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Ginzburg-Landau Theory: Simple Applications
References: de Gennes ch. 6, Tinkham ch. 4, AJL QL sect. 5.
Landau-Lifshitz (1936): 2nd order phase transition describable in terms of order parameter η, which is zero above Tc but takes finite value below, tending smoothly to zero as T → Tc (Example: spontaneous magnetization S of Heisenberg ferromagnet). η may be real scalar, complex scalar, real vector (e.g., ferromagnet), etc., but crucial feature is that η transforms under some symmetry group G which commutes with Hˆ (at least in absence of external ‘symmetry-breaking’ fields, e.g., Hˆ of FM invariant under rotation of S in absence of external magnetic field). This has the consequence that form of free energy, as a function of η, must be so constructed as to be invariant under G. Consider specifically case of complex scalar (appropriate to GL theory), then F must be invariant under rotation in Argand plane ⇒ in spatially uniform case, only even powers of |η|^2 can occur^1 : for small η, expand
F (η : T ) = F 0 (T ) + α(T )|η|^2 +
β(T )|η|^4 + O(|η|^6 ) (1)
Above Tc equilibrium value of η = 0 ⇒ α, β > 0. Below Tc, equilibrium value of η finite ⇒ α(T ) < 0. Simplest hypothesis:
α(T ) = α 0 (T − Tc), β ∼= β(Tc) ≡ β 0 (2)
In spatially varying case, expect one contribution to F is simply
F
η(r) : T
dr, with F (η : T ) as above, but in addition, expect gradient terms. Again, from analyticity and invariance under η → ηeiφ(φ 6 = f (r)) expect second-order terms (in η) to be proportional to |∇η(r)|^2 : Fgrad = γ(T )|∇η(r)|^2. [Note we can’t necessarily argue that gradient terms of order η^4 have this form: cf. below.] Thus, most general form of F
η(r) : T
for T near Tc is (since simplest hypothesis is γ(T ) ∼= γ(Tc) ≡ γ 0 )
F
η(r) : T
= F 0 (T ) +
α 0 (T − Tc)|η(r)|^2 +
β 0 |η(r)|^4 + γ 0 |∇η(r)|^2
dr (3)
Note a characteristic length (healing length) is defined by ξ(T ) = [γ 0 /|α 0 (T −Tc)|]^1 /^2 ∼ |T −Tc|−^1 /^2. Also note that theory is invariant under a constant rescaling of η: {η(r) → qη(r), q 6 = f (r)} provided coefficients α 0 , β 0 , γ 0 rescaled appropriately, i.e. absolute value of η has no significance. GL just choose the normalization so as to make γ 0 = ℏ^2 / 2 m. GL: take η(r) to be a Schr¨odinger-type wave function, denote ψ(r). Then the analysis goes through as above, except that we expect that in a vector potential A, ∇ → (∇ + 1 iℏ e
∗A(r)), e∗ (^) = effective charge associated with ψ(r). Anticipate result that e∗ (^) = 2e,
and choose normalization of∫ ψ(r) so that γ 0 = ℏ^2 / 2 m, m = single e−^ mass. (Thus, |ψ(r)|^2 dr 6 = 1 in general!) (^1) Odd functions of |η| are forbidden by the requirement of analyticity.
Thus, adding explicitly the magnetic field energy
2 μ^0 B(r)
(^2) dr (B ≡ curl A) we get
the ‘canonical’ GL free energy (Lecture 3.) (ψ now → Ψ)
F [{Ψ(r)}, T ] = F 0 (T )+ (4) ∫ { α(T )|Ψ(r)|^2 +
β|Ψ(r)|^4 +
2 m
|(−iℏ∇ − 2 eA(r)Ψ(r)|^2 +
μ− 0 1 (∇ × A)^2
dr
with α(T ) = α 0 (T − Tc), β ∼= const. The expression for the electric current, within this theory, is obtained by requiring that setting δF/δA(r) = 0 should give Maxwell’s equation, j(r) = ∇ × H(r) = μ− 0 1 (∇ × A(r)), thus
j(r) =
e m (Ψ∗(−iℏ∇ − 2 eA)Ψ + c.c.) (5)
Note in particular that if Ψ is constant in space, then j(r) = − 4 e 2 m |Ψ|
(^2) A(r).
Derivation of GL theory from BCS theory
It turns out that such a derivation is possible, if (a) normal component is in equilibrium (with static lattice, walls, etc.), (b) ‘number of Cooper pairs’ is small, and (c) all varia- tions in space are small on the scale of the Cooper pair radius.^2 In this case, the GL OP turns out to be, apart from the normalization which is a matter of convention, simply the COM Cooper pair wave function, i.e., the quantity
〈ψ↑(r)ψ↓(r′)〉r=r′=R ≡ F (R : ρ)ρ=0 (6)
that is ΨGL(r) = const · F (R, ρ)ρ=0, R=r (7)
Usual textbook derivation is via the GL equation which results from minimizing F with respect to Ψ(r) (cf. Lecture 3). I prefer to derive F directly: go through argument explic- itly for uniform case, but can be generalized provided scale of variation long compared to ξ 0. We define provisionally (i.e., forgetting about normalization)
Ψ = 〈ψ↑(r)ψ↓(r)〉 ≡
k
Fk, Fk ≡ 〈ak↑a−k↓〉 (8)
Consider the total free energy at finite T as a function of Ψ, where as usual we subtract a term μN. We have
F = 〈 Tˆkin〉 − μ〈 Nˆ 〉 − T S + 〈 Vˆ 〉 ≡ K + 〈V 〉 (9)
In the BCS model (Vkk′^ ≡ −|V 0 | within shell, zero outside), we have simply
〈V 〉 = −|V 0 |
k
Fk
≡ −|V 0 | |Ψ|^2 (10)
(^2) condition usually automatically satisfied in limit T → Tc since pair radius remains O(ξ 0 ) while both ξ(T ) and λ(T ) diverge for T → Tc
Thus, adding the potential-energy term, we finally get for the free energy F (Ψ : T )
F (Ψ : T ) = F 0 (T ) + {A−^1 (T ) − |V 0 |}|Ψ|^2 +
|B(T )||A(T )|−^4 |Ψ|^4 (18)
This is of the general form of the bulk terms in the GL free energy. It is clear that Tc is defined by the point at which the coefficient of |Ψ|^2 goes negative, i.e., at the temperature defined by 1 2
dn/d ln(1. 14 βcc) = 1/|V 0 | (19)
which is just the BCS equation for Tc. Eliminating |V 0 | and expanding for T close to Tc, we find
F (Ψ : T ) = F 0 (T ) + (dA−^1 /dT )Tc (T − Tc)|Ψ|^2 +
|B(T )||A(T )|−^4 ||Ψ|^4 (20)
We could perfectly well use this expression as our GL free energy. However, it turns out to simplify the ensuing formulae somewhat if we introduce the quantity |V 0 |Ψ = [A(Tc)]−^1 Ψ which since it has the dimensions of the energy gap ∆ and reduces to it for homogeneous equilibrium (see below), we will denote ∆: in terms of ˜ ∆ we have (since˜ d(ln A)/dT = − 1 /T ≈ 1 /Tc)
F ( ∆ :˜ T ) = F 0 (T ) + N (0)
− (1 − T /Tc)| ∆˜|^2 +
7 ζ(3) 8 π^2
(kB Tc)^2
| ∆˜|^4 +...
Needless to say, differentiation of F ( ∆ :˜ T ) with respect to ∆ gives back the BCS˜ result for T → Tc, ∆(˜ T ) = 3. 06 kBTc(1 − T /Tc)^1 /^2 (so in this case ∆ = ∆, the energy˜ gap). Now consider the gradient terms. The easiest way of deriving these is to consider a simple phase gradient and compare with the definition of the superfluid density. If we take vs ≡ 2 ℏm ∇φ, there the phenomenological expression for the energy due to flow is:
∆F =
ρs(T )υ s^2 =
ℏ^2
8 m^2
ρs(T )(∇φ)^2 (22)
while the GL expression is
∆F = γ(T )|∇ψ|^2 = γ(T )|Ψ|^2 (∇φ)^2 (23)
Equating these two expressions gives
γ(T ) =
ℏ^2 ρs(T ) 8 m^2 |Ψ|^2 (T )
which is of course valid for any normalization of Ψ. Suppose we choose the normalization so that Ψ = ∆˜ ∼= ∆(T ) (corrections are of higher order in the gradient term). Then we can use the result that for a pure system ρs/ρ = 1 − Y (T ) ∼=
7 ζ(3)/(4π^2 kB^2 T (^) c^2 )
∆^2 (T ),
so
γ(T ) =
nℏ^2 4 m
7 ζ(3) 8 π^2 k^2 BT (^) c^2
nℏ^2 4 m β
(Note: independent of T for T → Tc)
Inclusion of magnetic vector potential:
It is intuitively plausible that since Ψ is the wave function of the COM of a Cooper pair, in a magnetic vector potential the ∇ should be replaced by ∇ − 2 ieA(r)/ℏ. Formally this follows because
∑
i
p^2 i 2 m
i
pi − eA(ri)
/ 2 m →
2 m
i
|(−iℏ)∇i − eA(ri)|^2 (26)
Amplitude variations:
In the general case (Ψ not small) there is no good reason to suppose that the “bending energy” associated with variation in the amplitude should be simply related to that for phase variation (which we have just related to ρs). But for small Ψ, the analyticity argument indicates that the form γ|∇Ψ|^2 is unique, justifying the GL term. Thus we finally can write our GL free energy in the form:
F (Ψ(r), T ) = F 0 (T )+ (27) ∫ (^) { α(T )|Ψ(r)|^2 +
β(T )
Ψ(r)|^4 + γ(T )|(∇ − 2 ieA/ℏ)Ψ(r)
μ− 0 1 (∇ × A)^2
dr
and if we choose a normalization so that Ψ(r) is equal to ∆(˜ r), then the coefficients are given for the pure case as
α(T ) = −
(dn/d)(1 − T /Tc) (28)
β(T ) =
(dn/d) 7 ζ(3) 8 π^2
kB^2 T (^) c^2
∼ const.
γ(T ) =
nℏ^2 4 m (dn/d)
7 ζ(3) 8 π^2
k^2 BT (^) c^2 ∼ const.
For the dirty case, the values of α and β are practically unchanged, but γ(T ), like ρs(T ) is multiplied by a factor ∼ (l/ξ 0 ) 1.
The GL equations: differentiating the GL free energy functionally with respect to Ψ, we obtain α(T )Ψ(r) + β|Ψ(r)|^2 Ψ(r) − γ(∇ − 2 ieA/ℏ)^2 Ψ = 0 (29)
As already noticed, the functional differentation with respect to A(r) gives Maxwell’s equation [in the Landau gauge, div A = 0]
∇^2 A(r) = μ 0 j(r) (30)
provided that electric current j(r) is identified as
j(r) = e m
Ψ∗(−iℏ∇ − 2 eA)Ψ + c.c.
where ξ(T ) is the GL correlation (coherence, healing) length previously introduced:
ξ(T ) ≡ (γ/α(T ))^1 /^2 ∼ (Tc − T )−^1 /^2 (37)
(Note condition n · j = 0 , at walls automatically satisfied since Ψ is real!) It should be strongly emphasized that contrary to what one might perhaps think there is no general veto on superconductivity occurring in a sample of dimension ξ(T ) (provided boundary is e.g. with vacuum or insulator), or even of dimension ξ 0 (Cooper pair radius). The actual condition for complete suppression is more like ∆ 0 < (3D) single-particle energy splitting ∼ F /N , which is a great deal more severe. A second application: current-carrying state in a thin wire (d λ(T )). Under this condition, A can be neglected to a first approximation and we have by symmetry |Ψ| = const, Ψ = |Ψ|eiφ
j = 2 eℏ m
|Ψ|^2 (∇φ) (38)
or if we define vs ≡ 2 ℏm ∇φ, j = 4e|Ψ|^2 vs. The energy is
F = −α(T )|Ψ|^2 +
β 2 |Ψ|^4 + γ|Ψ|^2 (∇φ)^2 (39)
and minimizing this with respect to |Ψ|, we find
α(T ) − γ|∇φ|^2 β
1 − γ|∇φ|^2 /α(T )
≡ (1 − ξ^2 (T )(∇φ)^2 )^1 /^2 |Ψ∞| (40)
Thus the OP vanishes when the “bending” of phase over a length ξ(T ) is equal to 1: at this point, the bending energy becomes equal to the original (Ψ =const) condensation energy. (It is therefore not surprising that if we extrapolate ξ(T ) to zero temperature, it is always of the order of the pair radius: ξ(0) ∼ ξ 0 for a clean sample, ∼ (ξ 0 l)^1 /^2 for a dirty one). Because of the relation (40), j is actually a nonmonotonic function of ∇φ(vs), with a maximum at the point where ∇φ = ξ−^1 (T )/
3, i.e. υs = √^132 mξ^ ℏ (|Ψ|^2 = (2/3)|Ψ∞|^2 ). Thus, the critical current is given by
∇φ
jc =
2 eℏ m
α(T ) β
ξ−^1 (T ) (41)
Since α(T ) ∼ Tc − T and ξ(T ) ∼ (Tc − T )−^1 /^2 , this gives
jc(T ) ∝ (1 − T /Tc)^3 /^2 (42)
Note: at low temperature, in particular as T → 0, the behavior of j as a function of vs is rather different. See Tinkham (1996) p. 125.
- Isolated vortex line
Consider a thick (all dimension λ) flat slab with normal in the z-direction, and the external field at ∞ ‖ z. One possibility: magnetic field is either totally excluded, or turns macroscropic re- gions of material normal + penetrates through these. This is indeed what happens in a type-I superconductor. (Tinkham, Section 5.1.) However, there is a second possibility: the field may
z
Hext
penetrate through in microscropic regions. Let’s con- sider this possibility: field penetrates through a region of xy-plane centered at r = 0 (convention!) while ex- cluded from the bulk of the plane. Consider a single such region. The field will by Lon- don’s equation (B ∼ curl j) produce a screening current which will flow around the edges of the “hole”, since situation qualitatively similar to bulk 2D, expect length over which current falls off is ∼ λ. Beyond this point there is no current and no field, but there can still be a vector potential A. Recall
j ∝ ∇φ − 2 e ℏ
A (43)
so j = 0 ⇒ ∇φ = 2eA. But φ is gradient of phase of Ψ and thus must be single-valued modulo 2π, so ∮ ∇φ · dl = 2nπ →
A · dl ≡ φ = n(h/ 2 e) ≡ nφ 0 (44)
Hence, each such configuration (“vortex”) encloses n flux quanta φ 0. Restrict con- sideration to |n| = 1, since n = 0 is trivial and |n| ≥ 2 turns out to be unstable. Since the extent of the (2D) field is ∼ λ the central field is ∼ φ 0 /λ^2. Energy: The energy is composed of an “intrinsic” energy E 0 necessary to form the vortex in zero field, plus the (negative) energy saved by “admitting” the external field. Consider E 0 : this consists of (a) field energy,
(1/2)μ− 0 1 H^2 dr, (b) (minus) condensation energy due to deviation of OP from bulk value, (c) flow energy 1/ 2
ρsv^2 s dr where vs ≡ 2 ℏm (∇φ − 2(e/ℏ)A), (d) energy due to “binding” of amplitude of OP. The order of magnitude of (a) is (1/2)μ− 0 1 H 02 λ^2 ∼ (π/2)μ− 0 1 φ^20 /λ^2 , (b) is negligible in the limit ξ λ (see below). (c) can be estimated from the consideration that vs is of order (ℏ/ 2 m)(1/r)∂φ/∂θ ∼ ℏ/ 2 mr over a distance r ∼ λ, thereafter tends to zero (Meissner). Thus
(c) ∼
(1/2)ρsv s^2 d^2 rr ∼ (1/2)ρs(ℏ/ 2 m)^22 π
∫ (^) λ
r 0
dr/r ∼ (1/2)ρs(ℏ/ 2 m)^2 ln(λ/r 0 ). (45)
(r 0 ∼ lower cutoff). (d) is of order (c) without the logarithmic factor. But we have the general relation (in the local limit).
λ^2 = (μ 0 ρse^2 /m^2 )−^1 (46)
