Superconductivity and Superfluidity The London penetration depth but also F and H London suggested...
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Transcript of Superconductivity and Superfluidity The London penetration depth but also F and H London suggested...
Superconductivity and Superfluidity
The London penetration depthThe London penetration depth
but also
BB2
B
B2
F and H London suggested that not only
BB2
BB2
To which the solution is
LA xexpB)x(B LA xexpB)x(B
L is known as the London penetration
depth
It is a fundamental length scale of the superconducting state
x
xexpBA xexpBA
ABAB
Experiment had shown that not only but also within a superconductor 0B 0B 0B 0B
where 2soL enm 2soL enm
Lecture 4
Superconductivity and Superfluidity
Surface currentsSurface currents
So current flows not just at the surface, but within a penetration depth L
x
to equation Working backwards from the London equation
B
B2
BB2 7
gives 2ss enJcurlmB 2ss enJcurlmB
So, for a uniform field parallel to the surface (z-direction) the “new” equation becomes6
yoJxB
yoJxB
and as )xexp(B
xB
LL
A
)xexp(B
xB
LL
A
)xexp(B
J LLo
Ay
)xexp(
BJ L
Lo
Ay
LAy xexpJJ LAy xexpJJ or
LAy xexpJJ LAy xexpJJ
BA
Lecture 4
Superconductivity and Superfluidity
The London model - a summaryThe London model - a summary
The Londons produced a phenomenological model of superconductivity which provided equations which described but did not explain superconductivity
Starting with the observation that superconductors expel all magnetic flux from their interior, they demonstrated the concept of the Penetration Depth, showing that
So, in just one dimension we have
LAy xexpJ)x(J LAy xexpJ)x(J and
LA xexpB)x(B LA xexpB)x(B
Flux does penetrate, but falls of exponentially on a length scale,
Electric current flows only at the surface, again falling off exponentially on a length scale,
with the London penetration depth2soL enm 2soL enm
Lecture 4
Superconductivity and Superfluidity
Critical fieldsCritical fields
Onnes soon found that the normal state of a superconductor could be recovered by applying a magnetic field greater than a critical field, Hc=Bc/o
This implies that above Hc the free energy of the normal state is lower than that associated with the superconducting state
The free energy per unit volume of the superconductor in zero field isGS(T, 0) below Tc and GN(T,0) above Tc
The change in free energy per unit volume associated with applying a field Ha parallel to the axis of a rod of superconductor (so as to minimise demagnetisation)is
a
H
0
voa dHM)H(Ga
a
H
0
voa dHM)H(Ga
where Mv is the volume magnetisation
For most magnetic materials Mv is positive so the free energy is lowered when a field is applied, but if Mv is negative, the free energy increases
but for a superconductor MV is negative...
Lecture 4
Superconductivity and Superfluidity
Critical fieldsCritical fields
We have a
H
0
voSaS dHM)0,T(G)H,T(Ga
a
H
0
voSaS dHM)0,T(G)H,T(Ga
So, in the absence of demagnetising effects, MV= Ha = -Ha, and
a
H
0
aoSaS dHH)0,T(G)H,T(Ga
a
H
0
aoSaS dHH)0,T(G)H,T(Ga
When the magnetic term in the free energy is greater than GN(T, 0)-GS(T,0) the normal state is favoured, ie
2/1
SNo
c ))0,T(G)0,T(G(2
)T(H
2/1
SNo
c ))0,T(G)0,T(G(2
)T(H
HaHc
GN(T, 0)GN(T, 0)
GS(T, 0)GS(T, 0)
normal statenormal state
superconductingsuperconducting
2aoH2
1
2aoS H
21
)0,T(G 2aoS H
21
)0,T(G
Lecture 4
MV HaHc
Superconductivity and Superfluidity
Critical fields - temperature dependenceCritical fields - temperature dependence
Experimentally it is found that
2coc TT1H)T(H 2coc TT1H)T(H
Critical field
Lecture 4
Superconductivity and Superfluidity
Critical currentsCritical currents
If a superconductor has a critical magnetic field, Hc, one might also expect a critical current density, Jc.
The current flowing in a superconductor can be considered as the sum of the transport current, Ji, and the screening currents, Js.
If the sum of these currents reach Jc then the superconductor will become normal.
The larger the applied field, the smaller the transport current that can be carried and vice versa
In zero applied field
Current i
Radius, a
Magnetic field
Hi
idl.Hi iaH2 i so
The critical current density of a long thin wire in zero field is therefore
Typically jc~106A/m2 for type I superconductors
cc aH2i and
2c
ca
aH2j
aH2
j cc ieJc has a similar temperature
dependence to Hc, and Tc is similarly lowered as J increases
Lecture 4
Superconductivity and Superfluidity
The intermediate stateThe intermediate state
A conundrum:
If the current in a superconducting wire of radius a just reaches a value of
ic = 2aHc
the surface becomes normal leaving a superconducting core of radius a’<a
The field at the surface of this core is now
H’=ic/2a’
So the core shrinks again - and so on until the wire becomes completely normal
> Hc
But - when the wire becomes completely normal the current is uniformly distributed across the full cross section of the wire
Taking an arbitrary line integral around the wire, say at a radius a’<a, now gives a field that is smaller than Hc
as it encloses a current which is much less than ic ….so the sample can become superconducting again!
and the process repeats itself …………………………...this is of course unstable
Lecture 4
Superconductivity and Superfluidity
…….schematically.schematically
The sample is normal and current is distributed uniformly over cross section.
Current enclosed by loop at radius a’<a is i = ica’2/a2 < ic
also the line integral of the field around the loop gives H= i/2a’ = ica’/2 a2 < Hc
…..so the sample can become superconducting again
…..and the process repeats itself
Critical current is reached when the line integral of the field around the loop is ic = 2aHc
Current density is j c= i/a2.
Note current flowing within penetration depth of the surface.Superconducting state collapses
a
Lecture 4
Superconductivity and Superfluidity
The intermediate stateThe intermediate state
Instead of this unphysical situation the superconductor breaks up into regions, or domains, of normal and superconducting material
The shape of these regions is not fully understood, but may be something like:
isc sc scn
n
n
n
n
n
n
n
The superconducting wire will now have some resistance, and some magnetic flux can enter
Moreover, the transition to the normal state, as a function of current, is not abrupt
R
iic 2ic 3ic
Lecture 4
Superconductivity and Superfluidity
Field-induced intermediate stateField-induced intermediate state
A similar state is created when a superconductor is placed in a magnetic field:
Consider the effect of applying a magnetic field perpendicular to a long thin sample
The demagnetising factor is n=0.5 in this geometry, so the internal field is
Hi = Ha/(1-n) = 2Ha
So, the internal field reaches the critical field when
Ha = Hc/2
Ha
The sample becomes normal - so the magnetisation and hence demagnetising field falls to zero
The internal field must now be less than Hc (indeed it is only Hc/2)
The sample becomes superconducting again, and the process repeats
Again this is unphysical
Lecture 4
Superconductivity and Superfluidity
Field-induced intermediate stateField-induced intermediate state
Once again the superconductor is stabilised by breaking down into normal and superconducting regions
Resistance begins to return to the sample at applied fields well below Hc - but at a value that depends upon the shape of the sample through the demagnetising factor n
When the field is applied perpendicular to the axis of a long thin sample n=0.5, and resistance starts to return at Ha= Hc/2
For this geometry the sample is said to be in the intermediate state between Ha= Hc/2 and Ha=Hc
R
Ha/Hc
0.5 1.0 1.5
Lecture 4
Superconductivity and Superfluidity
HA=Hc(1-n)
Field distribution in the inetrmediate stateField distribution in the inetrmediate state
s
s
s
sn
n
n
When Hi=Hc the sample splits into normal and superconducting regions which are in equilibrium for Hc(1-n)<Ha<Hc
B at the boundaries must be continuous, and B=0 within superconducting region, so B=0 in both superconducting and normal regions
- the boundaries must be parallel to the local fieldH must also be parallel to the boundary, and H must also be continuous at the boundary, therefore H must be the same on both sides of the boundaryOn the normal side Hi=Hc, so on the superconducting side Hi=Hc
Therefore a stationary boundary exists only when Hi=Hc
Lecture 4
Superconductivity and Superfluidity
The Intermediate stateThe Intermediate state
A thin superconducting plate of radius a and thickness t with a>>t has a demagnetising factor of
n 1 - t/2a
So, with a field Ha applied perpendicular to the plate the internal field is
Hi = Ha/(1-n) = 2a.Ha/t
and only a very small applied field is needed to reach Ha= Hc
Generally, for elemental superconductors the superconducting domains are of the order of 10-2 to 10-1 cm thick, depending upon the applied field
The dark lines are superconducting regions of an aluminium plate “decorated” with fine tin particles
a
10-2cm
Ha
Lecture 4
Superconductivity and Superfluidity
Surface energySurface energy
The way a superconductor splits into superconducting and normal regions is governed by the surface energy of the resulting domains:
Surface energy >0
The free energy is minimised by minimising the total area of the interface
hence relatively few thick domains
Surface energy <0
Energy is released on formation of a domain boundary
hence a large number of thin domains
In the second case it is energetically favourable for the superconductor to spontaneously split up into domains even in the absence of demagnetising effects
Type IType I
Type IIType II
To understand this we need to introduce the concept of the “coherence length”
Lecture 4