Fujita S., Ito K., Godoy S. Quantum Theory of Conducting Matter

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126 9 Supercurrents and Flux Quantization

which is formally identical with the Schr

odinger equation of motion for a quantum

particle.

Since London’s macrowavefunction were introduced by the intuition of the

great men for a stationary state problem, it is not immediately clear how to de-

scribe its time-evolution. Our quantum statistical calculations show that the quasi-

wavefunction satisﬁes the familiar Schr

odinger equation of motion. This is a signif-

icant result. Important applications of Equation (9.68) will be discussed in Chapter

11.

9.5 Discussion

Supercurrents exhibit many unusual behaviors. We enumerate their important fea-

tures in the following subsections.

9.5.1 Supercurrents

The supercurrent is generated by a moving supercondensate composed of equal

numbers of ± pairons condensed at a ﬁnite momentum p. Since ± pairons, having

charges ±2e and momentum p, move with different speeds c

= v

(i)

/2, the net

electric current density:

j = en

(2)

−v

(1)

)/2

does not vanish. Supercurrents totally dominate normal currents. Condensed pairons

have lower energies by at least the gap 

than noncondensed pairons, and hence,

they are more numerous.

9.5.2 Supercurrent is Not Hindered by Impurities

The macroscopic Supercurrent generated by a moving supercondensate is not hin-

dered by impurities that are microscopic by comparison. The fact that no micro-

scopic perturbation causes a resistance (energy loss) is due to the quantum statistical

nature of the supercondensate; the change in the many-pairon-state can occur only

if a transition from one (supercondensate) state to another is induced. Large lattice

imperfections and constrictions can, however, affect the supercurrent signiﬁcantly.

9.5.3 The Supercurrent Cannot Gain Energy from a DC Voltage

The supercondensate is electrically neutral, and hence, it is stable against the exter-

nal electric force. The supercurrent cannot gain energy from the applied voltage.

9.5 Discussion 127

9.5.4 Critical Fields and Critical Currents

In the superconducting state, ± pairons move in the same direction (because they

have the same momentum) with different speeds. If a magnetic ﬁeld B is applied, the

Lorentz-magnetic force tends to separate ± pairons. Hence there must be a critical

(magnetic) ﬁeld B

. The magnetic force is proportional to pairon speeds v

(i)

/2. Thus

the superconductors having lower Fermi-velocities, like high-T

cuprates, are much

more stable than elemental superconductors, and they have higher critical ﬁelds.

Since the supercurrents themselves generate magnetic ﬁelds, there are critical cur-

rents J

9.5.5 Supercurrent Ring and Flux Quantization

A persistent Supercurrent ring exhibits striking superconducting properties: resis-

tanceless surface supercurrent, Meissner state, and ﬂux quantization.

The superconducting state ⌿

is represented by a momentum-state wavefunc-

tion: ⌿

(x) = A exp(−i

−1

x), p

≡ 2π n/L. Since the ring circumference

L is macroscopic, the quantized momentum p

is vanishingly small under normal

experimental conditions (n ∼ 1 − 100). The actual value of n in the ﬂux quanti-

zation experiments is determined by Onsager’s rule: ⌿ = n⌿

≡ nπ /e. Each

condensed pairon has an extremely small momentum p

and therefore an extremely

small energy  = v

(i)

/2.

9.5.6 Meissner Effect and Surface Supercurrent

A macroscopic superconductor expels an applied magnetic ﬁeld B from its interior;

this is the Meissner effect. Closer examination reveals that the B-ﬁeld within the

superconductor vanishes excluding the surface layer. In fact a ﬁnite magnetic ﬁeld

penetrates the body within a thin layer of the order of 500

A, and in this layer dia-

magnetic surface supercurrents ﬂow such that the B-ﬁeld in the main body vanishes.

Why does such a condition exist? The stored magnetic ﬁeld energy for the system in

the Meissner state is equal to VB

/2μ

. This may be pictured as magnetic pressure

acting near the surface and pointing inward. This is balanced by a Meissner pres-

sure pointing outwards that tries to keep the state superconductive. This pressure

is caused by the Gibbs free energy difference between super and normal states,

− G

. Thus if

− G

> VB

/2μ

, (9.69)

the Meissner state is maintained in the interior with a steep but continuously chang-

ing B-ﬁeld near the surface. If the B-ﬁeld is raised beyond the critical ﬁeld B

128 9 Supercurrents and Flux Quantization

the inequality (9.69) no longer holds, and the normal state will return. The kinetic

energy of the surface supercurrent is neglected here.

9.5.7 London’s Equation

The steady-state supercurrent in a small section can be represented by a plane wave-

function: ⌿

(r) = A exp(−i

−1

p ·r) = A exp(i

−1

px). The momentum p appears

in the phase. The Londons assumed, based on the Hamiltonian mechanical consid-

eration of a system of superelectrons, that the phase difference at two points (r

, r

)

in the presence of a magnetic ﬁeld B has a ﬁeld component, and obtained London’s

equation: j

=−e

−1

We assumed that the supercurrent is generated by the ± pairons condensed at a

momentum p, and used the standard Hamiltonian mechanics to obtain

=−2e

−1

A. (9.70)

The revised London equation has a proportionality factor different from that of

the original London equation.

9.5.8 Penetration Depth

The existence of a penetration depth λ was predicted by the London brothers [8, 9],

and it was later conﬁrmed by experiments. This is regarded as an important historical

step in superconductivity theory. The qualitative agreement between theory and ex-

periment established a tradition that electromagnetism as represented by Maxwell’s

equations can, and must, be applied to describe the superconducting state.

The Londons used their equation and Maxwell’s equations to obtain an expres-

sion for the penetration depth:

London



4πk



1/2

. (9.71)

If we adopt the pairon ﬂow model, we obtain instead

λ =



8πk



1/2

. (9.72)

Note: there is no mass in this expression, since pairons move as massless par-

ticles. The n

−1/2

-dependence is noteworthy. The supercondensate density n

ap-

proaches zero as temperature is raised to T

. The penetration depth λ therefore

increases indeﬁnitely as T → T

References 129

References

1. J. File and R. G. Mills, Phys. Rev. Lett. 10, 93 (1963).

2. J. Bardeen, L. N. Cooper and J. R. Schrieffer, Phys. Rev. 108, 1175 (1957).

3. B. S. Deaver and W. M. Fairbank, Phys. Rev. Lett. 7, 43 (1961).

4. R. Doll and M. N

abauer, Phys. Rev. Lett. 7, 51 (1961).

5. L. Onsager, Phys. Mag. 43, 1006 (1952).

6. L. Onsager, Phys. Rev. Lett. 7, 50 (1961).

7. N. Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961).

8. F. London and H. London, Proc. R. Soc. (London) A 149, 71 (1935).

9. F. London, Superﬂuids,Vol.I (Wiley, New York, 1950).

Chapter 10

Ginzburg–Landau Theory

The pairon ﬁeld operator ψ

†

(r, t) evolves, following Heisenberg’s equation of

motion. If the Hamiltonian H contains a pairon kinetic energy h

, a condensation

energy α(< 0) and a repulsive point-like interpairon interaction βδ(r

−r

), β(> 0),

the evolution equation for ψ is non-linear, from which we derive the Ginzburg-

Landau (GL) equation:

(r, −i∇)⌿



(r) +α⌿



(r) +β|⌿



(r)|

⌿



(r) = 0

for the GL wavefunction ⌿



(r) ≡



r|n

1/2

|σ



, where σ denotes the state of the

condensed pairons, and n the pairon density operator. The GL equation with α =

−

(T ) holds for all temperatures (T ) below the critical temperature T

, where



(T )istheT -dependent pairon energy gap. Its equilibrium solution yields that

the condensed pairon density n

(T ) =|⌿

(r)|

is proportional to 

(T ). The

T -dependence of the expansion parameters near T

conjectured by GL: α =

−b(T

− T ), β = constant are conﬁrmed. A new formula for the penetration depth:

λ = (c/e)[p

(1)

) + v

(2)

] is obtained, where c is the light speed, p the con-

densed pairon momentum, and v

(i)

= (2

)

1/2

are the Fermi velocities for

“electrons” ( j = 1) and “holes” ( j = 2).

10.1 Introduction

In 1950 Ginzburg and Landau (GL) [1] proposed a revolutionary idea that below

the critical temperature T

a superconductor has a complex order parameter (also

called a GL wavefunction) ⌿



just as a ferromagnet possesses a real order parameter

(spontaneous magnetization). Based on Landau’s theory of second-order phase tran-

sition [2], GL expanded the free energy density f (r) of a superconductor in powers

of small |⌿(r)| and |∇⌿(r)|:

f (r) = f

+α|⌿



(r)|

β|⌿



(r)|



|∇⌿



(r)|

, (10.1)

S. Fujita et al., Quantum Theory of Conducting Matter,

DOI 10.1007/978-0-387-88211-6

10,



Springer Science+Business Media, LLC 2009

131

132 10 Ginzburg–Landau Theory

where f

, α and β are constants, and m

is the superelectron mass. To include the

effect of a magnetic ﬁeld B, they used a quantum replacement:

∇→∇−(iq/)A, q = charge, (10.2)

where A is a vector potential generating B =∇×A, and added a magnetic energy

term B

/2μ

. The integral of the so-modiﬁed free energy density f (r) over the

sample volume V gives the Helmholtz free energy F. After minimizing F with

variations in ⌿

∗

and A

, GL obtained two equations:

|−i∇−qA|

⌿



(r) +α⌿



(r) +β|⌿



(r)|

⌿



(r) = 0, (10.3)

j =−

iq

(⌿

∗

∇⌿



−⌿



∇⌿

∗

) −

⌿

∗

⌿



A. (10.4)

With the density condition:

⌿

∗

⌿



(r) = n

(r) = superelectron density, (10.5)

Equation (10.4) for the current density j in the homogeneous limit (∇⌿



= 0)

reproduces London’s equation [3, 4]:

j =−⌳

A, ⌳

≡ e

−1

Equation (10.3) is the celebrated Ginzburg–Landau equation, which is quantum

mechanical and nonlinear. Since the smallness of |⌿



is assumed, the GL equation

was thought to hold only near T

, T

−T  T

.BelowT

there is a supercondensate

whose motion generates a supercurrent and whose presence generates gaps in the

elementary excitation energy spectra. The GL wavefunction ⌿



represents the quan-

tum state of this supercondensate. The usefulness of the GL theory has been well

recognized [5]. The most remarkable results are GL’s introduction of the concept of

a coherence length [1] and Abrikosov’s prediction of a vortex structure in a type II

superconductor [6], which was later conﬁrmed by experiments [7]. In their original

work GL adopted a superelectron model. The quantum ﬂux experiments [8–11],

however, show that the charge carriers in the supercurrent are pairons [12] hav-

ing charge magnitude 2e, which conﬁrms the basic physical picture of the BCS

theory [13].

In our text we take the view that ⌿



represents the state of the condensed pairons

(rather than the superelectrons). In this chapter we microscopically derive the GL

equation (10.3) with a revised density condition [14, 15]:

|⌿



(r)|

= condensed pairon density = n

(r). (10.6)

We obtain physical interpretation of the expansion parameters (α, β), and discuss

their temperature dependence. We further show that the revised GL equations are

10.2 Derivation of the GL Equation 133

valid for all temperatures below T

. The equilibrium solution of this equation

with no ﬁelds yields a remarkable result that the condensed pairon density n

(T )

is proportional to the pairon energy gap 

(T ), observed in the quantum tunnel-

ing experiments [16–18]. We propose a new expression (10.39) for the pairon ki-

netic energy. Using this, we obtain a revised expression (10.41) for the penetration

depth.

10.2 Derivation of the GL Equation

Dirac [19] and others [20, 21] showed that quantum ﬁeld operators, ψ(r, t) and

†

(r, t) for bosons (fermions) satisfying the Bose (Fermi) commutation (anticom-

mutation) rules:

[ψ(r, t),ψ

†



, t)]

∓

≡ ψ(r, t)ψ

†



, t) ∓ψ

†



, t)ψ(r, t) = δ

(3)

(r −r



)



ψ(r, t),ψ(r



, t)



∓

= 0 (10.7)

evolve in time, following Heisenberg’s equation of motion:

−i

⭸ψ(r, t)

⭸t

= [H,ψ(r, t)]

−

, (10.8)

where H represents the Hamiltonian of a system under consideration. In this section

the commutators (anticommutators) are represented by [A, B]

−

([A, B]

) rather

than by [A, B]({A, B}). If an interparticle interaction Hamiltonian



i=j

v(|r

−r



v(|r

−r

|)ψ

†

, t)ψ

†

, t)ψ(r

, t)

(10.9)

is assumed, the commutator [H

,ψ(r, t)]

−

generates (Problem 10.2.1)

−



v(|r −r

|)ψ

†

, t)ψ(r

, t)ψ(r, t), (10.10)

indicating that the evolution equation for the quantum ﬁeld ψ is intrinsically nonlin-

ear in the presence of the interparticle interaction [20,21]. Two pairons, each having

like charge, repel each other. We represent this by a repulsive point-like potential

v(r) = βδ

(3)

(r),β>0. (10.11)

134 10 Ginzburg–Landau Theory

We then obtain

,ψ(r, t)]

−

=−βn(r, t)ψ(r, t) ≡−βψ

†

(r, t)ψ(r, t)ψ(r, t), (10.12)

which vanishes for fermions. Hereafter, we shall consider bosons only.

If we assume a kinetic energy h

(r, p) and a constant condensation energy α

(< 0), then the following equation is obtained from Equation (10.8):

i

⭸ψ(r, t)

⭸t

= h



r, −i

⭸

⭸r



ψ(r, t) +αψ(r, t) +βn(r, t)ψ(r, t). (10.13)

Let us consider a persistent ring supercurrent. The wavefunction at the point r

in a small section along the ring is characterized by a discrete momentum (p

, 0, 0)

with

2π

ν, (ν = 0, ±1, ±2,...) (10.14)

where L is the ring circumference. We note that the absolute value |ν| charac-

terizes the number of ﬂux quanta ⌽

≡ π/e enclosed by the ring [10, 11].

Further note that the momentum eigenstate (p

, 0, 0) ≡ p deﬁnes the state σ of the

supercondensate.

Let us deﬁne a quasiwavefunction ⌿

(r)by

⌿

(r) ≡ TR{ψ(r)ρφ

†

}≡



r|n|σ



, (10.15)

where φ

†

is the condensed-pairon-state (σ ) creation operator, ρ the system density

operator, and the symbol TR denotes a grand ensemble trace; the pairon density

operator n is deﬁned through the position density matrix elements:

TR{ψ(r)ρψ

†



)}=



r|n|r





, (10.16)

normalized such that





r|n|r



≡

tr{n}=average pairon density, (10.17)

where the symbol tr means a one-body trace. The GL wavefunction ⌿



(r) can be

related with the pairon density operator n by

⌿



(r) =



r|n

1/2

|σ



, (n

1/2

)

= n. (10.18)

The two wavefunctions, ⌿



(r) and ⌿

(r), have different amplitudes; |⌿



(r)|



r|n

1/2

|σ



σ |n

1/2



≡ n

(r) is the condensed pairon density while ⌿

(r) itself is

proportional to n

(r).

10.3 Discussion 135

We multiply Equation (10.13) by ρφ

†

from the right and take a grand ensemble

trace. Writing the results in terms of ⌿

(r) and using a factorization approximation,

we obtain

i

⭸⌿

(r)

⭸t

= h

(r, −i∇)⌿

(r) +α⌿

(r) +β|⌿

(r)|

⌿

(r). (10.19)

For the steady state the time derivative vanishes, yielding

⌿

(r) +α⌿

(r) +β|⌿

(r)|

⌿

(r) = 0, (10.20)

which is precisely the GL equation, Equation (10.3).

In our derivation we assumed that pairons move as bosons, which is essential, see

the sentence following Equation (10.12). Bosonic pairons can multiply occupy the

net momentum state p while fermionic superelectrons cannot. The correct density

condition (10.6) instead of (10.5) must therefore be used.

Problem 10.2.1. Prove Equation (10.10), using Equations (10.7) and (10.9).

10.3 Discussion

We derived the GL equation from ﬁrst principles. In the derivation we found that

the particles that are described by the GL wavefunction ⌿



(r) must be bosons. We

take the view that ⌿



(r) represents the bosonically condensed pairons. This explains

the quantum nature of the GL wavefunction. In fact ⌿



(r) =



r|n

1/2

|σ



is a mixed

representation of the pairon square-root density operator n

1/2

in terms of the position

r and the momentum state σ . The new density condition is given by

⌿

∗

(r)⌿



(r) = n

(r) = condensed pairon density.

The nonlinearity of the GL equation arises from the point-like repulsive inter-

pairon interaction. In 1950 when Ginzburg and Landau published their work, the

Cooper pair (pairon) was not known. They simply assumed the superelectron model.

Our microscopic derivation allows us to interpret the expansion parameters (α, β)

appearing in the original GL theory. We shall see that α represents the pairon con-

densation energy, and β the repulsive interaction strength. The parameter β, from

its mechanical origin, is temperature-independent:

β = constant > 0, T

> T. (10.21)

BCS showed [13] that the ground state energy W for the BCS system is

W = ω

N(0)w

≡−2ω

{exp[2/(v

N(0))] −1}

−1

, (10.22)

136 10 Ginzburg–Landau Theory

where N(0) is the density of states per spin at the Fermi energy and w

the pairon

ground-state energy. Hence we can choose

α = w

< 0, T = 0. (10.23)

In the original work [1] GL considered a superconductor in the vicinity of the crit-

ical temperature T

, where |⌿



is small. Gorkov [22–24] used Green’s functions

to interrelate the GL and the BCS theory near T

. Werthamer and Tewardt [25–28]

extended the Ginzburg–Landau–Gorkov theory to all temperatures below T

, and

arrived at more complicated equations. Here, we derived the original GL equation

by examining the superconductor at 0 K from the condensed pairons point of view.

The transport property of a superconductor below T

is dominated by the condensed

pairons. Since there is no distribution, the qualitative property of the condensed

pairons cannot change with temperature. The pairon size (the minimum of the coher-

ence length derivable directly from the GL equation) naturally exists. There is only

one supercondensate whose behavior is similar at all temperatures below T

; only

the density of condensed pairons can change. Thus, there is a quantum nonlinear

equation (10.20) for ψ

(r) valid for all temperatures below T

The pairon energy spectrum below T

has a discrete ground-state energy, which is

separated from the energy continuum of moving pairons [29]. This separation 

(T ),

called the pairon energy gap, is T-dependent. This energy gap, as in the well-known

case of the atomic energy spectra, can be detected in photo-absorption [30, 31] and

quantum tunneling experiments [16–18]. Inspection of the pairon energy spectrum

with a gap suggests that

α =−

(T ) < 0, T

> T. (10.24)

Solving Equation (10.20) with h

⌿

= 0 (no currents, no ﬁelds), we obtain

(T ) =|⌿

= β

−1



(T ), (10.25)

indicating that the condensate density n

(T ) is proportional to the pairon energy

gap 

(T ).

We now consider an ellipsoidal macroscopic sample of a type I superconductor

below T

subject to a weak magnetic ﬁeld H

applied along its major axis. Because

of the Meissner effect, the magnetic ﬂuxes are expelled from the main body, and the

magnetic energy is higher by (1/2)μ

V in the super state than in the normal state.

If the ﬁeld is sufﬁciently raised, the sample reverts to the normal state at a critical

ﬁeld H

, which can be computed in terms of the free-energy expression (10.1) with

the magnetic ﬁeld included. We obtain after using Equations (10.6) and (10.22)

 (μ

β)

−1/2



(T ) ∝ n

(T ), (10.26)

Fujita S., Ito K., Godoy S. Quantum Theory of Conducting Matter - Superconductivity

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