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Физика конденсированного тела

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Eschrig - Theory Of Superconductivity, A Primer

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<<< < Предыдущая 1 2 3 4 56 / 66

9 MICROSCOPIC THEORY: PAIR STATES

asi-parti les.

In m an-

BCS theory, the

determined b

ompl te absen e of q

With the properti

of the Bog

iubov-Valatin tran formation,

ground state is

found

be the

onde

eldof Cooper pairs

plane-wstave K

= 0

states

rs of gravity. By o upying

quasi-p rti le

states a ording to

Fermi statisti s, thermodynamthistates of the BCS super ondu tor

are obtnsained.

9.1

The BCS ground

a ordingly.

Then, it was found after (162) that all

In^

= b

; and the b

wstate

Se tion 8.D the BCS ground

was assumed to be an o upation number

o upation numbers n^

are zero determinedthe ground state. This implies that h

eigenstate= 0; and

hen e

0i = 0

(167)

bk j

for all k : The (meaneld) BCS ground state is the

(uniquely

de ned) b-va uum. We show that

is the properly normalized state with properties (167). The normal metal ground state j0i of (130,

+ v

(168)

131) is

j0 =

ji =

(169)

and

has the form (168) too,

with u

= 0 for

< and u

= 1 for > and the opposite

behavior for v

k" k#

Whennowe demonstrate the properties of (168).

ji = u

+ v

= 1;

+ v ^

+ vSin^ e^

k k# k"

k" k#

0i of (168) is properly normalized. Moreover,

0 ^

v 0 ^

+ v

0 ^

ji =

" k

ji =

0 v

k0"

k(6 k )

+ v

k0#

ji = 0

(170)

1 ^

= 0. This ompletes our proof. Histori ally, Bardeen, Cooper and S hrie er

and analogously b

solved the BCS model with the ansatz (168), before the work of Bogoliubov and Valatin.

9.2

pair

Next wThend

w vefun t ons ontained in j

i: R all, th

a ondu tion ele tron in

the plane-wavthestatefunexp(tionk r)

(reates) is

(s) so that the ld

ator

exp(ikoper)

(rs) =

(s):

(171)

is, rs

the enter of gra

of the ele tron with its polarization loud whi h

toget

y and the

r makdenotesupthe

` ondu tion

elevitron'.)

spinand depending on these variables is

(Thathe

N-parti le wavefun tion

ontained

0(x1

; : : : ; xN ) = hx1

: : : xN j

= hj

uk + vk^

ji:

(172)

(xN )

(x1)

For the sake of

) =

y we

the spinors

, and nd

; : : ; x

kN rN

ikN 1 rN 1

ik1 r1

1 + g

ji =

k1:::kN

hjbrevit

suppress

N 1

k" k#

for i6 j

}

=0; if not ki

i=6 kj j

q q

N=2

i(k1 r1+ +kN rN )

hj ^

k1:::kN

kN N

k" k#

}

eik2 (r1 r2)

Æ 1

sum over all possible ontra tions

N ! =

(rN 1 rN )gkN Æ N 1

fk2ig

)

singlet

)

A (r

N 1

singlet

with

singlet

(174)

( ) X gk

ik ;

In the se ond line

(171) was insert

for the

f (172), and the

were fa tored out of

(x )

f (172) (leaving gk

= vk=u

behind in the se o

du ts

run

item of the fa tors). The k-pr

ver all grid points of the

k-m sh,

.g.

rmined by

boundary

for

volume

(173),while the sum runs over all

produ ts for

ets of N disjun t k-values out

of thamplesh. This disjun t natu(in nite)of

he k-sums is sibl

da h

the sum in the f

llowing

ines. Expansion of the rst

rodu

of whionditionsh nly

yields terms with 0, 2, 4, . .periodi. ^ -

the N ^-op

of an it m

f the k-

t rms with exa tly those N ^ -operators that orrespodet

left

the

produ e

non-zero r

between thei ated^-va uum

j ji.

results

mos

easily btained by

i ommuting a sult

the

perators to thstatesoperatorsrigh of all r ation op

nd are usually alled ano tra tions;

nding

iginal order of the operators,

a h

sum

1. The

produ tover

, whi h multiplies ea on

on and

h as

runs

ver

all

many k-valu s

y elds

normalizi

fa torwhi i h ispreviouslyindThesendenresultof the

f k

of the full

values

; : : : ; k

of the sums. Likdepea annihilationindividu

tribut

, it depends on the

hemi al potenratoial

are essen

-zero inside

andinprodunitelythet gap

. At this point onemesh,ust real ze that theng

the F

)

astiallynon-

gligible value

surfa e ( f. (161)). Hen e, the ontributionfa tor

; : : : ; x

only for all k -ve tors inside the Fermi surfa e, and

value

with an in reasing umb

h kermi-ve tors and de reases

gain,

an appre iable

umber of k -ve tors falls outside of the Fermi

surfa e:

the norm of (172) is maximalif

N-values suthisthat theinkreas2 o upy essentially all mesh points

inside the Fermi surfa e. That is, this norm is non-negligible only for those N-values orresponding to 52

the

umber

n the

normal Fermi

iquid: j

is a grandanoni al state with a sharp

The result is maximstate j

originalonsisting of pairs of

ele trons in the

(pair-orbital) ( )

In ordele tronto analyze what this pair-orbital ( ) looks lik

re all that u

and v

may be written as

parti leumber

the a

oni al N-v

lue. W

did0

not tra e normalizing fa tors in (173)

and us a sign of approximation

(174) ag

n, assumig

( ) to be normalized.

( )

d kg e

dkk g

d e

geminaldkk

singlet

fun tions of

and (

( f.

. Hen e, g

~(~v

k= ), and

~vFk

(161))

Æk

eik

e ik

(175)

kg~

sin k

dxxg~(x) sin

f( = 0);

By omparison

r of

in the pair, their distan e

(173), is the distan e ve

verage being of the order of 0

ile the pair-orbital

lenottronsdepend on the position R of

lane w v

with w v

ve tor K = 0.

Mor tover, again

due to (173), all N=2 10

ele tron

pairs o upy the

ame delo alized pair-o bital in the BCS ground state

: this ma ros opi ally

the enter of gravity of

he pair:

h respe t

the enter

of gravity the pair is delo alized,

weakly oupledsuperdeterminedypdependonduI tor,ondu tor this ratioeleis tronsypially

o upied

(th

is, onstanwitin R-

pair-

is the ondensate w vefun tion of the

re ly thedelo veragealized

distan e

of the

(173)

pair whi h

an be ompared to the

For a real

tate,

and

the

stru tur

oforbitalj

ensures that the solution of the BCS

the gap anspab

measurede)

(for instan e by measuring ther

superdistan ondue

ofting

arbitrary

x itationele trons in the solid given by the ele tron densitodynami. Forerage

model is

Fermi velo ity v

in the norm l state, this measurem Withyields

independen

quan ities whi h

thehen e

spe trum or dire tly by tu

neling spe tros opy.

the

103

: : : 104:

same delo

(176)

pair orrelation resulting in

ondensation of

ll ele trons intoolumeand

lized

In this ase, there

: : : 10

ele trons of

pairs in the v

betw en

given p

ir: The e

orbital in the super ondu ting

state,

however,otherpi ture of ele trons groupthed into

individual pairs

In order to reatemisleadingsuper urrent, the

ondensate w vefun tion, that is, the pair orbital must be

would be by

in order not o deform (and thus destroy) the pair orbital itself. Hen e, 0 has the meaning of the

provided withfar phase fa tor

( ; R) e

K R

: Obviously, it must be K (177)

he reation operators in (168) with ^

oherenrepla inglength of the sup ond

tor at zero temperature.

9.3 Non-zero temperature

k+K=2" k+K=2#

The

Hamiltonian (164) shows that the bogolons reated

of (166) and

y operators

havingtransformedenergy

of (165) are fermioni ex itations with harge

and s in above

Coope

pairs must be 2 Sinwhere is the re om

potential of bogolons. Hen e, at temperature T

BCS g ound statedispersionj :

they may

bine into Cooper pairs , the

hemi al potential of

the distribution of bogolons is

nk"

= nhemik# = al k=kT

+ 1

(178)

The gap equation (162) then yi lds ( f. the analysis leading to (163))

1 2 e k= T

+ 1 1

gN(0)

k Z

p!2

+ 2 ep!2

+2=kT +

tanh

+ 2

(179)

gN(0)

p!2

+ 2

2kT

where in the last equation0

he symmetry of the integrand with respe t to a sign hange of ! w

used.

For T ! 0, with the limes tanh x

1 for x ! 1, (163) is

For in reasing tempera ure, the numerator of (162) de reases, and hen e must also de rease. It

vanishes at the transition temperature

T ,

1 = gN(0)

d! tanh

reprodu! =2kTed

(180)

2kT

= gN(0)

dx tanh x:

Integration of the last integral by0 parts yields

ln x

=2kT

dx tanh x =

! =2kT

+ ln

tanh

2kT

ln x

osh4

2 !

;

osh

+ ln

2kT

= ln

+ ln

= ln

2kT

where ln

= C 0:577 is Euler's on tant.

The se ond line is

valid in the weak oupling ase

and the nal result for that ase is

)

(181)

2 0

=kT

exp(

with 2 = 1:13;

gN(0)

2 = 3:52:

The gap asand f n tion of

between zero and T

temperatureust be al ulated

numeri al y

from (178). However, the

simple

expression

(182)

(T ) 0s1

is a very good app

From

ene gy

expression

(158), the thermodynami quan

ties an be al ulated, on e (

)

hen e

k(T ) is giv

The

Figure 38: The gap as a fun tion of T .

main results are the

ondensation

energy at T = 0,

;

(183)

2(0)

N(0) 0

yielding the thermodynami riti al eld at zero temperature, the spe i heat jump at T

(184)

1:43;

and the exponential behavior of the spe i

at low temperatures,

Cs(T ) T

3=2 0=kT

for

T T :

(185)

heat

10 MICROSCOPIC THEORY: COHERENCE FACTORS

As a ready

last se ion,

states of

External elds,

are

by o -

ying qu si-parti le stat

with Fermi o upat on numbers.

wever,

most ases

ouple to t

done^-op

ator

elds. D

e tothermodynamioupling of ^-ex itationssuperin aondu torhondu obtainedtor terferen e

Let fk g be anthermodynamigiven disjun t set of quasi-parti le quantum numbers. Then,

terms appear in

response to su h elds.

10.1

The

state

(186)

P =

jthermodynami

(187)

is tate with

quasi-parti les

k 2fsuperk

ondu ting

ground s ate j 0

: If

here are

fk gabovk 2fk

many quasi-partithoseex itations pres nt,

hey in

ra t with ea h other and with the

ground state (the latter intera tion is in

erms of

operators P ^

and ^

P ), andondensateerathis in tion

l ads

to the temperature dependen ex ofited

energy

spersion law

+ 2

via the

temperature dependen e = (T ): The

state is

fk g

where the summation is over all possible sets of quasi-parti le quantum numbers,

fk =

exp(

(188)

=kT ) + 1

is the Fermi o upation number, and Z

Z(T; ) is the partition fun tion determined by

X Qk 2fk g fk

(189)

theusual,verage

quasi-

state k

numberexpeitselftationof

the thermody

= fk g

= 1:

value of any operator A is obtained as trAP : For instan e

o upation0 0

partij

the

Q f

fk g

Z Q f fk g

X k0

0 j

fk gi

(190)

The state k0 0 j

k gj k0 0

= fk0 trP = fk0

i is obtained from the state j

fk gi by removing the quasi-parti le k : Hen e,

if one fa tors out f

, the remaining sum is again P .

0 0

10.2

and spin momen

The hargek

^(q) = omponen^

+ele^ trons^

(191)

= densities^ ^

The operator of the q Fourier

of the

harge

y of

in a solid is

(192)

momen^ (q) =

dire^

tion)=

That of the spin

t density (in z-

k+q" k"

k# k q#

k+q

k q#

k+q

k+q"

55 k

The

to (187)

rom

P of a normal me alli state is omposed

^-op rators. The , in

al ulati

super o

averages, ea

the k -sum

(191) nd (192)operatoris veraged ind pendently. In

state, the item in r

last

of those expre sions ar

ou led

and

hen e

are not ananalogymore veraged

independen

herestatistippear on

due to their oheren

f ren

the super ondu ting

states

tly:

igenstatesfk

ThoseP rforming the Bogoliubov-Valatin transformationthermodynamif du tingharge density operator yields

fk g

tributiotributionsappear in the response of

sup

ndu ti

state to external elds whi thesesouple

ha ge and spin densities.

+ ^

k+q"

k q#

ujk+qjbk+q" + vjk+qjb

ukbk"

+ vkb k#

+ ukb k#

vkbk"

ujk+qjb k q# vjk+qjbk+q"

ujk+qjukbk+q"bk"

vjk+qjvkb k q#b k# + ujk+qjvkbk+q"b k#

+ vjk+qjukb k q#bk" +

+ukujk+qjb

b k q# + vkvjk+qjbk"b

k+q"

ukvjk+qjb

k+q"

vkujk+qjbk"b k q#

ujk+qjuk vjk+qjvk

k q#

bk+q"bk"

+ b k#b

jk+ jvk + ukvjk+qj

bk"b k q#

In obtaining the last equality some operator pairs were anti ommuted whi h leads to the nal result

k+q" k#

n^(q) = e k

ujk+qjuk vjk+qjvk

bk+q"bk

+ b k# k q#

An analogous al ulation yields

+ ujk+qjvk

+ ukvjk+qj

(193)

bk+q"b k#

bk"b k q#

m^ (q) = B

ujk+qjuk + vjk+qjvk

bk+q"bk"

b k#b k q#

+ ujk+qjvk

ukvjk+qj

(194)

k+q"

+ bk"b k q#

The

line of these relations re e ts the abov

mentioned oupli

g between ^-states, and the se ond

re e tsrsthe

oupling to the ondensate. Both lines

ontain oheren e fa tors omposed of u and v.

10.3

Ultrasoni

uation

. The

ouplingtera tion term of the Hamiltonian

the ele tri eld

auseldbyattenlatti e

As an example of

(external to the ele tron

to the

harge density we onsider

X p system)

(195)

phonon=

! orresponding^ + ^ ^(q);

its reation operator.

volume, ! is the phonon frequen y and a^y

where g is a ou

is the

ling onstant relating the qele tri eld of the phonon to its amplitude, V

We onsider the attenuation of ultrasound with ~!

< ; then, in lowest order pair pro es es do

not ontribute. A ording to Fermi's golden rule the phonon absorption rate may be written as

Ra(q)

tr HI Æ(Ef

Ei)HI P =

4 g2

!qnq

vjk+qjvk

k(1 fjk+qj)Æ( jk+qj k

~!q):

(196)

Here, E

and E

ujk+qjuk

are the total

states

forming the H -matrix el ments and n

from the rst term in the rst lenergiesthermodynamiof . Afte renaming k q ! k0; the se ond term yields the

umber of the

phononontai

o upation

state whi h in this ase in exte

sion of (187) also

phononi ex itations in thermi equilib ium. Half of the result of the last line is obtained

same

result

. The phonon emission rate is analogously

R (q)

H Æ(E

E(193)H P

4 g

!qnq

ujk+qjuk

vjk+qjvk

fjk+qj

(1 fk)Æ( jk+qj k ~!q):

(197)

With fk(1 fjk+qj) fjk+qj(1

kf ) = fk

fjk+qj, the attenuation rate is

4 g2

!qnq

(fk fjk+qj)Æ( jk+qj

k ~!q):

(198)

dtq

ujk+qjuk vjk+qjvk

The attenuation in the normal state is

4 g2

!qnq

(fk

fjk+qj)Æ( jk+qj k ~!q):

(199)

dtq

From a measurement of the di eren e, (Tk) an be inferred.

10.4

The spin

sus eptibility

With the intera tion Hamiltonian

= H( q)m^ (q) + : :;

(200)

where H is an external magneti eld,Ithe expe tation value of the energy perturbation is obtained

E(q) = tr

)

(201)

The sus eptibility is

P :

" I

E(q)

ujk+qjuk + vjk+qjvk

2 f

(q) = dH( q)

= 2 B

jk+qj

2 1 fk fjk+qj #

(202)

+ ujk+qjvk ukvjk+qj

jk+qj

The sus eptibility drops down below T

d vanishes exponenti

lly f

T ! 0.

Coheren e fa tors appear

s milar

manners

any

resp

nse fun tions as in the nu lear

relaxation time, in the diamagineti response, in the mi rowavmoreabsorption, and so on.

areAknowledgement:. I thank Dr. V. D. P. Servedio for preparing the gures ele troni ally with great

<<< < Предыдущая 1 2 3 4 56 / 66

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