Chap. 6 Type Ⅱ Superconductivity - super.skku.edusuper.skku.edu/lectures/Solid_Chap6(2012).pdf ·...
Transcript of Chap. 6 Type Ⅱ Superconductivity - super.skku.edusuper.skku.edu/lectures/Solid_Chap6(2012).pdf ·...
Chap. 6 Type Ⅱ Superconductivity
6.1 Introduction
6.2 The Vortex
6.3 The Modified Second London Equation
6.4 General Thermodynamic Concepts (skip)
6.5 Critical Fields
6.1 INTRODUCTION
Figure 6.1 The H-T phase space:
(a) type I and (b) type Ⅱ
superconductors.
In 1930
: W.J. de Hass & J. Voogd
– working with the superconducting alloy Pb-Bi.
The Tc (max) of this alloy (Tc
=8.8 K) does not differ much from its constituents.
(Lead Tc
= 7.2 K, Bismuth Tc
= 8.5 K)
But
Hc
(at 4.2K): Hc
(Pb) = 0.055 T, Hc
(Pb-Bi) = 1.7 T
H
T
Lower Critical Field
HC2
HC1
TC
Upper Critical Field
Normal State
Mixed State
Meissner State
Abrikosov Vortex Lattice
Temperature
Mag
netic
Fie
ld
★ This chapter develops the essential features of Abrikosov’s work
through the phenomenology of typeⅡ
superconductivity.
▶
In Section 6.2 : We see that flux is observed to enter a typeⅡ
superconductor in a discrete array of known
as vortices.
Each vortex has a single flux quanta Φ0
.
We will find the fields and currents associated with each vortex and show
that the radius of the vortex (the coherence length ξ).
The two type superconductors can be distinguished by comparing λ and ξ. ( type I superconductor –
λ < ξ, type Ⅱ
superconductor –
λ > ξ)~~
▶
In Section 6.3 : We see that B and J from a vortex are more
conveniently described by modifying the second London equation.
To fully understand vortex formation in typeⅡ
materials
▶
In Section 6.4 : We first review some basic concepts in equilibrium
thermodynamics (skip).
▶
In Section 6.5 : Section 6.4 concepts are then used to study the
various critical magnetic fields.
★
We therefore see that this chapter focuses on understanding the
vortices that give rise to the magnetic field phase boundaries, Hc1
(T) and Hc2
(T) for a typeⅡ
superconductor.
6.2 THE VORTEX
Above the lower critical field, Hc1
(T), magnetic flux will enter a typeⅡ
superconductor.
How we can detect the flux around a simple bar magnet
Consider typeⅡ
bulk superconductor (a ≫
λ)
H0
→ H0
> Hc1
A vapor of tiny micron-sized nickel particles is now evaporated over the top surface of the superconductor, perpendicular to the applied field.
The particles will coalesce on the surface along lines of constant flux density just as in the case of the iron filings and thus the surface of the superconductor is “decorated”, showing how flux enters.
Vortex and coherence length
(a) (b)
Figure 6.2
A type Ⅱsuperconductor that has been decorated with magnetic particle.
The applied field is perpendicular to the page.
(a) A photograph revealing the triangular array of vortices formed when a superconducting sample, in this case YBa2
Cu3
O7
, is placed in a magnetic field. (The anisotropic sample is oriented so that the c axis is parallel to
the applied field). The applied field strength is 40 Gauss and the distance 0.8μm. (b) The triangular array of patterns ; each patterns regions of constant flux density. C1
is a contour along the perimeter of a pattern, C2
is a contour around the center of one.
We see that the flux density is strongest at the center of each pattern. (the particles concentrate in regions of high-flux density)
The density of vortices nv
22 32
)2/3(1
A1
vn (α
: the distance between the
centers of nearest patterns)
ㅡ (6. 1)
The average flux density in the slab
0/ vnAB (Φ0
: the flux in each vortex) ㅡ (6. 2)
Since we can measure both <B> and α, the value of Φ0
is determined experimentally and found to always be single flux quantum (h/2e).
Φ = B · A =>
Let us find the consequences of this
experimental fact on the flux and current
densities in an isotropic typeⅡ
SC.
From the fluxoid
quantization condition
(section 5.5)
C Sstot dd sBl)J( ㅡ (6. 3)
along the contour C1
(shown in Figure 6.2b)
1 1
sBlJ200 C SS dd ㅡ (6. 4))(
0
The flux is greatest at center and so Bz
(x,y)
is at a minimum (constant)
along the contour C1
. Consequently,
0)()( ,,
ccZccZ yxBy
yxBx
where (xc
,yc
) are the loci of points that define the contour.
ㅡ (6. 5)
▽×B=μ0
Js
and combining this relation with Eq. (6. 5)
01
0 BJ
S along C1 ㅡ (6. 6)
Thus, the fluxoid
quantization condition of Eq. (6. 4) becomes
BBAdS
2
0 23sB
1
ㅡ (6. 7)
The surface S1
is the area of one of the patterns so direct integration recovers the fact that
densityvortex : 00
VVv nnA
NB
ㅡ (6. 8)
ki
00
kji
B
zz
z
Bx
By
Bxyx
The flux quantization condition for the second contour C2
, which is a circle centered in one of the flux patterns
sBlJ22
200 dd
SC S ㅡ (6. 9)
In the limit that the radius r of the circular contour C2
approaches zero,
2
lJlim 2000 C Sr
d ㅡ (6.10)
i1
2Jlim 2
0
0
0 rsr
We expect Js
to be constant along C2
and azimuthally directed(Φ).
From (6.10)
ㅡ (6.11)
rs1 J => J 0 sr
In Eq. (6.11)
)2(limlJlim 200
2000
2
rJd SrCSr
0)(limsBlim 00 2
BAdA
Sr
☜ Not physical !!!
i1
2J 2
0
0max s
If core has a radius ξ, then the maximum supercurrent
density is
ㅡ (6.12)
a bounded expression.
How we can overcome this problem??
ξ
is the radius of the normal region, we find the increase in the energy
δε
of each of the Cooper pairs ( the gap energy 2Δ) as the core is
approached.
We can overcome this problem by choosing
to center of the vortex as having a normal
core region.
The increase in energy comes from the increase in the kinetic energy of the Cooper pairs. To see this, recall
ss qn vJ ** ㅡ (6.13)
(vs
: the velocity of the Cooper pairs (superelectrons))
i1v *
max
ms
The maximum velocity (from Eq. (6.12))
ㅡ (6.14)
The increase in the energy of a Cooper pair is created by the increase in its velocity.
If there are no currents flowing, the electrons are all moving with some background velocity known as the Fermi velocity vF
(in Section 5.4).
See next page
12)-(6 --- i12
J 20
0max
s
Velocity of the Cooper pairs
YBCO.for /10~1031
1082.110v
i1*
v
i1*
i1* 2
2i1* 2
i1* 2
* *i1
* 2
* 2i1
* 2* v
i1* 2*)(*i1
*)(**2
)( i1Λ 2
i12
v**J
ynt velocitsupercurre ofn Calculatio
4930
34
0
20
2
0
0
02
0
0s
smmKg
Jm
mmmh
m
qqh
m
qe
h
mq
mqn
qnm
qn
s
s
s
s
)2(
)21()1(
,max,
2,
,
max,2
,2
,
max,2
,
yFsyF
yF
syF
yF
syF
vvv
vv
vvv
v
The average kinetic energy of a Cooper pair when no current is flowing;
)(21
21 2
,2
,2
,*2*0
kin zFyFxFF vvvmvm ㅡ (6.15)
=> Now consider the increased velocity due to the flow of current (Jy
).
])([21 2
,2max
,,2
,*1
kin zFsyFxF vvvvm ㅡ (6.16)
0kin
1kin
The difference in energy at the core
ㅡ (6.17)
ㅡ (6.18)max,,
* syF vvm=>
)( max,, syF vv
The additional velocity created by the current is much less than the
Fermi velocity so the linearizing
we find
Superconducting Energy Gap (2∆) and Coherence length
We expect that by averaging over all the electrons, we will find
2,
2,
2,
2zFyFxFF vvvv ㅡ (6.19)
(where the bracket denotes the average of the component of the velocity.)
The averaged velocity squared to be the same for each component
22, 3
1FyF vv ㅡ (6.20)
Substituting this averaged expression for and our relation for yFv ,max,sv
(Eq. 6.14) into Eq. 6.18 gives
13
1*
*max,,
*
mvmvvm FsyF
3Fv
=> ㅡ (6.21)
Since δε
≈
2Δ
at the radius of the core ξ, we find
32Fv ㅡ (6.22)
18)-(6 ----- max,,
* syF vvm
At zero temperature the energy gap is 2Δ0
and so the corresponding core radius, denoted ξ0
, is given by
000 46.3
132
FF vv ㅡ (6.23)
From the microscopic BCS theory
00
Fvㅡ (6.24)
(where ξ0
is known as the BCS coherence length)
Near the Tc
, the BCS theory shows that the energy gap goes to zero as
)/(1 CTT Therefore,
)/(1)0()(lim
cTT TT
Tc
ㅡ (6.25)
(where ξ(0) is the temperature independent coefficient of ξ
near Tc
)
Temperature dependence of coherence length (ξ)
Temperature dependence of coherence length
0.0 0.2 0.4 0.6 0.8 1.00.0
2.0
4.0
6.0
8.0
10.0
(T)
/(0
)
T/Tc
)/(1)0()(lim
cTT TT
Tc
We have found the coherence length by arguing that the Cooper pairs split apart of depair
when their increase in energy exceeds the gap
energy. Thus the maximum current density for Js,Φ
(Eq. 6.12) is referred to as the depairing critical current density Jpair
. Therefore,
20
0pair 2
J
This is the maximum current density
that can be sent through the
superconductor. This expression is only approximate because it depends on our model of the vortex core.
by the Ginzburg-Landau theory,
20
02
0
0pair 2.5
133
J
ㅡ (6.27)
If we apply a current density, J > Jpair
, the material becomes normal.
Depairing critical current density (Jpair)
r1
2J from 2
0
0s
※
Summary
⊙ The magnetic flux enters a type Ⅱ
superconductor for fields above
Hc1
in a regular triangular array of vortices. (experiment)
⊙ Each vortex has a single flux quantum.
⊙ The current density of the vortex increases as the center of the vortex
is approached until the increased kinetic energy causes the Cooper
pairs to unbind.
⊙ The vortex is thus modeled as having a normal core of radius ξ, the
coherence length.
M
H
M-H curve
Hc1 Hc2
0
Figure 6.3 shows our model of the density of superelectrons
(Cooper
pairs) for a single vortex. Inside the model core the conduction
electrons are in the normal state.
Ginzburg –
Landau theory
Density of superelectrons
Vortex-core model
* ξ ≥ λ : Type-I Superconductor
* ξ ≤ λ : Type-II Superconductor
* ξ ≪ λ : High–к Materials
What makes our approach reasonable is the fact that most practical type Ⅱ
superconductors have ξ
≪ λ. Thus, the core typically occupies
a very small fraction of the total volume of the vortex. In the Ginzburg –
Landau theory, the ratio of the two lengths if defined as
ㅡ (6.28)
( к : Ginzburg –
Landau kappa )
6.3 THE MODIFIED SECOND LONDON EQUATION
*Purpose : With vortex-core
model for how flux distributes itself in
a type II superconductor, we will find the full spatial dependence of
the flux density and currents
associated with the vortex.
In Figure 6.3, has a normal cylindrical core with a radius ξ. In the superconducting region outside the core, the local currents and fields are governed by the 2nd
London equation.
0B)J( s ㅡ (6.29)
Outside the normal core the magnetic flux density satisfies the vector Helmholtz
equation
0)r(B1)r(B 22
ㅡ (6.30)( for r ≥
ξ
)
012
2 zz BB
( for r ≥
ξ
) ㅡ (6.31)
Because of the cylindrical symmetry of the vortex structure with
its core centered along the z-
axis, , where r is the radius vector in cylindrical coordinates.zzB i)r(B
The solution to Eq. (6.31)
32)-(6 --- )sincos()(
)sincos)((),(
'
0
'
0
mDmDrI
mCmCrKrB
mmm
m
mmm
mz
* Cm
, Cm
’, Dm
, Dm
’: constant
* Im
: modified Bessel functions of the first order with m
* Km
:modified Bessel functions of the second order with m
Modified Bessel functions
m
x
m
x
xx
xmx
xm
x
xxx
)2( ! 2
1 )(K lim
)2
( !
1 )( I lim
ln )(K lim , 1 )( I lim
m0
m0
00 00
For m
> 0
For m
= 0
Bz is independent of the angle Φ for the flux of the single vortex.
Therefore, the solution is obtained with m = 0
)()()( 0000 rIDrKCrBz ( for r ≥
ξ
) ㅡ (6.33)
for because 0 0
BC.in D0 & C0 of values thefind We)1(
00
z
rIDBr
(6.34) ----- for )(
for )()(
for constant is Bz and normal is core The:core theof radius at the BCother The )2(
00
00
rrKC
rKCrB
r
z
)sincos()( )sincos)((),( '
0
'
0
mDmDrImCmCrKrB mm
mmmm
mmz
The coefficient C0
is determined by applying the fluxoid
quantization
condition. In section 6.2
0tot sBl)J( SC s dd ㅡ (6.35)
Let the contour of integration be a circle of radius Rc
in the x-y
plane.
Bz
decays exponentially to zero for r >>
λ
→ the associated current density will also decay in the same manner.
As Rc
→ ∞, Js → 0, the first term
in Eq. (6.35) →
0
Consequently,
The only contribution to the fluxoid
quantization condition comes from the
surface integral term where the chosen contour is the entire x-y plane.
(6.37) ------ )(K)(K21
2C
1
102
2
20
0
)](K)(K21[2C
)(K2C)(KC
)(K)(2C212)(KC
/ )( )(K2C 2 )(KC
2 )(KC 2 )(KC sB
102
22
0
12
02
00
12
00
200
000 00
000 00z0
rrr
drdrrdrrdrr
drrrdrrdsBdss
C0
can be determined by the Flux Quantization condition.
01 2
)(Klim
/)(K)(K
n
10
xx exx
rxxxdxxx
)( { r for )/(
r for )/(00
00
rKC
KCz rB
A considerable simplification is possible of most practical type
II
materials [High-κ
materials (κ = λ/ξ
≫ 1)]
Ex) Nb3Sn κ
≈
25 & HTS typically have κ
> 50
1
102
2
20
0 )(K)(K21
2C
Consider how C0
simplifies for κ ≫ 1.
0lnlim)(lim 2
002
0
xxxKx
xx
The first term
xxKx
ln)(lim 00
The second term
1)(lim 10
xxK
x
ㅡ (6.38)
ㅡ (6.39)xxK
x
1)(lim 10
Therefore, for κ ≫ 1, (from Eq.(6.34)
{i )(
2
i )(2
020
020
B(r)z
z
rK
K
ㅡ (6.40) { )(
)(
00
00
)(
rKC
KCz rB for r ≥
ξ
for r < ξ
20
2
(for κ≫1)
≈ 0 ≈ 1
The Js(r) associated with B(r), can be found from Ampère’s law (▽ Х B = μ0Js)
{ i )(20S
130
0
(r)J rK
for r ≥
ξ
for r < ξㅡ (6.41){
i )(2
i )(2
020
020
B(r)z
z
rK
K
iii ),()( ),(1)(1)( , , , zr100
00 xKxKxKdx
xdKdr
xdKdxdrdxdr xr
scoordinate lcylindricafor i1iizrr zr
Figure 6.5 shows how both the flux and current density decay exponentially far from the center of the vortex.
)/((r)J 1s rK)/(B(r) 0 rK
Js =0
Near the core , the magnetic flux density (B) in the
superconductor increases as;
zr ri ln
2Blim 2
0
ㅡ (6.42)
and current density (Js ) is given by;
i 1
2)/(limJlim 2
0
00S rrr
B ㅡ (6.43)
)ln()(lim )(for i )(2 0002
0rr
Kξ rr
KBxz
rxdxxd
drxddxdrdxdr xr 111ln1ln , , ,
The expression for the current density near the core is identical to Eq. (6.11),
which was obtained by considering the fluxoid
quantization.
11)-(6 i12
Jlim 20
0
0
rsr
iii zr
) ( r
We seek to modify the second London equation by adding a function V(r) as a source to insure fluxoid
conservation.
V(r)B)J( s ㅡ (6.44)←Modified second London equation
which is valid for all space assuming V(r) vanishes except along iz.
V(r) is in the same direction as the flux density
zi )r(V(r) V ㅡ (6.45)
Eq (6.44) is converted into the integral expression
S SC
S SS S
ddd
ddd
sV(r)sBl)J(
sV(r)sBs)J(
s
ㅡ (6.46)(by stoke’s
theorem)
Modified Second London Equation
SSC s
SC s
ddd
dd
s)r(VsBl)J(
sBl)J( 0 ㅡ (6.35)
ㅡ (6.46)
Rc
→ ∞, Js
→ 0: the first terms in Eq(6.35) & (6.46) →
0
We see that although V(r) is zero everywhere except at r = 0, its integral over that
point must be the constant Φ0 .
The only function that has this property is the two-dimensional delta function, δ2(r).
z20 i (r)ΦV(r) δ ㅡ (6.47)←
Vorticity
z20s i (r)B)J( ㅡ (6.48)
More generally, for N
vortices that located at the two-dimensional positions ri, the vorticity
is readily generalized to
N
ii
1z20 i )rr(V(r) ㅡ (6.49)
(for a vortex along the z-axis in an isotropic, high-κ superconductor)
Example 6. 3. 1
As an example of how to use the modified second London equation, we complete the solution of our single vortex problem. For an isotropic superconductor, from Eq.(6.48)
)r(122
02
2
zz BB ㅡ (6.50)
Particular solution, pzB
)(2 02
0
rKB p
z
ㅡ (6.51)
)(2
)B(113
0
0
0,
rKJ pzs
ㅡ (6.52)
This solution not only satisfies the original differential equation but also the boundary condition that the flux density vanishes far from the core. Moreover, this solution also satisfies the fluxiod
quantization condition for a high-κ
superconductor by construction.
Therefore, is the desired flux density distribution for the single vortex and the associated current density is
pzB
see next page.
)r(1i (r)BB
)/(
BB1 H)J(
HHH)(HJside;both for )( curl a Take
JH48)-(6 ---- i (r)B)J(
220
22
z2022
200
22
0
220
2s
22
z20s
zz BB
Derivation of Equ. (6.50)
{ i )(20S
130
0
(r)J rK
for r ≥
ξ
for r < ξㅡ
(6.41){i )(
2
i )(2
020
020
B(r)z
z
rK
K
(6.40) ㅡ
)(2 02
0
rKB p
z
(6.51) ㅡ )(2 13
0
0,
rKJs
ㅡ
(6.52)
These solutions match those of the finite-sized vortex core in the region r ≥
ξ.
However, for r < ξ, the solutions differ.
xxKx
ln)(lim 00
xxK
x
1)(lim 10
Eq(6.51) & Eq(6.52) diverge as r →
0. ← unphysical!!!
Consequently, to ensure that the solutions of the modified second London equation
match the physically correct ones, we restrict the validity of the solutions to the region
r ≥
ξ. Within the region r ≤
ξ, we will interpret the flux density to be a constant
defined by its value at r = ξ.
Example 6. 3. 2
Consider now the solution to the modified second London equation for an array of
vortices
defined by Eq(6.49). For an isotropic superconductor that fills
all of space,
Eq(6.44) leads to
)rr
(2
)( 020
p
pz KrB
ㅡ (6.54)
)rr(122
02
2i
izz BB
ㅡ (6.53)
z20 i )rr(V(r) ii
ㅡ
(6.49)V(r)B)J( s ㅡ
(6.44)
Since Eq(6.53) is linear, the solution is just the superposition of the particular
solution
for a single vortex centered at each array site; namely,izB
☜ Figure 6.9 Two vortices parallel to the z-axis and located along the x-axis at ±a in an infinite superconductor. The vortex at x=a has a positive vorticity
and the other at x=-a has a negative one. The contours of constant Bz
(x,y) are shown.
Example 6. 3. 4
We now examine the case of two vortices that individually generate a magnetic flux density in opposite directions.
zz yaxyax i )()(i )()(V(r) 00 ㅡ (6.59)
22
0
22
020 )()(
2),(
yaxK
yaxKyxBz ㅡ (6.60)
The flux density vanishes along the y-z
plane defined by x=0.(antisymmetry). However, the current density does not vanish in this plane. Instead, it is purely y-directed.
p (x, y)r
)(2 02
0
rKBi
z
We end this section by showing how it can also simplify finding the
electromagnetic energy associated with vortices.
If the vortices are modeled as having a normal state core, the total
electromagnetic energy W of the system will come from three places:
* Ws
: the energy of superconducting
region.
* Wc
: the energy of the normal cores
in the superconductor.
* Wn
: the energy of the normal regions
(no superconductor).
Integrating over the volume where there is no superconductor, Vn, we find
nVn dvW B
21 2
0ㅡ (6.64)
core
B 21 2
0Vc dvW
ㅡ (6.65)
where the integration is over the cores of the vortices & high-κ
superconductors
( λ≫ξ
), the contribution to the energy from the normal core is negligible . )0( cW
BH21
density energy Magnetic
m E
Vortex Energy (εv
)
dvW sVss
)]J()B(B[21 2
0
ㅡ (6.67)
Using the Ampère’s
law (∇ X
H = Js), Eq(6.66) becomes
The vector identity
D)(CD)(CC)(D ㅡ (6.68)
s)]J(B[2
1 )]J([BB2
1
)]J([B2
1 ))]J((B[B2
1
00
0
2
0
ddv
dvdvW
sV s
V sV ss
ss
ss
ㅡ (6.69)
where the integral over the surface that encloses is obtained by applying Gauss’
theorem.S SV
s sV ss ddvW s)]J(B[
21 VB
21
00 ㅡ (6.70)
by the modified second London equation V(r)B)J( s
dvW sV sss
JJB )]()([2
10
2
0
ㅡ (6.66)From 4-130,
The energy per unit length along the vortex direction will be denoted as and
similarly for each contribution and .
W nW
sW
zLWW / ㅡ (6.71)
(where Lz
is the length of the vortex in the z-direction.)
Therefore, after the integrating Eq(6.70) over z
ss C sSs ddaW n)]J(B[
21 VB
21
00 ㅡ (6.72)
☜ Figure 6.11
A vortex along the z-direction in a superconducting volume Vs
. The volume is enclosed by the surface ∑s
. The cross-
sectional area Ss
is parallel to planes of constant z and is encircled by the contour Cs
. Notice that Ss
is not a function of z.
dxdyda
s sV ss ddvW s)]J(B[
21 VB
21
00 ㅡ (6.70)
Let us use Eq(6.72) to find (the electromagnetic energy per
unit length for
a single vortex in an infinite isotropic superconductor).V
In Example 6.3.1, particular solution (the flux density) given as
)(2 02
0
rKBB p
zz
ㅡ (6.73)
Both the flux density and the current decay exponentially from the vortex core. Thus the second term in Eq (6.72) can be made vanishingly small
if the contour Cs
is taken far from the core.
ss C sSs ddaW n)]J(B[
21 VB
21
00 ㅡ (6.72)
)(4
(r)ii)(22
102
0
20
20020
0
KdarK
sS ZZV
ㅡ (6.74)
r @ i Φi (r)ΦV(r) z0z20δVortex energy per unit length
with
ss C sSs ddaW n)]J(B[
21 VB
21
00 ㅡ (6.72)
6.5 Critical Fields (Hc1
and Hc2
)
※ Purpose
: we calculate the upper [Hc2
(T)] and lower [Hc1
(T)] critical fields
for a type II super conductor where we will need to
combine the thermodynamics of a superconductor developed in section 6.4 .
We restrict ourselves to examining the properties of an isotropic superconducting slab of thickness 2a where a ≫
λ.
Gibbs free energy in a magnetic field is given by:
BHHHH VTUTTSTG ),(),(),( ㅡ (6.121)
where U is internal energy and V is volume.
Without applied field (H = 0), the free energy is given by:
),0(),0(),0( TUTTSTG ㅡ (6.122)
The increase in the internal energy (U) due to the presence of the field is the quantity W that was found in Section 6.3 :
WTUTU ),0(),(H ㅡ (6.123)
)],0(),([),0(
),0( ),(),0(
),0(),0( ),( ),()121.6(
TSTSTVWTG
TTSTTSVWTG
VWTTSTGTTSTG
HBH
HBH
BHHH
ㅡ (6.124)
The last term in this expression is just the heat exchange in the sample if the field is generated ∆Q = T∆S.
For the second-order phase transition
that no
heat is generated whenever we consider the phase boundaries like Hc1
and Hc2
for a type II superconductor.
0),( TSTQ H
Combining Eq. (6.121) through (6.123)
)123.6( from ),0(),0(),(
)121.6( from ),0(),0(),0(),0(),0(),0(
WTTSTGTU
TTSTGTUTUTTSTG
H
(6.161) --- ),(),(),( BHHHH VTUTTSTG
We can therefore rewrite Eq(6.124) as
BHH VWTGTG ),0(),( ㅡ (6.126)
The free energy in terms of the flux density B;
V
dvWTGTG BHH ),0(),( ㅡ (6.128)
s
s
V
Vss
dv
dvTGTG
B
BBB
H
H )()(12
1),0(),(0
2
0
ㅡ (6.129)
With W from Eq. (6.66: ) and the superconducting current
density given as ▽ Х
B=μ0Js
the Gibbs free energy for a superconductor is
dvW sV sss
)]J(JB[2
10
2
0
Likewise, for the normal material volume (Js
= 0), Vn,
nn VVnn dvdvTGTG B B HH 2
021),0(),(
ㅡ (6.130)
The lower critical field is found by considering the difference in the Gibbs free energy of the superconductor when the vortex is absent and when it is present.
),( TG H
Let us consider type II superconducting slab, shown in Figure 6.14.
[Figure 6.14
Field distribution in a type Ⅱ superconductor (a ≫
λ).]
),0(),( 00 TGTG ss H ㅡ (6.131)When there are no vortices B≈0.
(b) no vortex (c) single vortex in the slab(a) Superconducting slab
sVsss dvWTGTG B ),0(),( 01 HH ㅡ (6.132)
When one vortex is in the center of the slab the corresponding Gibbs free energy is),(1 TG s H
In Section 6.3 the electromagnetic energy of a single vortex of length Lz
was given by
vortex theoflength unit per energy the:V
ㅡ (6.134)zVs LW
appHH ㅡ (6.135)
The thermodynamic field for the slab is the applied field and soH
For a single vortex, we find that
S zzV
LBdxdyLdvs
0B HHH ㅡ (6.136)
Putting these expressions into Eq. (6-132) together yields the Gibbs free energy of the slab containing a single vortex
zzVss LLTGTG 001 ),0(),( HH ㅡ (6.137)
The difference in Gibbs free energies between having and not having a single vortex present follows from Eq(6.137) and (6.131) :
zVss LTGTG )(),(),( 001 HHH ㅡ (6.138)
The Gibbs free energy increased by introducing a vortex due to the electromagnetic energy εvLz .
(6.132) --- B ),0(),( 01 sVsss dvWTGTG HH
When , having a vortex is thermodynamically the lowest energy state and vortices will be present in the superconductor.
This happens as soon as the applied field exceeds the value
),(),( 01 TGTG ss HH
01 V
cH ㅡ (6.139)
where Hc1
is the lower critical field.
Using the expression for the energy per unit length of the vortex given by Eq (6.74), we find
)(4 02
0
20
KV
ㅡ (6.74)
)(4 02
0
01
KH c
ㅡ (6.140)
For high-κ
(λ/ξ≫1) superconductors
)ln(4
lim 20
01
cH ㅡ (6.141)
xxKx
ln)(lim 00
(6.138) --- )(),(),( 001
zVss LTGTG HHH Lower critical field (Hc1 )
Hence for H
< Hc1, no vortices enter the superconductor (B≈0 –
Meissner
state).
For H ≥ Hc1 vortices enter and the flux density is no longer zero.
Because both ξ
and λ
depend on temperature, so does Hc1(T), thereby forming the phase boundary in the H-T plane shown in Figure 6.15.
☜ Figure 6.15 The H-T phase diagram for a bulk superconducting slab in a uniform magnetic field for a type Ⅱ
superconductor.
☜ Figure 6.16
The top view of a superconductor with the field coming out of the page. The vortices form a triangular array with the separation α
between vortices.
As the applied magnetic field increases, the average flux density increases in the superconductor because the density of the vortices, nv, increase. According to Eq(6.2),
VVn B ㅡ (6.142)
where B=<B>.
Upper critical field (Hc2 )
Now let us try to estimate the upper critical field, Hc2. As the applied field increases, the vortices in the triangular lattice get closer together. The flux density B averaged over the sample is given by Eq(6.142)
20
32
B ㅡ (6.143)
As the applied field increases enough so that the cores begin to overlap, the whole superconductor begins to be covered with the normal cores and hence becomes normal. At this point B=µ0H because the material is normal.
Specifically with α ≈ 2ξ the cores begin to overlap so that at the upper critical field
20
02
0
02 32)2(3
2
cH ㅡ (6.144)
the material reverts to the normal state. Notice that our estimate of Hc2
depends on the model with which we describe the vortices.
From the Ginzburg-Landau calculation;
20
02 2
cH ㅡ (6.145)
)/(11~)(lim
cTT TT
Tc ㅡ (6.146)
ㅡ (6.147)
we find that
ccTT T
TTHc
11)(lim 22
This linear dependence on temperature near Tc
is also apparent in the figure.
☜ Figure 6.15 The H-T phase diagram for a bulk superconducting slab in a uniform magnetic field for a type Ⅱ superconductor.
cc T
TTH 1~)(2
Temperature dependence of Hc2
Since
20
02 2
cH