1. Shanghai Jiao Tong University
Lecture 4
Shanghai Jiao Tong University
Lecture 4
2017
Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign, USA
and
Director, Center for Complex Physics
Shanghai Jiao Tong University

2. Ginzburg – Landau (GL) theory of superconductivity
Formulated 1950 (pre – BCS): even in 2017 mostly adequate for “engineering” applications of superconductivity. (SC)
historically, takes its origins in general Landau-Lifshitz theory (1936) of 2nd order phase transitions, hence quantitative validity confined to 𝑇 close to 𝑇 𝑐 . Here, I will start by arguing “with hindsight” for its qualitative validity at 𝑇=0, and only later generalize to 𝑇≠0 and in particular 𝑇→ 𝑇 𝑐 .
Recall: could explain 3 major characteristics of SC state (persistent currents, Meissner effect, vanishing Peltier coefficient) by scenario in which fermionic pairs form effective bosons, and these undergo BEC. Suppose that 𝑁 fermions form 𝑁/2 bosons, which are then condensed into the same 2 – particle state, Neglect for now relative wave function and df.
COM wave function of condensed pairs ≡ 𝜑 𝒓
COM coordinate

3. “Standard” form of GL (free) energy (at T=0)
What is energy 𝐸 𝑜 (at 𝑇=0) of pairs as function(al) of 𝜑 (𝒓)?
At first sight, for a single pair,
𝐸 𝑜 𝜑 𝒓 =− 𝐸 𝑏 𝜑 𝒓 2 𝑑𝒓+ ℏ 2 2𝑚 𝜵− 2𝑖𝑒 ℏ 𝑨(𝒓) 𝜑 𝒓 2
binding energy of pair
2 electrons involved!
so , convenient to define
“order parameter” (OP)
Ψ 𝒓 ≡ 𝑁 2 𝜑 𝒓
𝛼 𝑇=0 ≡ 𝐸 𝑏
then energy of N/2 pairs is
𝐸 𝑜 Ψ 𝒓 =const. + 𝑑𝒓 −𝛼 𝑇=0 Ψ 𝒓 ℏ 𝟐 2𝑚 𝛁− 2𝑖𝑒 ℏ 𝑨 𝒓 𝛹 𝒓 𝟐
However, by itself this will not generate stability of supercurrents (lecture 3). To do so, must add (e.g.) term in |Ψ| 4 … Adding also EM field term:
Ε 0 Ψ 𝒓 =
𝑑𝑟 −𝛼 Τ=0 Ψ 𝑟 𝛽 Τ=0 Ψ 𝑟 ℏ 2 2𝑚 𝛁− 2𝑖𝑒 ℏ 𝑨 𝒓 Ψ 𝑟 𝜇 0 −1 𝛁×𝑨 𝒓 2
“Standard” form of GL (free) energy (at T=0)

4. and energy is Ε Τ=0 𝑒𝑞 = −𝛼 Τ=0 2 2𝛽 Τ=0
Notes:
Normalization of Ψ 𝑟 is conventional and arbitrary. Rescaling
Ψ 1 𝑟 ≡𝑞Ψ 𝑟 , 𝛼 Τ=0 ′ ≡ 𝑞 −2 𝛼 Τ=0 , 𝛽 Τ=0 ′ ≡ 𝑞 −4 𝛽 Τ=0 , ℏ 2 2𝑚 → ℏ 2 2𝑚 𝑞 2
leaves Ε 0 unchanged.
If 𝐴 𝒓 =0 and Ψ 𝑟 = constant ≡ Ψ Τ=0, 𝑒𝑞 then value is
Ψ Τ=0 𝑒𝑞 = 𝛼 Τ=0 𝛽 Τ=
and energy is Ε Τ=0 𝑒𝑞 = −𝛼 Τ= 𝛽 Τ=0
Minimization with respect to 𝑨 𝑟 𝛿𝐸 𝑜 𝛿𝐴 𝒓 =0 yields Maxwell’s
equation
provided that we identify
𝑗 𝒓 = 𝑒 𝑚 Ψ ∗ 𝑟 −𝑖ℏ𝛁−2𝑒𝚨 Ψ 𝑟 +𝑐.𝑐.
Minimization with respect to Ψ 𝒓 𝛿𝐸 0 𝛿Ψ 𝒓 =0 yields
Note that for 𝑨 𝒓 =0 this defines a characteristic length
ξ Τ=0 ≡ ℏ 2 2𝑚𝛼 Τ=
A second characteristic length follows if we put Ψ 𝑟 = constant ≡ Ψ 0
and compare terms in 𝐴 2 and 𝛻×Α 2 .
𝜆 Τ=0 ≡ 2𝑚/ 𝑒 2 𝜇 𝑜 Ψ − 1 2
or since with our normalization Ψ 0 = Ν/2𝑉 ,
𝜆 Τ=0 ≡ 𝑚/ 𝜇 0 𝑒 2 𝑛 − (≡ London penetration depth at Τ=0)
𝜵×𝜢=𝒋 𝒓
−𝛼 Τ=0 Ψ 𝑟 + 𝛽 Τ=0 Ψ 𝑟 2 Ψ 𝑟 − ℏ 2 2𝑚 𝜵−2 𝑖𝑒 ℏ 𝐴 𝑟 2 Ψ 𝑟 =0
electron density

5. standard form of GL free energy density
Generalization to Τ≠0:
Free energy
All we need do is to let Ε 0 →Ϝ 𝛵 and let the coefficients 𝛼 0 , 𝛽 0 be 𝛵 – dependent:
Ϝ Ψ 𝑟 :Τ = Ϝ 0 Τ + 𝑑𝑟ℱ 𝑟,Τ
ℱ 𝑟,Τ ≡
−𝛼 Τ Ψ 𝑟 𝛽 Τ Ψ 𝑟 ℏ 2 2𝑚 𝜵− 2𝑖𝑒 ℏ Α 𝑟 Ψ 𝑟 2
+ 𝜇 0 −1 𝜵×𝐴 𝒓 2
standard form of GL free energy density
Note that there is now a contribution to 𝛼 Τ (and possibly also 𝛽 Τ ) from the entropy term −𝛵𝑆 Ψ 𝑟 . (condensate itself carries no entropy, but electrons “liberated” from it do!)

6. Since Ψ 𝒓 describes S state, should →0 at some 𝑇 𝑐 , then 𝛼 𝛵 should change sign there. Most natural choice (corresponding to 𝐸 𝑏 ~ constant, 𝑆 Ψ 𝑟 ~ constant – constant Ψ 2 ): Ψ→ Ϝ↑
𝛵> 𝛵 𝑐
𝛼 𝛵 <0
𝛼 𝛵 = 𝛼 0 𝑇 𝑐 −𝑇
𝛽 𝛵 = 𝛽 0 Ϝ↑ *
Τ< Τ 𝑐
𝛼>0
For 𝛵≠0, all the above results go through with 𝛼 𝛵=0 →𝛼 𝛵 , 𝛽 𝑇=0 →𝛽 𝛵 .
In particular in limit 𝛵→ 𝑇 𝑐 from below: Ψ→
“Mexican-hat” potential
Ψ 𝑒𝑞 𝛵 = 𝛼 𝛵 /𝛽 𝛵 = 𝛼 0 / 𝛽 𝛵 𝑐 −𝛵 1 2
𝐹 𝑒𝑞 𝛵 − 𝐹 0 𝛵 =− 𝛼 2 𝛵 2𝛽 𝛵 =− 𝛼 𝛽 𝛵 𝑐 −𝛵 2
𝜉 𝛵 ∝ 𝛼 𝛵 − 1 2 ∝ 𝛵 𝑐 −𝛵 − 1 2
𝜆 𝛵 ∝ Ψ 𝑒𝑞 ∙ 𝛵 −1 ∝ 𝛵 𝑐 −𝛵 − 1 2
so ratio κ≡𝜆/𝜉 is independent of 𝛵 in limit 𝛵→ 𝛵 𝑐 . From 𝐹 𝑒𝑞 𝛵 − 𝐹 0 𝛵 ∝ 𝛵 𝑐 −𝛵 2 , entropy S has no discontinuity at 𝛵 𝑐 ,
but sp. ht. 𝐶 v ≡𝛵𝑑𝑆/𝑑𝛵 has discontinuity 𝛵 𝑐 𝛼 𝛽 0 ⇒ second order phase transition at 𝑇 𝑐 . 𝑆
𝐶 v 𝑁
𝛵 𝑐 𝛵→

7. Ψ=0 𝜉 Τ Ψ= Ψ ∞ 𝑧 Ψ 𝑧=0 =0 Ψ 𝑧→∞ = Ψ ∞ ≡ 𝛼 𝛵 𝛽 0 1 2
Some simple applications of the GL theory
Zero magnetic field 𝑨=0
Recovery of OP near wall (etc.)
Suppose boundary condition at 𝑧=0 is that Ψ must →0 (e.g. wall ferromagnetic).
Then must solve GL eq
Ψ=0
𝜉 Τ
Ψ= Ψ ∞ 𝑧
𝛼 𝛵 Ψ 𝑧 + 𝛽 0 Ψ 𝑧 2 Ψ 𝑧 − ℏ 2 2𝑚 𝜕 2 Ψ 𝑧 𝜕2 2 =0
subject to boundary conditions
Ψ 𝑧=0 =0
Ψ 𝑧→∞ = Ψ ∞ ≡ 𝛼 𝛵 𝛽
Solution:
Ψ 𝑧 = Ψ ∞ tanh 𝑧 2𝜉 𝛵
𝜉 𝛵 ≡ ℏ 2 2𝑚𝛼 𝛵 −1 2 ∝ 𝛵 𝑐 −𝛵 −1 2
Note: Ψ does not have to vanish at boundary with vacuum or insulator! [except on scale ~ 𝑘 𝐹 −1 where 𝑁 state wave functions do too] Thus, no objection to SC occurring in grains of dimension≪𝜉 𝛵 . However, contact with 𝑁 metal tends to suppress SC.

8. 2. Current – carrying state in thin wire
If 𝑑≪𝜆 Τ , can neglect A to first approximation
By symmetry, 𝑑 𝑗
Ψ= Ψ exp i𝜑, Ψ = constant, 𝒋= 2𝑒ℏ 𝑚 Ψ 2 𝜵𝜑
“Superfluid velocity”
𝐹=−𝛼 Τ Ψ 𝛽 Ψ ℏ 2 2𝑚 Ψ 2 𝛻𝜑 2
𝐯 𝑠 = ℏ 2𝑚 𝜵𝜑
and we need to minimize this with respect to Ψ for fixed 𝜵𝜑. Result:
Ψ = 𝛼 𝛵 − ℏ 2 2𝑚 𝛻𝜑 2 𝛽 ≡ Ψ ∞ 1− 𝜉 2 𝛵 ) 𝛻𝜑
⇒ Ψ →0 for 𝜵𝜑= 𝜉 −1 𝛵 (condensation energy changes sign). However, 𝑗 is nonmonotonic function of 𝜵𝜑, with maximum at point when 𝜵𝜑= 𝜉 −1 Τ . (at which point Ψ = Ψ ∞ ). Thus, critical current 𝑗 𝑐 given by
𝑗 𝑐 = 2𝑒ℏ 𝑚 ∙ 2 3 𝛼 𝛵 𝛽 𝜉 −1 𝛵 ∝ 𝛼 −1 𝑇 𝜉 −1 𝛵 ∝ 1−𝛵 𝛵 𝑐

9. Η 𝑐 𝛵 = 𝛼 2 𝛵 / 𝜇 0 𝛽 0 1 2 ∝ 1−𝛵/ 𝛵 𝑐 B. Behavior in magnetic field
Because of the Meissner effect, any superconductor will completely expel a weak magnetic field. If we consider just the competition between the resulting state and the 𝑁state, we have (per unit volume)
Δ𝐸 𝑚𝑎𝑔𝑛 = 𝜇 0 Η 2
Δ𝐸 𝑐𝑜𝑛𝑑 =− 𝛼 2 𝛵 /2 𝛽 0 𝑆 𝑁
vs.
and so (“thermodynamic”) critical field Η 𝑐 is given by
Η 𝑐 𝛵 = 𝛼 2 𝛵 / 𝜇 0 𝛽 ∝ 1−𝛵/ 𝛵 𝑐
However, in general this works only for “type-I” superconductors in form of long cylinder parallel to field. More generally:
(a) in type-I superconductors, “intermediate” state forms with interleaved macroscopic regions of
𝑁 and 𝑆.
(b) in type-II superconductors, magnetic field “punches through” sample in the form of vortex lines.

10. Isolated vortex line
𝜆 𝛵
𝜉 𝛵 𝛮
Consider 𝜆 Τ ≫𝜉 𝑇
(“extreme type-II”)
Magnetic field turns region ~ 𝜉 2 𝛵 normal, and “punches through” there. Field is screened out of bulk on scale 𝜆 𝛵 ≫𝜉 𝛵
At distances ≫𝜆 𝛵 , no current flows: however since
𝒋∝𝜵𝜑− 2𝑒 ℏ 𝜜
𝜵𝜑 and 𝜜 may be individually non zero, with 𝜵𝜑= 2𝑒 ℏ 𝜜.
But 𝜑 must be single-valued mod. 2𝜋, hence
𝜵𝜑∙𝓭ℓ=2𝑛𝜋⟹ 𝜜∙𝓭ℓ≡Φ=𝑛 Φ 0
trapped flux
ℎ/2𝑒 “superconducting” flux quantum
𝑛=0 is trivial (no vortex!) and 𝑛 ≽2 is unstable, so restrict consideration to 𝑛= ± 1.
Since field extends over ∼𝜆 𝛵 into bulk metal,
field at core ≡Η 0 ~ Φ 0 / 𝜆 2 𝛵 .

11. Ψ
Ψ ∞
Energetics of single vortex line (per unit length)
“intrinsic” energies:
circ.
currents
𝜉 𝛵
𝜆 𝛵 𝑟→
(a) field energy ~ 1 2 𝜇 0 Η 0 2 𝜆 2 ~ 𝜑 0 2 / 𝜇 𝑜 𝜆 2
(b) (minus) condensation energy ~− 1 2 𝜇 𝑜 Η 𝑐 2 𝜉 2
(c) flow energy: v 𝑠 ~ ℏ 2𝑚 1 𝑟 , for 𝑟≤𝜆 𝛵 , so
𝑐 ~ Ψ ∞ ℏ 2𝑚 2 𝜉 𝜆 𝑟𝑑𝑟 𝑟 2 ~ Ψ ∞ ℏ 𝑚 2 ℓ𝑛 𝜆/𝜉
but 𝜆 𝛵 = 𝜇 0 Ψ ∞ 2 𝑒 2 /𝑚 − 1 2 , so
𝑐 ~ Φ 0 2 / 𝜇 0 𝜆 2 Τ ℓ𝑛 𝜆/𝜉
dominant over
(a) and (b) for 𝜆 𝛵 ≫𝜉 𝛵
Thus “intrinsic” energy per unit length of vortex lines for 𝜆≪𝜉 is
Ε 0 𝛵 ~ Φ 0 2 / 𝜇 𝑜 𝜆 2 𝛵 ∙ℓ𝑛 𝜆/𝜉 .

12. (2)
On the other hand, the “extrinsic” energy saving due to admission of the external field is ~ 𝜇 0 Η 𝑒𝑥𝑡 Η 0 ~ Η 𝑒𝑥𝑡 Φ 0 / 𝜆 2 . Hence the condition for it to be energetically advantageous to admit a single vortex line is roughly
Η 𝑒𝑥𝑡 ~ Φ 0 / 𝜆 2 𝛵 ∙ℓ𝑛𝜅
𝜅 ≡𝜆 𝛵 /𝜉 𝛵 ≠𝑓 𝛵
This defines the lower critical field Η 𝑐1 .
What is the maximum field the superconductor can tolerate before switching to the 𝑁 phase?
Roughly, defined by the point at which the vortex cores (area ~ 𝜉 2 𝛵 ) start to overlap. Since for near-complete penetration we must have 𝑛 Φ 0 ~ Η 𝑒𝑥𝑡 , this gives the condition
number of vortices/unit area
Η 𝑒𝑥𝑡 ~ Φ 0 / 𝜉 2 Τ
This defines the upper critical field Η 𝑐2
Note that for 𝜆 𝛵 ≾𝜉 𝛵 , we have Η 𝑐1 ≳ Η 𝑐2 and would expect type-I behavior.

13. Results of more quantitative treatment:
condition for type-II behavior is 𝜅≡ 𝜆 Τ /𝜉 Τ > 2 − 1 2
in extreme type-II limit 𝜅≫1,
Η 𝑐1 𝛵 = Φ 0 / 4𝜋𝜆 2 Τ ℓ𝑛𝜅
in same limit,
Η 𝑐2 𝛵 = Φ 0 / 2𝜋𝜉 2 Τ .
Note that quite generally we have up to logarithmic factors
Η 𝑐1 𝛵 ∙ Η 𝑐2 𝛵 ~ Η 𝑐 2 𝛵
thermodynamic critical field
hence if Η 𝑐 𝛵 held fixed, Η 𝑐1 varies inversely to Η 𝐶2
Application: dirty superconductors
Alloying does not change Ε 𝑐𝑜𝑛𝑑 𝛵 , hence Η 𝑐 2 𝛵 , much. However, it drastically increases 𝜆 𝛵 ( 𝜉 𝛵 is also increased, but less). Hence 𝜅 is increased, and many elements which on type-I when pure became type-II when alloyed. As alloying increases, Η 𝑐2 increases while Η 𝑐1 decreases.

14. Summary of lecture 4
Ginzburg-Landau theory is special case of Landau-Lifshitz theory of 2nd order phase transitions: quantitatively valid only near Tc . Introduces order parameter (“macroscopic wave function”)Ψ 𝑟 which couples to vector potential 𝑨 𝑟 with charge 2𝑒:
𝐹 Ψ 𝑟 =const.−𝛼 Ψ 𝑟 𝛽 Ψ 𝑟 4 +
const. 𝛻− 2𝑖𝑒 ℏ 𝐴 𝒓 Ψ 𝒓 𝜇 𝑜 −1 𝜵×𝑨 𝑟 2
2 characteristic lengths:
healing length 𝜉 𝑇
penetration depth 𝜆 𝑇
both  𝑇 𝑐 −𝑇 −1/2 for 𝑇→𝑇 𝑐 .
Magnetic behavior depends on ratio 𝜅=𝜆 𝑇 /𝜉 𝑇 | 𝑇→𝑇 𝑐
For 𝜅> 2 −1/2 , (“type-I”) field completely expelled up to a thermodynamic critical field 𝐻 𝑐 𝑇 , at which point turns fully normal.
For 𝜅< 2 −1/2 , (“type-II”) field starts to penetrate at 𝐻 𝑐 in form of vortices, continues to do so up to 𝐻 𝑐2 where turns completely normal.