The formation of the order parameter in BCSmodel near Tc

S.V.Iordanskiy

June 22, 2007

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The classical selfconsistency equation used in most papers is validif the formation time of the pair condensate is much shorter thenthe other characteristic times.This assumption is not valid in theproblem of the emergence of the superconducting state. In anumber of works the kinetic properties are considered using thereduced hamiltonian of BCS model and the electron-electroncollisions are completely disregarded. However we emphasize thatthe superconducting pairing of electrons is a small effect againstthe background of stronger interaction in the electronic Fermiliquid of the metall. In this work,we assume that collisions betweenexcitations are sufficiently frequent and lead to the thermalizationof excitations,whereas the order parameter varies quite slowly. Inthe present work the physical picture for the emergence of thesuperconding state is considered in a supercooled electron gas.

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Assuming a slow rate for the phase transition it is possible toconstruct approximately stationary states with the superconductivenuclei of finite size. Its evolution rate is defined by the collisions ofexcitations changing the number of electrons in Cooper pairs. Theproblem of the uniform relaxation of the order parameter along thislines was considered by Iordanskiy,Saptsov,Brener (2007)in [8] .

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The main difficulty in the construction of the quasystationarystates is the impossibility to satisfy the selfconsistency equation forthe equilibrium distribution of quasiparticles. The physical base forthe consideration of quasystationary states with the nonequilibriumvalues of the order parameter is given by Landau assumption of thelarge relaxation time in the vicinity of the critical temperature. It ispossible to consider the states with the nonequilibrium values ofthe order parameter as approximately stationary states usingLagrange method to find the minima of BCS hamiltonian at givenmodulo square of the order parameter.The original BCS hamiltonian has the form

H0bcs =

d3r [+ (r)(0 )(r) 0++(r)+(r)(r)+(r)]

(1)where 0 = (i~)2/(2m)()2 , = (pF )2/2m is the electronchemical potential, = (+), () are spin indices, 0 is theinteraction constant.

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Assuming a given value of |(r)|2 we introduce the properLagrange multiplier (r) and a new effective hamiltonian

H bcs = H0bcs

d3r(r)++(r)

+(r)(r)+(r) (2)

That is equivalent to the considering of BCS model with thechanging in space interaction constant eff (r). It is possible totreat the effective hamiltonian (2) by the standard mean fieldtheory trying to find stationary states close to Tc . Following [9]one can use the perturbation theory in for Gorkov Matsubarafunctions

G(r, ; r, ) = < T+ (r, )(r, ) > (3)

F(r, ; r, ) = g < T(r, )(r, )

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The equations for the Green contain the gap function defined bythe selfconsistency equation

(r) = eff (r)F (r, = + 0, r, ) (4)

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It is essential that the Green functions are periodic in and areunambigously defined by the equations for their Fourriercomponents

inGn(r, r) = (0(r) )Gn(r, r) + (r)F+n(r, r) (5)

inF+n(r, r

) = (0(r) )F+n(r, r) + Gn(r, r) (6)

and depend only on the gap function . Really we need theexpansion of F+n near Tc up to

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F+n(r, r) =

d3sG 0n(r, s)

G 0n(s, r) (7)

d3sd3s d3s G 0n(r, s)(s)G 0n(s, s

)(s)

G 0n(s, s)(s)G0n(s

, r)

Here G 0 is the corresponding Green function for the noninterectingelectron gas

G 0n(r, r) = m

2piRexp

[(ipF sign n |n|

vF

)R

](8)

where R = |r r|.

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The expression for the modified BCS energy(2) can be alsoexpressed in terms of

< H bcs >= 2

d3r(0(r )G (r, ; r, )|( + 0, r r (9)

d3r(r)F+(r, r; , )|( +0),(rr)

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We can use the first equation (5) for G to exclude differentiationin the first term of (9). Thus we obtain more convenientexpression for the thermodynamic energy

< H bcs >= 3

d3r(r)F+(r, r, + 0) (10)

+2Tn

Gn(r, r; + 0)

here

Gn =

d3sG 0n(r, s)(s)F

+n(s, r)

Because Green function (8) strongly oscillates on the distances ofthe order of vF/T ~pF therefore it is possible to neglect the lastterm as a small quantity of the order of T/ compare to the first.Using expansion (7) it is possible to calculate the expansion in for the thermodynamic energy of quasistationary states.

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We assume that the gap function has a small change on thedistances of the order of 0 = vF/T and the gradient energy issmall.In the vicinity of Tc the gap is always small and the non localcorrections are essential only for qudratic terms in energyneglecting such corrections for the quartic terms,which can becomputed in the local approximation.The quadratic terms in BCS thermodynamic energy (10) has theform

H(2)bcs = 3

d3rd3s||2(r)T

n

G 0n(r, s)G0n(s, r) (11)

3

d3rd3s(r)

1

2

2

rirk(ri si )(rk sk)G 0n(r, s)G 0n(s, r)

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Linear in derivatives terms vanish due to isotropy. The sum in thefirst term has a logarithmic divergence. Really we can use theknown results from [9] for all terms in thermodynamic BCS energyup to quartic in

H bcs = 3mpF2pi2

[ln2DpiT

]

d3r |(r)|2 + (12)

21(3)mpF v2F

32pi4T 2

d3r |(r)|4 7(3)mpF v

2F

4pi4T 2

d3r

2

rk2

Here is the Euler constant,(3) is the zeta Rieman function,D isthe Debay energy.

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This expression has the standard form of Ginsburg-Landau freeenergy giving the possibility to find stationary states with a definitenumber of electrons in Cooper pairs proportional to

=

d3r |(r)|2 (13)

minimizing the new thermodynamic potential

() = H bcs()

d3r |(r)|2 (14)

with Lagrange multiplier . Therefore the stationary states with adefinite number with 6= 0 in a finite volume of electron gas aredefined by a kind of nonlinear Schroedinger equation

H bcs

= 0 (15)

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The value of can not be smaller than the quantum energeticcorrection due to the relaxation process > ~/r .At smallervalues a rapid relaxation of to zero occured by the fluctuationsdestroying the coherence of the order parameter. Therefore theequation (??) is valid only at || > 1 where 1 is somephenomenological constant. Because of the rapid change of inspace at the smaller values there is some surface tension (1)which must be taken into account in the stability condition for thesolutions of (??) . Assuming the spherical symmetry of thesuperconcting nuclei it is possible to obtain a kind of Laplaceformulae at its boundary r = R by the the variation of (14) in R

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|r|2 |1|2 + 2(1

R= 0 (16)

Here the first two terms are defined by the expansion in (14).Two boundary conditions (16) and = 1 and the equation (??)determine R(),(, r) assuming given small value of1. Thuswe have a family of solutions depending on one real parameter .

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The convenient parametrization of the obtained family ofquasistationary states is given by the electron number in Cooperpairs of the superconducting nuclei. In the absence of theprocesses changing this number these states can be considered asstationary. For the description of the relaxation process one needsto find the connection of the electron number in Cooper pairs Nswith the normalization constant in(13).

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Assuming the large scale for the variation of in thesuperconducting nuclei it is possible to neglect the nonlocalcorrections and obtain the electron density by Bogolyubovtransformation in the form

ne = 2

d3p

2pi3v2p + 2

d3p

2pi3(u2p v2p )n(p, r) (17)

where the coherence factors up(r), vp(r) are defined by thestandard formulae

u2p + v2p = 1, u

2p v2p = p/(p)

where p = p2/2m , excitation energy (p) =

||2 + (p)2

and n(p, r) is the distribution function of excitations.

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The first term in (17) is the number density of electrons in Cooperpairs whereas the second term gives the electron number inexcitations.The electron density in Cooper pairs is proportional to ||2 withthe proportionality coefficient found in ([8]) ns = mpF/pi

2~Tc ||2.The total number of electrons in Cooper pairs of thesuperconducting nuclei is

Ns =mpFpi2~3Tc

d3r ||2 (18)

This quantity can be changed by the processes of the annihilationof two excitations or by the scattering of excitations both with thephonon emission. The phonon is needed to take or supply theextra energy. We do not consider the processes of triple excitationcollisions.

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The total energy of the quasistationary states for thesuperconducting nuclei is given by the original BCS Hamiltonian(1) with the interaction constant0 and can be expressed in theterms of the Matsubara Green functions as

< H0bcs >=

d3r(0r )G (r, ; r, + 0)|r=r (19)

0

d3rF+(r, ; r, + 0)F (r, ; r, + 0)

This quantity can be represented neglecting small terms of therelative order T/F like (10) in the form

E 0bcs =

d3r(2 + 0F+(r, ; r, + 0)F (r, ; r, + 0

)(20)

and can be calculated by the expansion in of the proper Greenfunctions.

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Because the collisions with the change ofNs are rare one needs thechange of the energy E 0bcs at relatively small variations of Ns atthe collision

=E 0bcsNs

Ns (21)

The value of Ns depends on the type of the relaxation process. Inthe annihilation of two excitations the change of the electronnumber in Cooper pairs according to (17) isNs = u

2p v2p + u2|qp| v2|qp|.At the scattering of the excitation

with the momentum p with the phonon emission with themomentum q it is Ns = u2p v2p u2|pq| + v2|pq|. The energybalance at the annihilation process is defined by(p + |qp| (p,q) q) = 0 where q is the phonon energy.In the process of phonon scaterring the energy balance in thescattering is p |pq| q (p,q) = 0

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The further consideration of the annihilation and the scaterringprocesses does not alter essentially compare to the uniform case[8].The vertex for the probability of the proper collision can befound in ([12]) with the changed law for the conservation of theenergy. The evolution of the Ns in time will be described by theequation

dNsdt

=

d3rW r (p,q)[u2(p) v2(p) + u2(|q p|) v2(|q p|)]d

3p

2pi3d3q

2pi3+ (22)

2

d3rW s(p,q)[u2(p) v2(p) u2(|p q|) + v2(|p q|)]d

3p

2pi3d3q

2pi3

The values of the scattering probabilities W r ,W s in the unit ofthe time and the volume are calculated by the perturbation theoryin electron phonon interaction.

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Taking into account the change in the energy conservation lawthey acquire the form

W r (p,q) = 2pi3~2cqmpF 1TE0bcsNs

(23)[u(|q p|)v(p) + u(p)v(|q p|)]2[u2(p) v2(p) + u2(|q p|)v2(|q p|)][1 n(p)][1 n(|q p|)]N(q)[(p) + (|q p|) cq]

and a close expression for W r Here c is the sound velocity, is thematerial constant in the electron-phonon interaction [5], N(q) isthe phonon distribution function. These formulae are obtained inthe linear approximation in E 0bcs/TcNs .

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In metals c

The equation (24) simplifies for the uniform relaxation of the orderparameter in the whole volume of the superconductor where the

quantityE0bcsNs

can be calculated in the terms of the current valueof || [8]. In this case (24) can be represented in the form ofevolution equation for ||

(|| |eq|)t

= || |eq|r

(25)

here1

r=||+ |eq|

8Tc

pi

ph.

This equation coincides up to numerical constant with the real partof the so called equation of TDGL obtained in [13] at a differentconditions(e.g. the magnetic impurities).Obtained in the presentedwork the relaxation time is approximately 0,34 of the relaxationtime in TDGL equation.

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