Alexandrov A.S.,Theory of Superconductivity - From Weak to Strong Coupling

Подождите немного. Документ загружается.

Weak coupling theory

Then using equation (2.194), we obtain

G(k,ω) = P



ω − 

ω + 



−iπ tanh





δ(ω−

) +v

δ(ω+

)]

(2.199)

where we have applied the relation 1/(ω +iδ) = P(1/ω) −iπδ(ω).Heretheﬁrst

term is understood as the principal value of the integral when integrating with

respect to ω. For zero temperature, tanh(ω/2T ) = sign(ω) and

G(k,ω) =

ω − 

+ iδ

ω + 

− iδ

(2.200)

as it should (see equation (2.165)).

2.16 Microscopic derivation of the Ginzburg–Landau

equations

Using the Green’s function formalism near T

, Gor’kov [45] derived the

Ginzburg–Landau equations from BCS theory. Let us consider the BCS

superconductor in a stationary magnetic ﬁeld with the vector potential A(r).

The Fourier transform of equations (2.183) and (2.184) with respect to the

thermodynamic time yields



iω

[∇ + ie A(r)]

+ µ



(r, r



) = δ(r − r



) + (r)

(r, r





−iω

[∇ − ie A(r)]

+ µ



(r, r



) =−

∗

(r)

(r, r



) (2.201)

where

(r, r



) =



1/T

dτ (r, r



,τ)exp(iπ nTτ)

(r, r



) =



1/T

dτ

(r, r



,τ)exp(iπnTτ)

and



∗

(r) =−2E



(r, r).

If the temperature is close to T

, the order parameter is small. Then we can

expand the GFs in powers of (r). The zero-order GF is a temperature GF of

an ideal Fermi gas,

(n)

(r, r



) in the magnetic ﬁeld, which satisﬁes the following

equation:



iω

[∇ + ie A(r)]

+ µ



(n)

(r, r



) = δ(r − r



). (2.202)

Microscopic derivation of the Ginzburg–Landau equations

If there is no magnetic ﬁeld,

(n)

(r, r



) =

(0)

(r − r



) and its Fourier transform

is found as

(0)

(k,ω

) =

iω

− ξ

. (2.203)

Transforming back to real space we obtain

(0)

(ρ) =

(2π)



ik·ρ

iω

− ξ

(2.204)

where ρ = r − r



. Calculating the integral over the angles in the polar spherical

coordinates yields

(0)

(ρ) =

(2π)

iρ



∞

k dk

ikρ

− e

−ikρ

iω

− ξ

. (2.205)

We can replace k by k = k

+ξ/v

because only the states near the Fermi surface

contribute to the integral in equation (2.205), when ω

is of the order of T

 E

The result is

(0)

(ρ) =

(2π)

iρ



∞

−∞

dξ

iξρ/v

− e

−ikρ

−iξρ/v

iω

− ξ

. (2.206)

Using the contour in the upper or lower half-plane, we obtain

(0)

(ρ) =−

2πρ

exp[isign(ω

ρ −|ω

|ρ/v

]. (2.207)

The Fourier component of the normal state GF falls exponentially with the

characteristic length

ρ ≈ ξ

(2.208)

where ξ

= v

/(2π T

) is the zero temperature coherence length. GFs in a

magnetic ﬁeld are not translation invariant and the exact calculation of the normal

(n)

(r, r



) using equation (2.202) is a challenging problem. But we can assume

that the spatial variations of the vector potential are small over the characteristic

distance ξ

. If we neglect these variations, then the ﬁeld operator in the magnetic

ﬁeld can be readily expressed via the free-electron operator. It satisﬁes the

equation of motion

∂ψ

(r,τ)

∂τ



[∇ + ie A(r)]

+ µ



(r,τ). (2.209)

If A(r) is a constant then the solution is

(r,τ) =



exp[i(k − e A) · r − ξ

τ)] (2.210)

Weak coupling theory

which is veriﬁed by direct substitution into equation (2.209). Hence, if the vector

potential varies slowly, the GF differs from the zero-ﬁeld GF only by the phase:

(n)

(r, r



) = e

−ie A(r)·ρ (0)

(ρ). (2.211)

This quasi-classical approximation with respect to the magnetic ﬁeld is applied

if the phase change over the distance ξ

is small. |A(r)| varies by Bξ

over

this distance, and the phase changes by eBξ

. The maximum ﬁeld is (see

section 1.6.4)

≈



2πξ

− T

(2.212)

so that the quasi-classical approximation is applied, if

− T

 1. (2.213)

It is the very same region where the Ginzburg–Landau theory is applied. Using

the normal state GF in the magnetic ﬁeld (equation (2.211)), we can transform

Gor’kov equations (2.201) into the integral form:

(r, r



) =

(n)

(r, r



) +



(n)

(r, x)(x)

(x, r



) (2.214)

(r, r



) =−



(n)

−ω

(x, r)

∗

(x)

(x, r



Now we expand

(r, r



) to the terms of the second order and

(r, r



) to the

terms of the third order in (r) inclusive:

(r, r



) =

(n)

(r, r



) −



d y

(n)

(r, x)(x)

(n)

−ω

( y, x)

∗

( y)

(n)

( y, r



) (2.215)

(r, r



) =−



(n)

−ω

(x, r)

∗

(x)

(n)

(x, r



)



d y



(n)

−ω

(x, r)

∗

(x)

(n)

(x, y)( y)

(n)

−ω

(z, y)

∗

(z)

(n)

(z, r



Using this expansion, we obtain the following equation for the order parameter:



∗

(r) = 2E





(n)

−ω

(x, r)

∗

(x)

(n)

(x, r)

− 2E





d y



(n)

−ω

(x, r)

∗

(x)

(n)

(x, y)( y)

(n)

−ω

(z, y)

∗

(z)

(n)

(z, r). (2.216)

Microscopic derivation of the Ginzburg–Landau equations

Let us consider the kernel in the ﬁrst term

K (x − r) = T



(n)

−ω

(x, r)

(n)

(x, r) = K

(0)

(x − r)e

−2ie A(r)·(x−r)

(2.217)

where

(0)

(x − r) = T

(2πρ)



exp[−2|ω

|ρ/v

]

(2πρ)

sinh(2π Tρ/v

)

(2.218)

and ρ =|x − r|.ThekernelK

(0)

(ρ) has a characteristic radius about ξ

, while

(x) changes over a much larger distance of the order of ξ(T )  ξ

near T

Therefore in the integral over x we can expand all quantities near the point x = r

up to the second order in ρ inclusive:



dx K

(0)

(x − r)e

−2ie A(r)·(x−r)



∗

(x)

≈ 

∗

(r)



dρ K

(0)

(ρ) +



dρ K

(0)

(ρ)ρ ·[∇

− 2ie A(r)]

∗

(r)



i, j=1



dρ K

(0)

(ρ)ρ



∂

∂r

− 2ieA



∂

∂r

− 2ieA





∗

(r).

(2.219)

Here the second term is zero because the function under the integral is odd. The

remaining integrals are:



dρ K

(0)

(ρ) = T





dρ

(0)

−ω

(ρ)

(0)

(ρ)

= T



+ ξ

= N(E

)



dξ

tanh(ξ/2T )

(2.220)

and



dρ K

(0)

(ρ)ρ



dρ K

(0)

(ρ)ρ

4(π T )

N(E

)



∞

sinh x

7ζ(3)v

8(π T )

N(E

(2.221)

We cut the divergent integral in equation (2.220) at |ξ |=ω

as usual and

introduce the DOS at the Fermi level: N (E

) = mk

/(2π

). The second term

Weak coupling theory

on the right-hand side of equation (2.216) is cubic in (r). Here we can neglect

the space variations of (x) and the phase due to the vector potential under the

integrals. Then applying the Fourier transform of the zero-ﬁeld ideal gas GF, we

obtain the following expression for this term:

−2E

T 

∗

(r)|(r)|



(ω

+ ξ

)

=−λ

7ζ(3)

8(π T )



∗

(r)|(r)|

(2.222)

We note that T

is determined from the linearized BCS equation:

1 = λ



dξ

tanh(ξ/2T

)

Then close to T

, we can expand it as



dξ

tanh(ξ/2T )

≈

− T

(2.223)

where we use the integral, equation (B.28) from the appendix. Introducing the

‘condensate wavefunction’ φ(r) as

φ(r) = (r)



7ζ(3)n

8π

(2.224)

and collecting all integrals in equation (2.216), we obtain the phenomenological

Ginzburg–Landau equation (section 1.6.1):

−

[∇ +i2e A(r)]

φ(r) + β|φ(r)|

φ(r) =−αφ(r).

Now the coefﬁcients α and β are determined microscopically as

α =

12π

7ζ(3)mv

(T − T

) (2.225)

β =

12π

7ζ(3)mv

Here n

= k

/(3π

) is the electron density. Finally let us show that the second GL

equation for supercurrent follows from the expansion of GFs (2.215) in powers

of (r). The current–density operator can be expressed in terms of the ﬁeld

operators:

ˆ (r) =





(∇

−∇



→r



†



)

(r) −

A(r)

†

(r)

(r)



(2.226)

Microscopic derivation of the Ginzburg–Landau equations

Its expectation value is

j(r) =

(∇

−∇



→r



(r, r



) −

A(r). (2.227)

There is no current in the normal state in the stationary magnetic ﬁeld, and the

ﬁrst (normal) term in the expansion of GFs (equation (2.215)) cancels the second

term on the right-hand side of equation (2.227). The remaining supercurrent is

quadratic with respect to the order parameter:

j(r) =

(∇



−∇



→r





d y

(n)

(r, x)(x)

(n)

−ω

( y, x)

∗

( y)

(n)

( y, r



(2.228)

The normal state GF in the magnetic ﬁeld

(n)

(r, r



) = exp(−ie A · ρ)

(0)

(ρ)

is the product of a slowly varying phase exponent and the zero-ﬁeld GF, which

oscillates in real space with the electron wavelength (≈1/k

). Differentiating the

oscillating part yields

j(r) =





d y (x)

∗

( y)e

2iA·(x−y)

(0)

−ω

(|y − x|)

×[

(0)

(|r − x|)∇

(0)

(|y − r|) −

(0)

(|y − r|)∇

(0)

(|r − x|)].

(2.229)

Expanding the slowly varying order parameter and the magnetic phase exponent

in powers of |y − r| and |x − r|, which are of the order of ξ

(x) ≈ (r) + (x − r) · ∇(r)



∗

( y) ≈ (r) + ( y − r) · ∇

∗

(r)

exp[2ie A ·(x − y)]≈1 + 2ie A · (x − y)

we obtain

j(r) =

C[

∗

(r)∇(r) − (r)∇

∗

(r)]−

|(r)|

A(r). (2.230)

Here the constant C is given by

C =





d y

(0)

−ω

(|y − x|)

(0)

(|r − x|)[(x − y) ·∇

]

(0)

(|y − r|).

(2.231)

Weak coupling theory

Using the Fourier transform of the ideal gas GF in calculating the integrals over

x and y in C leads to

C =



iω

+ ξ

∂

∂ k

· k

(iω

− ξ

)



(iω

+ ξ

)(iω

− ξ

)

. (2.232)

Integrating with respect to k yields

C =

N(E





∞

−∞

dξ

− ξ

(ω

+ ξ

)

(2.233)

and we obtain

C =

N(E



7ζ(3)n

16π

Replacing the order parameter in equation (2.230) by the ‘condensate

wavefunction’ leads to the second GL equation (see equation (1.40)):

j(r) =

∗

∗∗

[φ

∗

(r)∇φ(r) − φ(r)∇φ

∗

(r)]−

∗2

∗∗

A(r)|φ(r)|

(2.234)

where e

∗

= 2e and m

∗∗

= 2m are the charge and mass of the Cooper pair,

respectively.

The BCS theory is a mean-ﬁeld approximation, which is valid if the volume

occupied by the correlated electron pair is large compared with the volume per

electron. As an example, in aluminium the size of the pair (i.e. the coherence

length at T = 0) is about ten thousand times larger than the distance between

electrons. The Cooper pairs in the BCS superconductor disappear above T

and the one-particle excitations are fermions. When the coupling constant λ

increases, the critical temperature increases and the coherence length becomes

smaller. Hence, one can expect some deviations from the BCS behaviour in the

intermediate coupling regime. At the ﬁrst stage, deviations from BCS theory arise

in two ways: (1) the BCS approximation for the interaction between electrons

does not provide an adequate representation of the retarded nature of the phonon

induced attraction; and (2) the damping rate becomes comparable with the quasi-

particle energy. Both the retardation effect and the damping are taken into account

in the Eliashberg extension of the BCS theory to the intermediate coupling regime

[36], which is discussed in the next chapter. However, if λ is larger than 1, the

Fermi liquid becomes unstable even in the normal state above T

because of the

polaron collapse of the electron band (chapter 4). Here we encounter qualitatively

different physics [11].

Chapter 3

Intermediate-coupling theory

3.1 Electron–phonon interaction

The attraction between electrons in BCS theory is the result of an ‘overscreening’

of their Coulomb repulsion by vibrating ions. When the interaction between

ion vibrations and electrons (i.e. the electron–phonon interaction) is strong, the

electron Bloch states are affected even in the normal phase. Phonons are also

affected by conduction electrons. In doped insulators (like high-temperature

superconductors), ‘bare’ phonons are well deﬁned in insulating parent compounds

but a separation of electron and phonon degrees of freedom might be a problem

in a metal. Here we have to start with the ﬁrst-principle Hamiltonian describing

conduction electrons and ions coupled by the Coulomb forces:

H =−



∇



i=i



− r



− Ze



− R



j =j



− R



−



∇

(3.1)

where r

, R

are the electron and ion coordinates, respectively, i = 1,...,N

;

j = 1,...,N; ∇

= ∂/∂r

, ∇

= ∂/∂ R

, Ze is the ion charge and M

is the ion mass. The system is neutral and N

= ZN. The inner electrons

are strongly coupled to the nuclei and follow their motion. Hence, the ions

can be considered as rigid charges. To account for their high-energy electron

degrees of freedom we can replace the elementary charge in equation (3.1) by

√



∞

,where

∞

is the phenomenological high-frequency dielectric constant.

One cannot solve the corresponding Schr¨odinger equation perturbatively because

the Coulomb interaction is strong. The ratio of the characteristic Coulomb energy

to the kinetic energy is r

= m

/(4πn

/3)

1/3

≈ 1 for the electron density

= ZN/V = 10

−3

(further we take the volume of the system as V = 1,

unless speciﬁed otherwise). However, we can take advantage of the small value

Intermediate-coupling theory

of the adiabatic ratio m

/M < 10

−3

. The ions are heavy and the amplitudes

|u| 

√

1/Mω

of their vibrations near the equilibrium R

≡ l are much

smaller than the lattice constant a = N

−1/3

|u|

≈





1/4

 1. (3.2)

In this estimate we take the characteristic vibration frequency ω

of the order

of the ion plasma frequency ω



4π NZ

/M. Because the vibration

amplitudes are small we can expand the Hamiltonian in powers of |u| up to

quadratic terms inclusive. Any further progress requires a simplifying physical

idea, which is to approach the ground state of the many-electron system via

a one-electron picture. This is called the local density approximation (LDA),

which replaces the Coulomb electron–electron interaction by an effective one-

body potential

(r):

(r) =−Ze



|r − R

+ e





n(r



)

|r − r



+ µ

[n(r)] (3.3)

where µ

[n(r)] is the exchange interaction, usually calculated numerically

or expressed as µ

[n(r)]=−βn

1/3

(r) with the constant β in a simple

approximation.

(r) is the functional of the electron density n(r) =





†

(r)

(r). As a result, the Hamiltonian takes the form in the second

quantization:

H = H

+ H

e−ph

+ H

e−e

(3.4)

where





dr 

†

(r)



−

∇

+ V (r)





(r) (3.5)

is the electron energy in a periodic crystal ﬁeld V (r) =



v(r −l) which is (r)

calculated at R

= l and with the periodic electron density n

(0)

(r +l) = n

(0)

(r),





−

∇

∂

∂l



dr n

(0)

(r)V (r)





l,m,α,β

lα

mβ

αβ

(l −m)

(3.6)

is the vibration energy. Here α, β = x, y, z and

αβ

(l − m) =

∂

∂l

∂m







=m



− m





dr n

(0)

(r)V (r)



(3.7)

is a dynamic matrix. The electron–phonon interaction is given by

e−ph



∂

∂l









†

(r)

(r) − n

(0)

(r)



V (r)

Electron–phonon interaction



l,m,α,β

lα

mβ

∂

∂l

∂m









†

(r)

(r) − n

(0)

(r)



V (r)

(3.8)

and the electron–electron correlations are described by

e−e





|r − r











†

(r)

†



)



)

(r)



−









(0)



)

|r − r



+µ

(0)

(r)]







†

(r)

(r)



l=m

|l − m|

. (3.9)

We include the electrostatic repulsive energy of nuclei in H

e−e

, so that the average

of H

e−e

is zero in the Hartree approximation.

The vibration Hamiltonian H

is a quadratic form and, therefore, can

be diagonalized with a linear canonical transformation for the displacement

operators



q,ν

qν



2NMω

qν

exp(iq · l) + H.c. (3.10)

∂

∂u



q,ν

qν



Mω

qν

exp(iq · l) − H.c.

where q is the phonon momentum, d

qν

is the phonon (Bose) annihilation

operator, e

qν

and ω

qν

are the unit polarization vector and the phonon frequency,

respectively, of the phonon mode ν.ThenH

takes the following form



q,ν

qν

†

qν

+ 1/2) (3.11)

if the eigenfrequencies ω

qν

and the eigenstates e

qν

satisfy

Mω

qν



αβ

qν

(3.12)

and



∗α

qν

= Nδ

αβ

. (3.13)

The last equation and the bosonic commutation rules [d

qν

†



]=δ

νν



from (∂/∂u

− u

(∂/∂u

) = δ

αβ

.Here

αβ



exp(iq · m)D

αβ

(m) (3.14)