state

g
sc state

2
2
0
c
H
g
m



Magnetic field


energy density



Extract free energy difference
between normal and SC states
with H
c






Know magnetic response
important; use R = 0 + Maxwell’s
equations … ?

London Theory


1


Newton’s law (inertial response) for applied electric field



S
J
dt
d
E


2
e
n
m
s











e
n
J
dt
d
m
eE
s
S


s
v
dt
d
m
F

s
s
s
ev
n
J

dt
dJ
m
E
e
n
S
s

2
dt
J
d
m
E
e
n
S
s









2
dt
J
d
dt
B
d
m
e
n
S
s







2
0
2










B
m
e
n
J
dt
d
s
S



Supercurrent density is

B
m
e
n
J
s
S



2




We know B = 0 inside superconductors

Faraday’s law

Fritz & Heinz London, (1935)

London Theory


2



S
J
dt
d
E


2
e
n
m
s


B
m
e
n
J
s
S



2






London Equations





t
E
J
B









0
0
0

m
m
J
B











0
m


B
m
e
n
B
B
s






2
0
2
m







B
m
e
n
B
s



2
0
2
m


Ampere’s
law

=0; Gauss’s law
for electrostatics

Magnetic Penetration Depth
-

l

B(z)

l

z

2
0
2
e
n
m
s
m
l

B
B



2
2
1
l



Screening not immediate;
characteristic decay length



Typical
l

~ 50 nm



m,e fixed


l
uniquely specifies
the superconducting electron
density n
s


Sometimes called
the “superfluid
density”

l
/
0
)
(
z
e
B
z
B


B
0

SC

Ginzburg
-
Landau Theory
-

1

4
2
2







n
s
f
f

First consider zero magnetic field



Order parameter




Associate with cooper pair
density:



Expand
f

in powers of |

|
2





To make sense,


> 0,
  
(T)


Free energy of

superconducting state

Free energy of

normal state

2


s
n
Need


>
-
Infinity; B > 0

Free energy of
SC state ~ #
of cooper pairs

Ginzburg
-
Landau Theory
-

2

4
2
2







n
s
f
f





4
2
2







n
s
f
f


0
2
2








n
s
f
f
d
d







2



For


< 0, solve for minimum
in
f
s
-
f
n











http://commons.wikimedia.org/wiki/File:Pseudofunci%C3%B3n_de_onda_(teor%C3%ADa_Ginzburg
-
Landau).png




Know that
f
n
-
f
s

is the condensation energy:









Ginzburg
-
Landau Theory
-

3







2
4
2
2







n
s
f
f


2
2



n
s
f
f
2
0
2
1
c
s
n
B
f
f
m


s
n
f
f

2
0
2
1
c
B
m


m
2
0

c
B
Ginzburg
-
Landau Theory
-

4



qA
i
p






Momentum term in H:




Now


include magnetic field



Classically, know that to include
magnetic fields …



0
2
2
4
2
2
2
2
1
2
m





B
eA
i
m
f
f
n
s















i
p
V
m
p
H
,
2
2
0
2
2
m
B
f
magnetic


4
2
2







n
s
f
f
Ginzburg
-
Landau Theory
-

5


Free Energy Density











0
2
2
4
2
2
2
2
1
2
m





B
eA
i
m
f
f
n
s











0
2
2
2
1
2
0
2
2
4
2
















dV
B
eA
i
m
m





d

0

F
d


0
2
2
1
2
2











eA
i
m







eA
i
m
e
J
2
Re
2
*





Ginzburg
-
Landau Theory
-

6



0
2
2
1
2
2











eA
i
m

Take


real,

normalize







2
0
2
2
2
3







































m






0
2
)
(
3
2
2







m
T


Define

m
T
T
2
)
(
)
(
2

x


0
)
(
2
2
2





T
x
Linearize in


Superconducting coherence length
-

x

x


(x)

Vacuum

SC

Superconductor

x



0
2
2
1
2
2











eA
i
m

0
)
(
2
2
2





T
x




Characteristic length scale for SC
wavefunction variation


London Theory



magnetic penetration depth
l



Ginzburg
-
Landau Theory


coherence length
x







l  x



two kinds of superconductors!

Pause

Surface Energy and “Type II”

H(x)

l

x

x


(x)

H(x)

x


(x)

x

l

x
l

l
x

Surface Energy:
x  l

H(x)

l

x


(x)

g
magnetic
(x)

2
2
0
c
cond
H
g
m

2
2
0
c
cond
H
g
m

energy penalty for excluding B

energy gain for being in SC state

g
sc
(x)

SC

Surface Energy:
x  l

H(x)

l

x


(x)

g
magnetic
(x)

2
2
0
c
cond
H
g
m

2
2
0
c
cond
H
g
m

energy penalty for excluding B

energy gain for being in SC state

net energy penalty at a surface / interface

g
net
(x)

g
sc
(x)

SC

Surface Energy:
x  l

H(x)

l

x


(x)

g
magnetic
(x)

2
2
0
c
cond
H
g
m

2
2
0
c
cond
H
g
m

energy penalty for excluding B

energy gain for being in SC state

net energy gain at a surface / interface

g
net
(x)

g
sc
(x)

SC

Type I




Type II

H(x
)
l
x

(x
)
g
magnetic
(x
)
g
net
(x
)
g
sc
(x
)
H(x
)
l
x

(x
)
g
magnetic
(x
)
g
net
(x
)
g
sc
(x
)

predicted in 1950s by Abrikosov


elemental superconductors

x
l

2
1

k
x
l

2
1

k
x 
nm


l
(nm)

T
c

(K)

H
c2

(T)

Al

1600

50

1.2

.01

Pb

83

39

7.2

.08

Sn

230

51

3.7

.03

x 
nm


l
(nm)

T
c

(K)

H
c2

(T)

Nb
3
Sn

11

200

18

25

YBCO

1.5

200

92

150

MgB
2

5

185

37

14

x
l
k

Type II Superconductors
x  l

H

Normal state cores

Superconducting region

http://www.nd.edu/~vortex/research.html


London Theory



magnetic penetration depth
l



Ginzburg
-
Landau Theory


coherence length
x







l  x



two kinds of superconductors

The End