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Eschrig - Theory Of Superconductivity, A Primer

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where the a^ mean either ferm oni operators ^

or bosoni operators b . The eld operators

(x) and

(x) obey the anoni al (anti-) ommutation relations

y 0

(x )

= Æ(x x );

[

= 0 = [

(x);

(x )

(126)

[ (x);

(x); (x )

They provide a spatial parti le density

(r; s)

(127)

n^(operator) =

having the properties

X X

y s

(128)

)

(r; s);

r ^(r) =

hn(

( ; s)ha^i a^ji j

a^i a^i:

These relations are

readily

from

those of the reation

and

annihilation

operators, and by

In terms of eld

obtaihe

Hamiltonian

(119) reads

taking into a ount the omple

ness and

y 06) of the orbitals

(r ; s ): (129)

r d

rorthonormalit; s ) ; s

0 (r ; r )

(r ; s )

ss0

operators,d

; s hss0 (r

(

; s

(109)+

X Z

It is obtained by ombining

(119) with (125) and (111, 112).

2 s1s0

s2s0

1 1

s1s1;s2s2

1 2

8 MICROSCOPIC THEORY: THE BCS MODEL

The gr at adv tage of the

and

eld op rators

in the fa t that we

an use

. W

quantum stat

inannihilationphysially omprehensible w y

ithout expli ly

themwav

an think of modi ed operators

whi h w know little

than

theirwingalebrai p operti s.

Thusepoi

is that the

upation

umber formal sm applies for ev ry orbital

set (106).

transition from one set of operat

anoni al (anti-) ommutation rmorelations to

another suThesetefunismanipulatelltiond anon alreationtr

obeyingquantum theory.

8.1

The normal

Fermi liquidansformationas

parti-

gas

spe trum

h behav

ondu ting Fermi liquid

fermioni

very

inlik-sp

though this

isquasinot sential

here). Non-intera ting fermions

would

gas of independent parti les with en rg es

(forextheitationsakof simpli itywhi assume

havnormalground state with all orbit hasl with < o upied

and all

with > empty; es

isotropithe hemi

al po

ntial.

or removing

fermion with = new

states with N 1

fermions

are obtain

By addingas fermiumptionwith > an ex ited state is obtained w th

ex itation

and thean removex ited it without hanging theexharaitationter of the state any more

(Figs. 33 and 34).

energy . By removing

reatingorbitalsholegroundthe original ground

fermion with < | that is,

state |

state is

obtained

with

energy j j: rst lift the fermion to the level

quasi-ele tron

j k j

quasi-hole

FIG.

34:

Ex itation spe trum of a Fermi gas.

FIG. 33: Creation of an ex ited ele tron and of a hole, resp.

same

The ground

tate j0

nor

metal has mu

bov . These are

s with

> and holes with < may be

with

ene gies

up the metal together with the atomi nu lei.

they ar

ons or

missing ele trons surr

nded

y polarization loudsexof itationher elepropertiet ons and ondu tionin whi

all

gro

nd state j0i

(although absorbedquite ela

orate

theory

exists

for them whiRatherwu lei

here).nearlyW just

assume that they makingy be represented by

operators

likignorethose in the gas:

the Coulomb inter ti

. Wex iteddo not p e isely kno

these ex itations

doelew troknow the

fermioni^ j0 =

0; ^

^y withj0 =propertiesj0 ;

(130)1

k is the

veve tor and the sp

s ate of the quasi-

Sin e

all

presen

in the

ground statepartij0

already absorbed in the quasi-

energies

, o

d ondu

ion ele trons or holes exert a remainder intera tion. Hen e, wpartimayle

write down an e e tiv

Hamiltonian

^ 0

0 ^

Htera= tions^ (

k> ; ; q

0 0

k+q k0 q 0

kk q k k

k> ;k ; q

0 0

k+q

k< ;k

(132)

0 0

e rst and

k+q k

t line and attra linestiv in the se ond

line;

ele trons

and holespredominanhavopposite harges.

The matrix elements

the three

are qualitatively di rent: they are

tly repulsive

With the relations (130, 131) one nds easily

(133)

h0jHj0i =

( k ) = E0

Hj0i = j0iE0:

and have

We subtra t this onstant energy from the

(134)

H H E ;

Hamiltonianh0j j0 = 0

Hj0i = 0:

We may reate a quasi-parti le above

0in this ground state:

> :

jk i = ^

j0 ;

(135)

Hjk i = jk i (

< :

jk i = previous^ j0 ;

Hjk i = jk

(136)

1 1

k1 1

1 1

The last relation is easily veri ed with our

formulas. For k =6 k , we hav

, and together with ^

= j0 one nds the above result. Likewise

k k

1 k

Hen e, the single-partik1 1lekex1 1 itationk1 1

spe trum of our e e tive Hamiltonian above the state j0i is just

yk1

1 1

is obtained. Here, ^

= 0, so that one term has

be removed from the sum of (133).

= j

(137)

er the w vefu

of j0

migh

be.

and the ex it d states are jk i of (135, 136),

T say

truth, this

is only approximately right. There

re n

fermioni

op r

tors for wh h

the relations (130, 131) ho

true

y for thewhatevue ground

state

j0tion. T

fore, the

rst

(133)

thelast

lations (135,

136)

re also not rigorous. The quasi-

have

nite lifetime

whi h may be

expr ssed by ompl

energies k. However, for j kj partihelesapproximation relationis quit

Coandsidernormal,w a state orrelatedwith w

ex ited

good in

weakly

meta tls.

kparti

les:=^

j0 :

(138)

To be spe i we

two

with h les or with an ele tron and a hole

ele trons, the

onsider^ j0i = ex^ ited^

k2 2

H ^

j0i (

j0i w

(139)

) +

are ompletely analogous. The appli ation of the e e tivasesHamiltonian yields

k1 43 k2

X y

k1k2q

k1 1

k2 2

k1 1 k2 2

k1+q 1 k2 q 2

The intera tion term is

with the rule k

= j0 . One on

appears from

k = k : The minus signobtainedthis

t ibu

is removed

y anti ommutingtributionthe wo ^ -operators,

d then, by

with q under

the q-sum and

observing

whi h derives

rom w

;

)repla=wing(; r ), this se ond

ontribution

equal

to the rst one, wh n e om tting

k ; k0

= k

; and another on

from k0 0

= k

;

+ 2

k2k1q

fa t

terms

k1 q 1

(1/2) in fron . (This is how ex

ppear aut

with ^-op ra ors sin e their

tisymm try

k2k1

k1k2q

u ation rules

the

y of

f states.) F

quasi-parti les

with

equal spin di ers from that

ftatio quasi-parti les with opp

site spin. The spin-

omitted

(132) the spin d pende

e of the intera tion matorixthelsimplimen . itIt is always

and in

presenti omm

quasi-parti le hange

and

it may matialw

ysallybe added afterwards

onfusion.

(Weremighe tiv automatiintrodu e allyshort-hand an

n k for k .)

The e e tive

intera tionwithoutofwriting

s attering of quasi-parti les | anretaintera tion

with

hanging

and

into

and 0

| may

often be negle ted. Then, the q-sum of (139) need not be ompleted by additional spin sums.

8.2

The Cooper problem

state of the e e tive Hamiltonian,

From (139) an be seen that the state (138) is not any more an eige

andwithinus formappro

orrelated pair

try to nd

is pair

of low

energy

wterael t trons

( > 0) within our approximate

pproa h. subseSin etionw expe t

this

staeste

formed

the

ximations made in

. The

x ited quasi-parti les

quasi-parti le life ime b omes in nite f

j j ! revious0:) W

that

the

state

lowest in energy has

not of quasi-parti le

with

0 our ap

roximations thatnnot be riti al.

(The

zero total momentum,exhenitationse w

build

jenergies= Xquasijk k 0i:

with k

= k

(140)

it out of

-partiexpele

airst

where w

assume a xed om

and the y

wn expansion

ts to

of and

depend on k

ly, be ause

thebinationsough s

te is to be expe ted to

de nite total spin. (Re

that w

are on

metal in

hapter.)

Wet wha

that

pair stateoeÆj ienis

eigenstate

of H:

X y

H j

i = jsideringE; H j

j0this2 a

unkno

a :

(141)

^ 0

j0i = Æ

0 h0j(Æisotropi^

0 )j0i h0j^

0 0 ^

(Æ

0 Æ

^ 0

)j0i =

k k 0

k k

+q q 0

k kq k

the last relation with h0j

0 ^ 0

and observe h j^

^ 0

j0i = h0j^

0 (

Multiply y

k k

k 0

k 0 k

k +q

k q; k +q;q k q

h0j(Æ

0 )Æ

0Æ

0 j0i = Æ

to obtain

0 a

= 2

(142)

In the last

used again w k0

q;k0+q;q

= wk0

+q; k0

q; q and then repla ed the sum over

q by a sumterm,v

angularnonvial solution wi h ev

ite parity. It is immediately seen that

hen e

should havalsode

F r

Due to the isotropy

f our problem we expe t the solution to be an

momentum

a spin triplet 0

onlymomena no -

solution with odd

parity (odd angular

momeneigenstate,um) is

a ity a k

= ak

(even angular

tum) is only possible, if Æ 0

= 0, that is

for

singlet 0

= .

trivial=ase

Y(Thek=k)

(143)

possible. To be spe i , onsider the singlet

triplet ase is analogous.) Assume

1L. N. Cooper, Phys. Rev. 104, 1189 (1956)

k44lm

with even l: In (142),

ename k0

! k, k0

q ! k0

. The matrix

lement w

;k k

determines the

s attering amplitude from states k; k into states k

; k

. we use an expansion

wk0

; k0

;k k0

Ylm(k)Y

(144)

This redu es (142) to

l lm

wkC

C =

(145)

ak = Elm 2 k ;

wk0 ak0 :

Inserting the left relation into the right one yields

1 = l

= F (Elm):

(146)

jwkj

Elm 2 k

) = 1= .

The sought lowest pair energy orresponds to the lowest solution of F (E

F (E

)

The

fun tion

has poles

for E

where it

jumps

that

+1.

R all

-values

are

all

positivthe

start

from zfrom.

For

and

)

! 1, F (E

proa hes

zero from nega

ues.

Hen ,

if >

then

lowest

olu

on E of (146) is p si-

andlm the

ound

0,j

the

normal

metal

stable.

stateon

tiv

negativ

(at

ral

ter

tion),

then

re is

inavoidablyleast

solution E

of (146):

2 k

he ` x ited pair' has nega

ative

and the norm

ble agai state

unst

of pairs

of baluenergyd

quasi-par

no ma

how sm ll j j

i les,

is (h w weak the attra

tive intera formingti is).

Pairs

Figure 35: The fun tion F (E) from Eq. (146).

spontaneously

f rmed

grounda the ground state

If the intera tion is ut of at some energy ! ,

stru ts. This is the

onreteont

;

of Cooper's theorem.

(147)

wk =

for 0 < k < !

and the density of states for

is nearly

elsewhere

onstant in this interval, N( ) = N(0); then, with negative

Elm,

Elm

N(0)

"jElm

+ 2!

hen e,

jwkj

2 k

= N(0)

jElmj + 2

jElmj

;

jElmj =

(148)

This yields

exp

N(0)j

jElmj

2! exp

# l

for N(0)j lj

(149)

N(0)j j

: N(0)j

in the weak a

only onsider the weak oupling limit

d strong oupling limits. In this hapter

j is exponentially s

ll.

pair has a non-zero total momentum

The whole ana ysis may be repeated for the ase

that a e the denominator of (146) is to be repla ed with E

(q)

k+q=2

wherewhereno

jk q=2j mus

be larger than k

. For small q; t

is ondition redu es the density of stat s in e t in

an Ini

is the Fermi velo ity.

rval of thi kness j = kjq=2 = vF q=2 at the lowhere-integration limit; vF

The result is

(q) E

+ v

q=2:

(150)

In the weak oupling limit, E

(q)

n only be negative for expon ntially small q.

We performed the analysis with a pair of parti les. It an likewise be done with a pair of holes

with an analogous result.

8.3

BCS

poin

out that the ele tron-phonon intera tion is apable of providing an

Fr•ohl

was the rst

e e t vTheattra ti

be w

ele trons in the energy r

f phonon energies.

e tiv for

From Cooper's analysis it follows that, if there is a weak

a tra tion,

it a

only be

pa rs with zero total mom

tum, that is, betw

k and k. With the assumption that th

attra tion

the

= 0 spin Hamiltoniansingleen onduhannel,tionthis ledBardeen, Cooper and S hrie er

to the

simple model

Hamiltonian

< ;

0 < +!

X y

(

(151)

BCS

k# k"

. Sin e

d nsity of

Here, g > 0 is the BCS oupl ng onstant, and V

is the norm

plane-w v

states

k-sp

is V=(2

= V=(2 )3

R d3k

atrixolumeelmen

of an n-parti le

intera tion

in an

-fold k-sum) must

proportionalizationV ( 1)

order ththethe Hamil-

(appearingV ). The modeled attra tivbeintera tion is assumed in

in energy

nge of width

2! around

hemi al pot

tial (Fermi level in

ase T =

0), where !

phonon

en rgy for whi

the Deby

energy of the

an be taken.

The s extensivate j0 of (130, 131) annot

ny more bethe grou

tate of this H miltoni n sin e Cooper's

toniorem

ells us that this state is unsta

again t spontaneous

of boundharap irsteristiwith the gain

of their binding energy. The problem to solvlatti e

w to nd the g ound

state

and the q

asi-parti le

rum of

BCS-Hamilto

ian. This

ans

s lved by Ba

deen,

Cooper and S hrie er,

shor

thereafter

and

independently

bley

ofas

anoni altransformation,

Bogoliubov and,

Valatin. Bardeen,

and S hrie er

us provided the rst mi ros opi

theory

of super ondu -

tivity, 46 years after the dis overy of

theproblemthenomenon.

1H. Fr•ohli h, Phys. Rev. 79, 845 (1950).

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

8.4

The Bogoliubov-V

its partner

the ground state ontaalatinns bound pair. Ex iting one parti le of that pair leav

imultaneously its partner. Led by this transformationthe quasi-

operators in the ground

behind, and hen e also in an ex ited state. If one w

ts to ex ite only one parti l , one muest annihilate

Supposetate of (151) an ansatz

= u ^

onsideration,v ^ ; b

+ v parti^ le

theirlea in

minute. Sin e

; onsider^ ; b

;

an r-independent p as

orbital annihilated by ^

is ma

. u

d v

are v

k k"

k k#

parameters. Again w

the isotr pi p oblem and

dependen e

on k =ariationaljkj ly.

The reason of

di eren

signs in

he tw

relations be omes

fa t r

may

hosen,

and v

may

real

without

loss

generality. Thensee

may be summarized as

Bogoliubov-Valatin transformations together with their Hermitian

arbitrarily^

= u ^

;

be^

k# k" k#

(152)

onjugatev ^ :

We want these transformations to be

anoni al, that is,assumedw want the new operators b ; b again to

k k

be fermioni operators. One easily

;

^ 0

; b

= alvulates0 [^

; ^

+ [^

ely.onsideredAnal

ihilation and reati

n operators ^

^ ,

In the rst equality it was already

for the b-

and

b -operators,

k +

k k +

k k

= ukvk0

0Æk k0

Æ 0

+ Æ kk0

( + ) = 0:

Æ 0

= kvk

Æ kk0

the same w y. The sign fa tor inthemselvtra

rmation ensures that the anti ommutation is retained

respe tively, anti ommute among

es. The analogous result for the b

perators is obtai

gously0 2

0 Æ

0 ;

; b

; ^

+ v

= urespe[^ ;tiv^

+ v

k +

(153)

hen e the ondition

+ v2

ensures that the

ransform

rs are again fermioni operators.

is anoni al and the new

Multiplying

rst relation

(152) by u

in the s ond one

with k , multiplying

it with vk

, andthehen adding

results

with (153)operatth inverse transformation

(154)

=bothu

+ v b

repla;

ing^

= u

+ v b

yields

k k

Observe the reversed sign

u b

+ v

+ v b

fa^ tor=

The next step is to transform the Hamiltonian (151). With

k k

+ ukvk

+ v

and the

= b bk

b k b

b b

+ b k bk

ti ommutation rules it is easily seen that the single-parti le part of the BCS-Hamiltonian

transforms into

k k

( k )vk

( k )(uk vk)

( k )ukvk

bk bk + 2

bk"b k# + b k#bk"

It has also been used that under the k-

k may be repla ed by : Further, with

=sumb

+ v b

k# k"

k k"

+ ukvk

;

(155)

ukb k#bk"

vkbk"b k#

b k#b k#

bk"bk"

N. N. Bogoliubov, Nuovo Cimento 7, 794 (1958); J. G. Valatin, Nuovo Cimento 7, 843 (1958).

the full transformed BCS-Hamiltonian reads

( k )vk

( k

)(uk

vk)

HBCS

bk b

;

(156)

( k )ukvk

bk"b k#

+ b k#bk"

Bk0 Bk

kk0

where for

y w

omitted thekb

of the last sum.

Now, wbrevittroduin

e the o upatioundsumber operators

(157)

= bk bk

state j i of the BCS-Hamiltonian is an o upation number

. Then, its energy is found to

of the b-operators and assume an logous to (130, 131) that for properly hosen b-op rators the ground

eigenstate

ukvk

(1 nk"

k#)

(158)

E = 2

)vk

)(uk

vk)(

k" + nk#) V

( k

(153), when e v = u

= u =v . For givthen

o upation

nummbers, (158) has its minimum for

The nk

are the eig

values

r 1) of

ber operators (157)

the ground state

0This.

variational para

and v

whi h are onne ted by

energy expression still ontains

eters u

o upationv

2 #

nk#

= 0;

= 4( k )uk + 2

1 nk"

(159)

Hen e, u

and v

ukvk

1 nk"

nk#

are determined by 153) and

= u2

(160)

)

Their ombination yields a biquadrati equation with the solution

( k )

= 2

1 p(

+ 2

;

2ukvk = p(

+ 2 :

(161)

Insertion into (159) results in the self- onsisten y

1 onditionn

whi h determines as

fun tion of the BCS

oupling

g, the

relation k

+ 2 ;

(162)

1 =

( k

normal state ^-quasi-parti les (in esse e

he Fermi velo ity), and the o dispersionupationumbers nk of the

b-quasi-parti les of the super ondu ting state (in essen e onsttemhe

perature as seen later).

The

par mete s

and

pi ted

on the righthe.

For-

mally,

inter hang ng

with

would

solut on

also

the

biquadrati

quation. It is, how-

ever, easi y seen that

his would

not lead

minim

. With (161)

readily

een

thait

Figure 36: The fun tions u and v.

se o

sum

(158) is positiv

def-

inite. Hen e, the abs lute minimum of energy (ground state) is attained, if all o upation numbers of

the b-orbitals are zero.

redu es to

For the ground state, the self- onsisten y

1 =

+ 2

gN(0)

+ 2

gN(0)

ln p!2

+ 2

p( k )2

ondition! d! p

gN(0)

resulting in

(163)

= 2! exp

(162)0

for the v

of in

he ground state (at zero

gN(0)

If one re

la es the last term

(g=V )

of the tr

med BCS-Hamiltonian (156) by the

meaneldaluepproxima ion

(re temperature)a hat w nsfointrodu d as = (g=V )h

0i,

k Bk

+Bk

k Bkj

relation (160) makes the ano

f. (155, 159)), than

is readily seen

hat

^ ^

alous terms (terms b b

or bb) of this Hamiltonian vanish: In mean-theld approximation the BCS-Hamiltonian is diagonalized

by the Bogoliubov-Valatin transformation,

(

resulting)( v

m-f

(

)(u

) + 2 u

(164)

onst. + k k bk bk

with the b-quasi-parti le energy dispersion relation

+ 2

(165)

= p(

obtained by inserting (161) into the se ond line of (164).

The

rsion

with the

normal state

energyk

see

jdispdepdispers

the righogether. It

at the physi al mearelationing of is the gapand

normal

stat

relation

b-quasipar i le ex itation

spe trum

f the

super on

u ing

The name bdispersiongol

j k j

The fa t thatthese Bogoliubov-Valatin trans-

diagonalizes the BCS-

form tion

jus i es

oftenle

mean-stateld appro

bogol

partially

f a normal-state

F m (152) one ould arrivogetherHamiltoniann lusion that

used for

quasi-parti les.

the

litthe

37: Quasi-parti le dispersion relation in the

o upati

umber

operators. In

Figu e

statosteriorimay be foundassumptionan eigensximationof

diago

alizing the meaneld BCS-

ground

our

super ondu ting state.

ture, the key rela ions (160) arethatof en derivedras

instead

minimizing

nergy

(158). Inofa t both onne ions are equally im-

Hamilt

ian. Clearly, the BCS-theory

por an

and prov de onlythe

expresslution of

that solution is

- e d

n e would not

rry

tegeronsistsh rge qu

tum. However,based

ele tron and partiallymeanof

hole, a

rst was pointed out by Josephsotheoryn, he true b-quasi-parti leannihilation and

reation operators are

nnihilating an el tron pair and

repla ing

it withpartiallynormal-state ele tron. The se ond part of the

= u ^

;

= u ^

v ^

(166)

v P ^

P ;

ed, that is, a hole-bogolon is r ated, by

reating a no mal-stateNohle and partially

a positivereatesarge, when e

the ho

-bo olon (with jkj

< k

) arries

te er

k k

wherein th

P anni ilates and P

bound pair wi h zero mom ntum and z

o spin

onsidered

Cooper problem. Its w vefun tion will

onsidered

n xt hapt

w,as

bogolon is

sitiv

quantum. Lik

an ele tron-bogolon (with jkj > k ) is

Fpartially by

ensures

operators mak

b-bogolon be surrounded byarriesuper ondu tin

ba k ow of

harge whi

that an integerreatesharge quantum travels with the bogolon. It is

egativfor jkj < k

and

reatingfor

alsoat

and partially by reating

ele tron pair and simultan ously

tum.

nronormaless-stahargel le tron.

Agaiewise,the

golon

an integer

(

e) reatedharg

The P -

Of ourse, it remains to show that the new ground state j

positivis indeed super ondu annihilatinti egativ.

jkj > k

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