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8.5 The Theory of Superconductivity (A)	487

where ωq is the energy of the created phonon (with = 1 and ωq = ω−q′). Using

n	+1 a† n		=	n		+1 ,		(8.123)
q	q q			q
we find
nq +1 S nq = −i ∑ BqCk†		+qCk				nq′ +1	δq−′q
nq +1 S nq = −i ∑ BqCk†		+qCk		εk		−εk +q′ −ωq′	δq−′q
	k,q			εk		−εk +q′ −ωq′		(8.124)
						nq′ +1		(8.124)
= −i∑ B−q′Ck†			−q′Ck			nq′ +1	.
= −i∑ B−q′Ck†			−q′Ck				.
	k				εk −εk −q′ −ωq′

In a similar way we can show

nq′ S nq′ +1 = i∑ Bq′Ck†	+q′Ck	nq′ +1
nq′ S nq′ +1 = i∑ Bq′Ck†	+q′Ck	.
k		εk −εk +q +ωq′

Now, using

H S = H 0 + 12 [H epS − SH ep ] +…,

with

H ep = i ∑ BqCk†+qCk (aq − a−†q ) k,q

(8.125)

(8.126)

(8.127)

(X has now been set equal to 1), and taking phonon expectation values for a particular phonon state, we have

n H S n =	n H 0	n +	1	∑[ n H ep m m S n − n S m m H ep n ]
			2	m
=	n H 0	m +	1	[(H ep )n,n−1Sn−1,n + (H ep )n,n+1Sn+1,n (8.128)
			2	− Sn,n−1(H ep )n−1,n − Sn,n+1(H ep )n+1,n ].
				− Sn,n−1(H ep )n−1,n − Sn,n+1(H ep )n+1,n ].

Since we are interested only in electronic coordinates, we will write belownq|HS|nq as HS, and nq|H0|nq as H0, and hope that no confusion in notation will arise. Using

(H ep )nq ,nq−1	= −i∑ B−qCk†−qCk nq ,		(8.129)
	k
and
(H ep )nq ,nq+1	= i∑ BqCk†	+qCk nq +1 ,	(8.130)
	k

488 8 Superconductivity

the effective Hamiltonian for electrons is given by combining the above. Thus,

H S = H 0 +

∑ C†

CkC†

Ck′nq

k,k′ k −q

′+q

εk′ −εk′+q +ωq

+ C†

C†

k′

+1)

k′−q

εk′ −εk′−q

−ωq

(8.131)

−C†

C†

′−q

k′

k +q

εk′ −εk′−q −ωq

−C†

C†

+1)

εk′ −εk′+q

′+q

k′

k −q

+ωq

Making dummy variable changes, dropping terms that do not involve the interaction of electrons (i.e. that do not involve both k and k′), and using the commutation relations for the C, it is possible to write the above in the form

H S = H 0 +

∑ Ck†′+qCk′Ck†−qCk

k,k′

(8.132)

−

−ε

−ω

−ε

+ω

k −q

k′

k′+q

In order to properly interpret Hamiltonians such as the above equation, which are expressed in the second quantization notation, it is necessary to keep in mind the appropriate commutation relations of the C. By Appendix G, these are

CkCk†′ + Ck†′Ck = δkk′ ,

(8.133)

C†C†	+ C†	C† = 0 ,		(8.134)
k k′	k′	k
and
CkCk′	+ Ck′Ck		= 0 .	(8.135)

The Hamiltonian (8.132) describes a process called a virtual exchange of a phonon. It has the diagrammatic representation shown in Fig. 8.19.

k – q

k′ + q

k	q	k′

Fig. 8.19. The virtual exchange of a phonon of wave vector q. The k are the wave vectors of the electrons. This is the fundamental process of superconductivity

8.5 The Theory of Superconductivity (A)	489

Note that (8.132) is independent of the number of phonons in mode q, and it is the effective electron Hamiltonian with phonons in the single mode q. To get the effective Hamiltonian with phonons in all modes, we merely have to sum over the modes of q. Thus, the total effective interaction Hamiltonian is given by

H I =

∑ ∑

2 Ck†′+qCk′Ck†

−qCk

q k,k′

(8.136)

−

−ε

−ω

−ε

+ω

k −q

k′

k′+q

By dropping further terms that do not involve the interaction of electrons (terms not involving both k and k′) and by making variable changes, we can reduce this Hamiltonian to

H I = ∑ ∑	Bq	2	ωq		Ck†′+qCk†	−qCkCk′ .	(8.137)

			(εk −εk −q )2	−ωq2
q k,k′

From the above equation, we see that there is an attractive electron–electron interaction for |εk − εk−q| < |ωq|. We will assume, for appropriate excitation energies, that the main interaction is attractive. In this connection, most of the electron energies of interest are near the Fermi energy εF. A typical phonon energy is the Debye energy ωD (or cutoff frequency with = 1). Many approximations have already been made, and so a very simple criterion for the dominance of the attractive interaction will be assumed. It will be assumed that the interaction is attractive when the electronic energies are in the range of

εF − ωD < εk < εF + ωD ( ≠1here) .

(8.138)

The states that do not satisfy this criterion are not directly involved in the superconducting transition, so their properties are of no particular interest. Hence, the effective Hamiltonian can be written in the following form (fourth major approximation):

H I ≡ −∑ ∑ VqCk†′+qCk†	−qCkCk′ .	(8.139)
q k,k′

For simplicity, we will assume that Vq is positive and fitted from experiment, that Vq = V−q and Vq = 0, unless q is such that (8.138) is satisfied. We assume that any important interactions not included in the above equation can be included by renormalizing (i.e. changing) the quasiparticle mass.

8.5.3 Cooper Pairs and the BCS Hamiltonian (A)

Let us assume that εk = 0 at the Fermi level. The total effective Hamiltonian for the electrons is then

H = ∑εkCk†Ck −	∑ VqCk†′+qCk†	−qCkCk′ .	(8.140)
k	k,k′,q

490 8 Superconductivity

By Appendix G, the Fermion operators satisfy

n …n

… = (−)Pj n

n …(1− n

)… ,

(8.141)

C† n …n

… = (−)Pj (1− n

) n …(1+ n

)…

(8.142)

where

j −1

Pj = ∑nP .

(8.143)

P=1

It is essential to notice the alternation in sign defined by (8.142). This alternation is very important for discovering the nature of the lowest-energy state. When we begin to guess a trial wave function, if we pay no attention to this alternation of sign, the presence of the interaction will result in little lowering of the energy. What we need is a way of selecting the trial wave function so that most of the matrix elements of individual terms in the second sum in (8.138) are negative. The way to do this for the ground state is by grouping the electrons into Cooper pairs. (These will be precisely defined below.)

There are several assumptions necessary to construct a minimum energy wave function [60, p. 155ff]. For the ground-state wave function, it will be assumed that the Bloch states are occupied only in pairs. In fact, the superconducting ground state is a coherent superposition of Cooper pairs. The Hamiltonian conserves the wave vector, and only pairs with equal total momentum will be considered, i.e.,

k + k′ = K ,

(8.144)

where K is the same for each pair. It is reasonable to suppose that K is zero for the ground (noncurrent carrying) state of the pairs.

Cooper Pairs8

Before proceeding, let us discuss Cooper pairs a little more. A large clue as to the nature of the unusual character of the superconducting state was obtained by L. Cooper in 1956. He showed that the Fermi sea was unstable if electrons interacted by an attractive mechanism—no matter how weak.

Consider the normal Fermi sea of electrons with a well-defined Fermi energy EF. Now add two more electrons interacting with an attractive interaction V(1, 2) and suppose the only interaction with the other electrons is via the Pauli principle.

We write the Schrödinger wave equation for the two electrons as

	2	2		2	2
−			−			+V (1, 2)	ψ(1, 2)	= Eψ(1, 2) .	(8.145)
−			−	2m		+V (1, 2)	ψ(1, 2)	= Eψ(1, 2) .	(8.145)
	2m 1			2m	2

8 See Cooper [8.10].

		8.5 The Theory of Superconductivity			(A) 491

We seek a solution of the form
ψ(1, 2) =	V	A(1, 2)	1	∫ eik (r1 −r2 ) f (k)dk ,	(8.146)
	(2π)3
		V
where A(1, 2) is the antisymmetric spin zero spin wave function
A(1,2) = 1		[α(1)β(2) −α(2)β(1)] ,
2

with α, β being the usual spin-up and -down wave functions (note A†A = 1) and f(k) = +f(−k) so that the spatial wave function is symmetric (it can be shown that the ψ with spin 1 and antisymmetric wave function yields no energy shift, at least in our approximation, and in any case such wave functions correspond to p-state pairs that we are not considering). Note that the spatial wave function pairs off the electrons into (k, −k) states.

Inserting (8.146) into (8.145) we have

ik (r1

−r2 )

(2π)3

A(1, 2)

∫

f (k)e

(8.147)

=A(1, 2) ∫ Ef (k)eik (r1 −r2 )dk . (2π)3

Now multiply by

A† (1, 2) V1 e−ik′ (r1 −r2 ) ,

and integrate over r1 and r2 and we obtain (r = r1 − r2, V(r1, r2) = V(r1 − r2) = V(r),

and Ek = 2k2/2m)
∫∫ e−ik′ r [2Ek +V (r)] f (k)eik r drdk = ∫ Ef (k)ei(k −k′) rdk .							(8.148)
Using
		1		∫eik r dk = δ (k) ,
	(2π)3			∫eik r dk = δ (k) ,
	(2π)3
and
Vk′,k		=	1	∫ e−ik′ rV (r)eik rdr ,
Vk′,k		=		∫ e−ik′ rV (r)eik rdr ,
		V
we obtain
[2Ek′ − E] f (k′) +					V	∫ f (k′)Vk′,k dk = 0 .	(8.149)
[2Ek′ − E] f (k′) +					(2π)3	∫ f (k′)Vk′,k dk = 0 .	(8.149)
					(2π)3

492 8 Superconductivity

We suppose

Vk′,k = −V0 < 0 for EF < Ek , Ek′ < EF + ωD .

= 0 otherwise.

Notice we are using the ideas that led us to (8.138), divide by 2Ek′ − E and integrate over k′ and obtain (after canceling)

1 =V0	V	∫	dk′	.	(8.150)
	(2π)3		2Ek′ − E

Note that in the limit of large volumes

V / N	∫ dk′( ) ↔	1	∑k′ ↔ ∫ N (E′)( )dE′ ,
(2π)3		N

where N(E) is the density of state for one spin per unit cell (N unit cells). Thus with Epair = E

=V0 ∫EF + ωD

N (E′)

dE′ .

2E′ − E pair

Note we can replace N(E′) N(EF) because

ωD << EF so we obtain

V N (E

)

2EF + 2

ωD

− Epair

1 =

2EF − Epair

Let δ = 2EF − Epair so

−1

exp V N

)

δ =

ωD

sinh

V N (E

)

(8.151)

(8.152)

(8.153)

and in the weak coupling limit

	− 2
				(8.154)

δ = 2 ωD exp V N (E		F	) .
	0

We note in particular, the following points:

1.A pair electron wave function that is independent of the direction of r1 − r2 is said to be an s wave function, which is consistent with an antisymmetric spin wave function.

2.δ is not an analytic function of V0 so ordinary perturbation theory would not work.

3.In the BCS theory one considers pairing of all electrons.

4.For δ > 0 then the Fermi sea is unstable with respect to the formation of Cooper pairs.

8.5 The Theory of Superconductivity (A)	493

BCS Hamiltonian

Returning to the mainstream of the BCS argument, the above reasoning can be used to pick out the best wave function to use as a trial wave function for evaluating the ground-state energy by variational principle. For mathematical convenience, it is easier to place these assumptions directly in the Hamiltonian. Also, due to exchange, the spins in the Cooper pairs are usually opposite. Thus, the interaction part of the Hamiltonian is now written (fifth major approximation) with K = 0,

H I = −∑VqCk†	+q↑C−†k −q↓C−k↓Ck↑ .	(8.155)
k,q

Next, assume a “BCS Hamiltonian” for interacting pairs consistent with (8.155), with k + q → k, k → k′, Vq = Vk−k′ = −Vk,k′

H = ∑εkCk†σ Ckσ + ∑ Vk,k′Ck†↑C−†k↓C−k′↓Ck′↑ ,					(8.156)
kσ	k,k′
where
εk	=	2k 2	− μ ,		(8.157)
εk	=	2m	− μ ,		(8.157)
		2m
and where μ is the chemical potential. Also
H ≡ H 0 + H I ,					(8.158)
and note
V	=V	k′,k	=V *	.	(8.159)
k,k′		k′,k	k,k′

As before C are Fermion (electron) annihilation operators, and C † are Fermion (electron) creation operators. Defining the pair creation and annihilation operators

b†

= C†

C†

(8.160)

k↑

−k↓

bk = C−k↓Ck↑ ;

(8.161)

and defining

Tr(e−βH b

)

bk =

(8.162)

Tr(e−βH )

†

using Tr(AB) = Tr(BA). We can also show in the representa-

we can show bk

= bk

tion we use that bk

= bk. We define

k = −∑Vk,k′

bk′ = *k .

(8.163)

k′

494 8 Superconductivity

As we will demonstrate later, this will turn out to be the gap parameter. We can write the interaction term as

H I = ∑Vk,k′bk†bk′ .

(8.164)

k,k′

Note

bk′ =

−

(8.165)

bk′ + δbk′ = bk′ + (bk′

bk′) ,

and

b†

+δb†

+ (b†

−

* ) ;

(8.166)

k′

b† =

* +δb† =

+δb† ;

(8.167)

and we will neglect (δbk′)×(δbk†) terms. (This is sort of a mean-field-like approximation for pairs.) Thus, using (8.166) and (8.167) and neglecting O(δb2) terms, we can write

bk†bk′ = (

bk +δbk† )(

bk′ +δbk′ )

†

(8.168)

k k′

k′

bk bk′ +

bk′bk† −

bk bk′ ,

assuming bk is real. Also,

H I = ∑Vk,k′(

bk′bk† +

(8.169)

bk′bk −bk′bk ) .

k,k′

Thus,

H = ∑εkCk†σ Ckσ − ∑ (

kbk† +

kbk − k

(8.170)

bk ) .

kσ

We now diagonalize by a Bogoliubov–Valatin transformation:

Ck ↑ = ukαk + vk βk† ,

(8.171)

C−k↓ = uk βk − vkαk† ;

(8.172)

where uk2 + vk2 = 1 (to preserve anticommutation relations), uk and vk are real, and the α and β given by

αk† = ukCk†↑	− vkC−k↓ ,	(8.173)
βk† = ukC−†k	↓ + vkCk↑ ,	(8.174)

8.5 The Theory of Superconductivity (A)	495

are Fermion operators obeying the usual anticommutation relations. The αk†, and βk† create “bogolons”. The algebra gets a bit detailed here and one can skip along unless curious,

bk = C−k↓Ck↑ = (−vkαk† + uk βk )(ukαk + vk βk† )

bk† = Ck†↑C−†k↓ = (ukαk† + vk βk )(−vkαk + uk βk† )

Ck†↑Ck↑ = (ukαk† + vk βk )(ukαk + vk βk† )

C−†k↓C−k↓ = (−vkαk + uk βk† )(−vkαk† + uk βk )

bk† = −ukvkαk†αk + ukvk βk βk† − vk2βkαk + uk2αk† βk†

bk = −vkukαk†αk + uk vk βk βk† − vk2αk† βk† + uk2 βkαk

Ck†↑Ck↑ = uk2αk†αk + vk2 βk βk† + uk vkαk† βk† + uk vk βkαk

C−†k↓C−k↓ = uk2βk†βk + vk2αkαk† −ukvk βk†αk† − vkukαk βk

H= ∑k [2εkukvkαk† βk† + 2εk βkαkuk vk

+εk (uk2 − vk2 )(αk†αk + βk†βk ) + 2εk vk2

+kukvk (αk†αk + βk†βk )

− kuk vk + k vk2βkαk − kuk2αk† βk†
+	kukvk (αk†αk + βk†βk ) − kukvk
+	kvk2αk† βk† − *kuk2βkαk ] + ∑k k
		bk .

Rewriting this we get

H = ∑k {( 2

kukvk −εk ( vk2 −uk2 ))(αk†αk + βk† βk )

+( 2ε

( v2

−u2

))(α† β† + β

)} +G,

k k

where

G = ∑(2εk vk2 − 2 kuk vk + kbk ) .

Next, choose

2εkuk vk + k (vk2 −uk2 ) = 0 ,

(8.175)

(8.176)

(8.177)

(8.178)

(8.179)

(8.180)

(8.181)

(8.182)

(8.183)

(8.184)

(8.185)

496 8 Superconductivity

so as to diagonalize the Hamiltonian. Also, using uk2 + vk2 = 1 let

		vk2 =			1 − a ,							(8.186)
				2
	vk2 − uk2 = −2a ,											(8.187)
ukvk =						1	− a2 ;					(8.188)
						4
2εk		1 − a2					=		k 2a ,			(8.189)
		4
2		1	− a	2			=	2		2	.	(8.190)
εk		4	− a				=	k a			.	(8.190)
		4
Thus
a =					εk			.				(8.191)
			2	εk2 +				2k
Rewriting,
H = [2 kuk vk −εk (vk2 − uk2 )]×(αk†αk + βk†βk ) + G .												(8.192)
But, define
Ek =				εk2 +					2k ,			(8.193)
			a =		εk			,				(8.194)
			a =		2Ek			,				(8.194)
					2Ek
and thus
	2uk vk =							k	.			(8.195)
	2uk vk =								.			(8.195)
							Ek
Thus, after a bit of algebra,
2 kukvk −εk (vk2 − uk2 )] = Ek .												(8.196)
So
H = ∑k Ek (αk†αk + βk†βk ) + G ,												(8.197)

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