587
Lecture 6.1
Ginzburg-Landau theory I:
Free energy
The complete fundamental equations for macroscopic superconductivity were written by
Ginzburg and Landau in 1950. 1 A superconductor is simply a charged superfluid. A
superfluid is a Bose-condensed state of interacting bosons. (These are pairs of fermions
in every known case of a charged superfluids, and for some neutral superfluids.)
The GL free energy is the macroscopic elastic theory appropriate for any superconductor. The degrees of freedom are two fields, each varying smoothly in space. The
first field is the complex-valued “order parameter” field, Ψ(r), which takes on complex
values. The complex order parameter is what defines superconducting order. It will
often be convenient to separate the complex phase angle:
p
Ψ(r) ≡ |Ψ(r)|eiθ(r) ≡ ns (r)eiθ(r)
(6.1.1)
Here ns (r) has dimensions of a number density, and is called the “superfluid density”
(you will shortly see why). Here the arbitrary choice of θ; expresses the spontaneous
breaking of the continuous (gauge) symmetry, while the magnitude |Ψ| expresses the
strength of superconductivity (with |Ψ| = 0 if and only if the state is not superconducting.) The second field is the vector potential, representing the electromagnetic degrees
of freedom. It is easiest to start with the neutral case.
6.1 A
Ginzburg-Landau free energy: neutral superfluid
Neutral
We begin by considering In a neutral superfluid, such as 4 He, the GL free energy
depends only on the {Ψ(r)} field.
The Ginzburg-Landau total free energy has the form
Z
Ftot (Ψ(r), A(r)) = d3 r FGL (r),
(6.1.2)
1 L. D. Landau and V. L. Ginzburg, Zh. Eksp. Teor. Fiz. 20, 1064 (1950); an English translation is
printed as an appendix in Men of Physics: L. D. Landau I, by D. ter Haar (Pergamon, 1965).
c
Copyright 2011
Christopher L. Henley
588
LECTURE 6.1. GINZBURG-LANDAU THEORY I: FREE ENERGY
where the free energy density at r is
FGL (r) = FL (Ψ(r)) + Fgrad (r)
(6.1.3)
LATER The physical meaning of the terms is best understood by pretending they
refer to a charged Bose superfluid with particles of mass m∗ and charge e∗ (which are,
in fact, pairs.) A superconductor is nothing but a charged superfluid.
Terms in Ginzburg-Landau free energy
The first portion of the GL free energy density is the Landau term
FL (Ψ) = α|Ψ|2 +
β
|Ψ|4 + . . . .
2
(6.1.4)
This is the same free energy density as in the Landau mean field theory of any kind of
ordering (as seen in stat mech courses, or in Lec. 1.4 C). It represents the free energy
density in a spatially uniform state in which the order parameter is constrained to be
|Ψ|.
The minimum of FL (Ψ) occurs at Ψ0 = 0, or at some nonzero Ψ0 , depending whether
the equilibrium state is normal or superconducting. Here α > 0 or α < 0 corresponding
to whether T > Tc or T < Tc . The assumption leading to the form (6.1.4) is that
FL depends analytically on the components of Ψ, and that it has the the symmetry
assumed (namely, rotation of Ψ by a phase factor). Landau also assumes analyticity as
a function of temperature, so we expect
α∼
= α0 (T − Tc )
(6.1.5)
in the vicinity of Tc . In the case with Ψ0 6= 0, it would be more exact to say that, on the
complex plane of possible Ψ values, the minimum occurs at all points around a circle
of radius Ψ0 . Choosing the point with a particular phase θ is a spontaneous symmetry
breaking of the phase symmetry.
The gradient term is
~2
Fgrad =
|∇Ψ|2 .
(6.1.6)
2m∗
From the viewpoint of standard Landau theory,
Near Tc , and only near Tc , we can neglect the further terms in the expansion of
FL (Ψ); at lower temperatures, it does not have this exact shape, but it keeps the
qualitative property of a minimum at Ψ0 . Similarly, this gradient term is literally valid
only near Tc where |Ψ| is small. In a more complete expansion, the coefficient of |∇Ψ|2
has corrections of O(|Ψ|2 ); furthermore, that coefficient is different for gradients in
the magnitude or phase of Ψ(r). Nevertheless, this basic version of G-L with just three
material-dependent parameters (α, β, and m∗ ) is qualitatively correct for most purposes
even at T = 0.
6.1 B
Microscopic rationalization of GL free energy
To motivate the GL theory, I’ll justify it as a mean-field approximation of a microscopic
toy model of Nb 1 bosons with mass m∗ interacting by a pair potential u(R) at
zero temperature. What’s the (many-boson) ground state? If we turned off u(R), all
bosons occupy the (normalized) single-boson ground state wavefunction ψ0 (r). Such an
occupation of a state by a macroscopic number of bosons is called Bose condensation.
6.1 C. G-L FREE ENERGY: CHARGED SUPERFLUID
589
So let’s assume the interacting ground state is still Bose-condensed and approximate
it by placing all Nb bosons in the same state ψ(r), to be optimized variationally: this is
the Bose version of Hartree-Fock theory (compare Lec. 1.4X). It turns out that Ψ(r) is
proportional to ψ(r) and the GL free energy is the variational expectation. The relation
of Fgrad ((6.1.10)) to the kinetic energy term is evident; the |Ψ|4 term comes from the
interaction term.
The bosons have a chemical potential µb which is, as always, just a Lagrange multiplier to control the final number Nb in the minimization. In the case of a superconductor
or paired superfluid, you might view µb = µb (T ) as controlling the propensity of fermions
to pair: in this case, the density of “bosons”, i.e. Cooper pairs, is certainly not fixed.
We’ll show next that, after another approximation or two, the expectation (6.1.7) is the
Ginzburg-Landau free energy. If we approximate the interaction as being short-ranged
compared to the scale on which ψ(r) varies, and represent its strength by u. Then
hHi = Nb
Z
dd r
~2
1
|∇ψ(r)|2 + Nb (Nb − 1)
2m∗
2
Z
dd ru|ψ(r)|4
(6.1.7)
Let Ψ(r) ≡ Nb 1/2 ψ(r). Then the local number density is
ns (r) ≡ |Ψ(r)|2
(6.1.8)
and thus Ψ(r) has a sensible limit as the system size is increased. Relation (6.1.8) will
be fundamental within the GL theory as the definition of the “superfluid density” ns .
Identify Ψ(r) with the GL order parameter field, then hHi = Ftot , the total GinzburgLandau free energy, defined by (6.1.2) (with β = vcell U ). This is the grand result. The
gradient term comes from the kinetic energy, the |Ψ|4 term represents an interaction, and
the |Ψ|2 term represents the combination of true potential and (temperature-dependent)
chemical potential.
6.1 C
G-L free energy: charged superfluid
Now we incorporate as a second classical field the vector potential A(r), – a real, 3component vector with the usual gauge freedom. This enters the GL free energy because
the current in a charged superfluid couples to the magnetic field. Now the free energy
density becomes
FGL (r) = FL (Ψ(r)) + Fgrad (r) + Umag (B(r))
(6.1.9)
with FL unchanged,
The gradient in Fgrad is replaced by a gauge-invariant gradient,
Fgrad =
2
ie∗
~2 A)Ψ
(∇ −
2m∗
~c
(6.1.10)
where A is the vector potential.
Finally, the magnetic field energy term comes straight from the elementary theory
of magnetostatics and does not depend on the material:
Umag =
1
|B|2 .
8π
(6.1.11)
590
LECTURE 6.1. GINZBURG-LANDAU THEORY I: FREE ENERGY
London approximation
The London approximation is analogous to the fixed-length spin approximation in
statistical mechanics of lattice models. It means we set |Ψ(r)| to a fixed constant, but
still allow θ(r) to vary. Historically this was introduced when only Type I superconductors were recognized, but in fact its main regime of validity is for extreme Type II
superconductors (such as high-Tc cuprates, see Lec. 6.6 ). Sometimes it also happens to
apply in thin wires or films when the boundary condition constrains Ψ(r) to be nearly
constant throughout the cross-section.
6.1 D
Phase symmetry and phase gradients
The symmetry broken in superconductivity (or superfluidity) is the phase of a pair wave
function: the so-called gauge symmetry. What physical phenomena correspond to this?
First of all, within G-L theory the supercurrent is the (gauge-invariant) gradient of the
phase. (That makes sense: the equilibrium state of uniform phase is the state when no
currents are moving.) Secondly, there should be a Goldstone mode – a kind of sound –
in any neutral superfluid.
Supercurrent operator
From the same kind of correspondence as in Sec. 6.1 B the expectation of the current
operator gives the supercurrent
ie∗
∗ ~i
[∇ −
A(r)]Ψ(r)
(6.1.12)
Js (r) = Re Ψ(r)
m∗
~c
I have restored the gauge-invariant derivative for a charged superfluid. In a more useful
form Was this was written elsewhere?
i
h
~
e∗
Js (r) =
ns ∇θ − A(r) .
(6.1.13)
m∗
~c
You can verify that this conservation equation is satisfied,
d
ns (r) + ∇ · Js (r) = 0
dt
(6.1.14)
by substituting ns ≡ ΨΨ∗ into (6.1.18).
Gauge symmetry
Evidently the GL free energy is invariant under the same gauge transformations that
are applied to a Schrödinger wavefunction.
A special case of the gauge symmetry is a uniform phase change,
Ψ(r, t) → Ψ(r, t)ei∆θ
(6.1.15)
while A(r, t) is unchanged. (If we had a neutral superfluid, there is no gauge freedom
(6.1.17), but we do have the global phase symmetry (6.1.15).) Eq. (6.1.15) is the
continuous symmetry which is spontaneously broken in the choice of an arbitrary phase
angle θ.
6.1 X. TIME-DEPENDENT GINZBURG-LANDAU EQUATION
591
More generally, FGL is invariant under the following change of gauge :
A(r, t) → A(r, t) +
e∗
∇χ(r, t)
~c
Ψ(r, t) → Ψ(r, t)eiχ(r,t)
(6.1.16)
(6.1.17)
(This is obvious since both changes affect only the Fgrad term.) Recalling (6.1.1), we
see that θ̌ = θ + χ.
6.1 X
Time-dependent Ginzburg-Landau equation
What about the dynamics of the fields? I will give the answer first (its form is fairly
obvious), and then work on justifying it. The time-dependent Ginzburg-Landau equation
~i
d
~2 2
Ψ(r, t) = −
∇ Ψ(r) + α + β|Ψ(r)|2 Ψ(r)
dt
2m∗
(6.1.18)
for a neutral superfluid.
For a charged superfluid, the gradient in (6.1.18) is upgraded to the gauge-invariant
one, as you might have guessed. In addition, we expect second equation for B/dt; in
fact, we actually get the full Maxwell equations, with the Js (r, t) as the current.
Number and phase are canonically conjugate
We can rewrite (6.1.18) as
~i
d
δ
Ψ(r, t) =
Ftot ({Ψ(r, t); Ψ∗ (r, t)}).
dt
δΨ∗ (r, t)
(6.1.19a)
A somewhat suspect (but standard) twist in the recipe is that one must expand |Ψ(r, t)|2 →
Ψ(r, t)Ψ∗ (r, t) everywhere, and then take variational derivatives as if Ψ and Ψ∗ were
independent coordinates. Of course, we also have the conjugate,
−~i
d ∗
δ
Ψ (r, t) =
Ftot ({Ψ(r, t); Ψ∗ (r, t)}).
dt
δΨ(r, t)
(6.1.19b)
Then these are Hamilton’s equations from classical mechanics, where Ftot plays the role
of the classical Hamiltonian, and Ψ(r) and Ψ∗ (r) are canonically conjugate. What if we
decompose the order parameter as
p
Ψ(r, t) = ns (r, t)eiθ(r,t) .
(6.1.20)
Simply substituting this into (6.1.19a) (and its complex conjugate), then using eqs. (6.1.19a6.1.19b) plus the chain rule, you can obtain
~
δFtot
d
θ(r, t) = −
dt
δns (r, t)
(6.1.21a)
~
δFtot
d
ns (r, t) = +
dt
δθ(r, t)
(6.1.21b)
and
Then Eqs. (6.1.21a) and (6.1.21b) are also Hamilton’s equations of motion, and
furthermore the phase θ and the (charge) density ns are canonically conjugate. It turns
out they’re quantum-mechanically conjugate, too, a fact which we come back to in
Lec. 6.5 .
592
6.1 Y
LECTURE 6.1. GINZBURG-LANDAU THEORY I: FREE ENERGY
Goldstone mode?
A continuous symmetry like (6.1.15) normally implies the existence of a gapless Goldstone mode (as stated in Lec. 1.5 ): namely, if we make phase gradients at arbitrarily
large wavelengths, they will have arbitrarily small restoring forces leading to arbitrarily
slow motions, hence the dispersion should satisfies ω(q) → 0 as q → 0.
Zero sound in a neutral superfluid
Such a mode exists in a neutral superfluid. The time-dependent G-L equations are
all the machinery needed to find the small oscillation modes, analogous to lattice waves
or spin waves. They consist of oscillations of phase and (90◦ out of phase) superfluid
density – recall we found those are canonically conjugate. You can obtain
−~δ θ̇(r) = βδns (r)
~δ ṅ(r) =
~2 n 0 2
∇ δθ(r)
m∗
(6.1.22a)
(6.1.22b)
with n0 = |Ψ0 |2 . (Actually, in (6.1.22a), a ∇2 δns term also appears on the right-hand
side, but it is negligible in the long-wavelength limit.) It is because of the continuous
phase symmetry that only gradient terms can appear on the right-hand side of (6.1.22b).
When we put (6.1.22a) and (6.1.22b) together, we obtain
d2
δn(r) = −v02 ∇2 δn(r)
dt2
(6.1.23)
v02 = n0 β/m∗
(6.1.24)
with
But (6.1.24) can be written to look exactly like the equation for the ordinary sound
velocity in a fluid: v02 = B/ρ, where B = n20 β = n20 ∂ 2 FL /∂ns 2 is simply a bulk modulus
corresponding to the condensate density, and ρ ≡ m∗ n0 is the mass density of the
condensate. So our gapless mode is simply zero sound (sound which exists in a zerotemperature fluid). Such a sound mode is present with essentially the same velocity in
the normal state; in the case that the normal state is a Fermi liquid (i.e. 3 He), zero
sound was mentioned in Lec. 1.8. The quanta of zero sound are phonons. Such phonons
are also the dominant gapless excitation in a Bose superfluid such as 4 He. 2
Gapped modes in a superconductor
Of course, in a superconductor the superfluid density fluctuations carry charge density fluctuations, which have long-range Coulomb interactions, whereas Goldstone’s theorem only applies to local interactions. In the charged fluid, this sound mode is in fact
the familiar plasma oscillation (as in any metal) and the frequency is high.
6.1 Z
Table of useful formulas
2 At T > 0, the Goldstone excitation is properly second sound which is a distinct mode from first
sound. Second sound is well defined only at temperatures high enough that the “gas” of excitations
can keep in equilibrium; it becomes ill-defined at low termperatures when the mean-free path exceeds
the sample dimensions.
6.1 Z. TABLE OF USEFUL FORMULAS
Name
Effective (pair) mass
Pair charge
Ginzburg-Landau coefficients
Critical temperature
Condensation (free) energy
[Order parameter magnitude]
Superfluid density
[Gradient stiffness in FGL ]
Symbol
m∗
e∗
α, β
Tc
Fcond
Ψ0
ns n s ~2
m∗
593
Formulas
2me ?
2e
(
0
α ≈ α (T − Tc )
β>0
–
(
α2 /2β
Hc2 /8π (cgs units?)
(−α/β)1/2
Lecture
(Lec. 6.1 ?)
above
|Ψ0 |2 = −α/β
Lec. 6.2
Fcond ξ 2
Lec. 6.0
? Lec. 6.0
Lec. 6.2 ?
Lec. 6.2
1/2
above
m∗ c 2 · 1
4πe∗ 2 ns
1/2
~2 · 1
2m∗ |α|
Penetration depth
λ
(GL) Coherence length
ξ
Ginzburg-Landau parameter
κ
λ/ξ
Lec. 6.2 (and Lec. 6.6 )
(pair) flux quantum
Φ∗0
2π~c/e∗
Lec. 6.4
(Thermodynamic) Critical field
Hc
Lower critical field
Hc1
Upper critical field
Hc2
GL critical current
JcGL
2 ns e ∗ ~
√
3 3 m∗ ξ
Lec. 6.3
vortex line tension
v
4π|Fcond |ξ 2 ln κ
Lec. 6.6
√ ∗
2 Φ0
4π
∗ λξ
ln κ H
ln κ Φ0 = √
c
4π λ2
2κ
∗
2 Φ0 = √2κH
c
4π ξ 2
Lec. 6.2
Lec. 6.2
Lec. 6.0 (?), Lec. 6.3
Lec. 6.6
Lec. 6.6
Table 6.1.1: Ginzburg-Landau parameters and useful formulas. Values of α0 , Tc and β are
material dependent; formulas for them can be derived only within the microscopic (BCS)
theory, see Lec. 7.8 [omitted]. Numerically Φ∗0 = 2. − 7 × 10−7 gauss − cm2 .