Download Superconductivity in a magnetic field 1 Ginzburg

Phys 735 Superconductivity
Lecture 6 - Sept. 25, 2013
Superconductivity in a magnetic field
Lecturer: Ed Taylor
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Ginzburg-Landau theory in the presence of an external magnetic field
In the last lecture, we considered the Ginzburg–Landau free energy
"
#
2
Z
2
B
β
1 ~
e∗
3
2
4
Fs = Fn + d r α(T )|Ψ| + |Ψ| +
∇ − A Ψ +
2
2m∗ i
c
8π
(1)
in the absence of an external magnetic field H and derived the Ginzburg–Landau equations by minimization
of Fs with respect to the order parameter Ψ as well as the vector potential A. This free energy was actually
a Helmholtz free energy, F = U − T S. In contrast, when there is an external magnetic field H, it is much
more convenient to work with the Gibbs free energy
B·H
.
(2)
4π
In doing so, one does not have to worry about the energetics of the induced electromotive force (emf) that
arises when magnetic flux enters the system under consideration1 . As before, in this expression B is the
magnetic flux. In contrast to H which is just the applied magnetic field, the flux is the total magnetic field,
including contributions from screening currents, etc.
We now wish to compare the Gibbs free energies of the superconducting and normal phases when a
magnetic field H is applied. In the normal state there are no screening currents and B = H. On the other
hand, in the superconducting phase, because the magnetic field is screened inside the superconductor, B ' 0.
Equation (2) shows that there is a (Gibbs free) energy cost associated with the expulsion of a magnetic field.
When the field exceeds the thermodynamic critical field (f ≡ F/Ω is the free energy density)
G=F−
Hc2
α2
≡ fs − fn =
,
8π
2β
(3)
the superconducting state is no longer favourable, and superconductivity is destroyed.
As it turns out, the foregoing analysis is only valid for so-called type I superconductors. In arriving
at (3), we assumed that no magnetic fields entered the superconductor below Hc . In fact, for type-II
superconductors, the magnetic field can penetrate into the superconductor through vortices at a field (Hc1 )
below Hc . It was a student of Landau’s — Alexei Abrikosov — who first had this insight. Abrikosov was
trying to understand some peculiar experimental results concerning the critical field in thin superconducting
films[1]. The results seemed to disagree with predictions of the Ginzburg–Landau (GL) theory. Abrikosov
refused to believe that the theory could be wrong and looked for alternative solutions of the GL equations.
To see where the new solutions come from, it is helpful to put the Ginzburg–Landau free energy (1)
into dimensionless form, as Abrikosov
originally did. Using the same order parameter scaling as before
p
but
now also scaling the position r0 ≡ r/λL by the London penetration
Φ(r) ≡ Ψ(r)/Ψ∞ with Ψ∞ ≡ |α|/β,
√
0
depth, the vector potential
√ A = A/ 2Hc λL by the upper critical field and the penetration depth, and the
magnetic field B0 = B/ 2Hc , the dimensionless free energy becomes (dropping primes)
2
fs − fn
1 4 1
2
= −|Φ| + |Φ| + ∇ + A Φ + B 2 .
(4)
2
Hc /4π
2
iκ
1 For
details see F. Reif, Fundamentals of Statistical and Thermal Physics, (McGraw Hill, New York, 1965).
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Here, the only parameter is
κ≡
λL
,
ξ
(5)
the ratio of the London penetration depth and the coherence length. This tells us that the magnetic properties
of superconductors are completely determined (within GL theory at least) in terms of this parameter. It
leads, for instance, to the classification alluded to above:
√
(6)
κ < 1/ 2; Type I
and
√
κ > 1/ 2; Type II.
(7)
√
As it turns
out, for κ < 1/ 2, the “surface energy” at a normal and superconductor interface is positive;
√
for κ > 1 2, it’s negative. The thermodynamics of type-II superconductors is thus different than that for
type I. Instead of a single thermodynamic critical field, there are two critical fields. Above the lower critical
field Hc1 , it is energetically favourable for vortices to form. Each of these vortices carries magnetic flux.
As the magnetic field is increased, the density of vortices increases. At some higher critical field, Hc2 , the
vortices become so densely packed that they begin to overlap. At this point, superconductivity in a type-II
superconductor is destroyed.
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Surface energy
Suppose that an external magnetic field equal to the thermodynamic critical field Hc is applied to a superconductor. The Gibbs free energy of the superconducting and normal phases will be equal and the two
phases can in principle coexist. The crucial scale that determines the state in this region is the surface energy
between the normal and superconducting regions. If it’s positive, coexistence is not energetically favourable
and superconductivity will vanish for H > Hc (type I). On the other hand, if it’s negative, superconductivity
will continue to exist for H > Hc with superconducting domains interspersed with magnetic-field carrying
normal domains. We use the GL functional to estimate this surface energy.
The GL equation resulting from the variation of (4) with respect to Φ∗ is
2
i
∇ + A Φ(r) + |Φ(r)|2 Φ(r) − Φ(r) = 0.
(8)
κ
We imagine that the magnetic field H is pointing along the z axis while the superconducting-normal interface
is perpendicular to the x axis. Using the choice of gauge
A = A(x)ŷ = H(x)xŷ,
(9)
2 2
fs − fn
1 4
1 dΦ dA
2
= −|Φ| + |Φ| + 2 +
+ A2 |Φ|2 .
Hc2 /4π
2
κ dx
dx
(10)
the GL free energy (4) simplifies to
The resulting GL equations are
2
δFs
−2 d Φ(x)
=
0
=
−κ
+ |Φ(x)|2 Φ(x) − Φ(x) + A2 Φ(x)
δΦ∗
dx2
and
(11)
δFs
d2 A
=0=
− |Φ(x)|2 A.
(12)
δA
dx2
The surface energy Σ is the difference between the Gibbs free energies of the coexistence phase and the
phase where there is either only a superconductor or only a normal metal (remember that at H = Hc , these
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are the same). Using the same dimensionless units as led to (4), the dimensionless Gibbs free energy density
when H = Hc is
2
√
gs − gn
1 4 1
2
(13)
= −|Φ| + |Φ| + ∇ + A Φ + B 2 − 2B.
2
Hc /4π
2
iκ
√
√
The factor of 2 here comes from the scaling B0 ≡ B/ 2Hc . The surface energy is thus
"
"
# Z
2
2 √ #
Z ∞
∞
√
1 4 1
1
2
2
2
√
dx −|Φ| + |Φ| + Σ=
dx
∇ + A Φ + B − 2B −
−√
2
iκ
2
2
−∞
−∞
"
2 2 #
Z ∞
1
1
1
dx −|Φ|2 + |Φ|4 + =
∇ + A Φ + B − √
.
(14)
2
iκ
2
−∞
If we multiply (8) by Φ∗ and integrate over x, one sees that (14) can be simplified to
"
2 #
Z ∞
1
1
4
Σ=
.
dx − |Φ(x)| + B − √
2
2
−∞
(15)
Equations (11) and (12) must be solved numerically in order to calculate the surface energy (15). Without
carrying out such a calculation, we can make some analytic observations. Deep in the superconducting region
where Φ = 1, the magnetic field is zero, B = 0, and the integrand of (15) is small. Likewise,
deep in the
√
normal region, Φ = 0 and the magnetic field is equal to Hc (in our scaled units, B = 1/ 2). Here again the
integrand is zero. Taken together, this means that the integrand in (15) is localized about the superconductornormal interface as one would expect for a surface energy. Now, when
√ κ 1, the magnetic field penetrates
far into the sample, well beyond the interface, and hence B ∼ 1/ 2 in the interface region, meaning that
the surface energy is negative for κ 1.
In the opposite limit, when κ 1, the magnetic field barely penetrates into the superconducting sample
and B ' 0 over most of the interface region. Since Φ(x) < 1 in this region, the surface energy will be positive.
From this we conclude that past a critical value κc of κ, the surface energy between a superconductor and
normal metal will be negative, while for κ < κc , it will be positive. To estimate this value, we note that the
integrand in (15) will vanish identically when
q√
Φ(x) =
2B(x) − 1.
(16)
√
One can show that this is an exact solution of the GL equations (11) and (12) when κ = 1/ 2. This leads to
the conclusion we stated earlier: for type I superconductors, the surface energy between a superconductornormal interface is positive. Abrikosov’s Nobel-worthy discovery was that there was a second class of superconductors – type II – characterized by a negative surface energy. The negative surface energy means that for
type II superconductors, when H is on the order of Hc , rather than destroying superconductivity completely,
it is energetically favourable to form a mixed state in which superconducting domains coexist with normal
regions in which the magnetic field is substantial. Solving the GL equations for type-II superconductors,
Abrikosov realized that (for typical magnetic field configurations), these domains take the form of a lattice
of vortices, usually called Abrikosov lattices in his honour (see Fig. 1)
While most elemental superconductors are type-I, in the e.g., high-Tc cuprates, κ is typically on the
order of 10-40, and hence these are type-II superconductors, a fact which gives rise to a very rich array of
effects. Importantly for practical applications, superconducting Niobium (and related Niobium-Titanium
compounds) is type-II with a very high Hc2 , on the order of 15T. This is a crucial feature since it allows
wires of Niobium-Titanium to carry large currents. In contrast, the thermodynamic critical field of most
elemental superconductors is on the order of 10−3 − 10−1 T.
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Lower critical field
The foregoing considerations suggest that there should be a lower critical field Hc1 such that for H > Hc1 ,
normal domains first appear in a superconductor. As it turns out, the lowest-energy normal configuration
which can support a magnetic field is a vortex. Thus, Hc1 corresponds to the magnetic field at which it is
energetically favourable to nucleate a single vortex.
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Upper critical field
As the magnetic field is increased above Hc1 , the number of vortices increases. At some point, the density
will become so great that superconductivity will simply vanish. This occurs at the upper critical field Hc2 .
We can estimate this field by noting that the “size” of a vortex is determined by the coherence length ξ.
Thus, Hc2 is the magnetic field at which the mean distance d ∼ ξ between vortex cores equals the coherence
length. In the scaled units we use in (4), this corresponds to d = κ−1 . The magnetic field carried by a vortex
is one flux quantum per d2 ; i.e., κ. In the normal phase B = H so this gives the estimate (in unscaled units)
√
(17)
Hc2 = 2κHc .
Solving the GL equations, one can see that this is actually an exact result.
References
[1] A. Abrikosov, Nobel lecture;
http://www.nobelprize.org/nobel_prizes/physics/laureates/2003/abrikosov-lecture.pdf.
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Figure 1: The first reported observation of an Abrikosov lattiice. From U. Essmann and H. Trauble Physics Letters,
v. 24A, p. 526, 1967.
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