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Физика конденсированного тела

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Eschrig - Theory Of Superconductivity, A Primer

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The lines (of kness ) indeed form nearly individually (Fig. 24). Sin e B 2

= B 12 2

= ln ; the

latti e onstanthia

at B

= a1

& :

(89)

2 2 =

The ores of the ux lines (of thi kn ss ) tou h ea h other while the eld is already quite homoge-

neous (Fig. 25). Sin e j

j 1 in the

ore, the Ginsburg-Landau equations apply, and j

j may rise

ontinuously from zero:

the phase transition

at B 2

is se ond order.

solution

j j

a &

a2j&j

of (4.3)

FIG. 24: Mixed1phase for B

ext

FIG. 25: Mixed phase for B

ext

F r long ylindri rod the stray eld reated by super urrents outside of the rod may be negle ted,

and one

y express the eld inside the rod as

B = B

ext

by magnetization density M: The hange in Free Energy at xed T and V by tuning up the external

magneti

eld is

dF = MdB

ext

Fn Fs =

B 2

0 B

dBextM = B

B 2

Bext

B 1

2 0

00000001111111 00000001111111

00000001111111

jump or

00000001111111

000000011111110011

0011

in nite slope

Figure 26: Magnetization urve of a type II super ondu tor.

is only a theoreti al

The d erently dashed areas in Fig. 26 are equal. For type II super ondu tors, B

quantity.

EFFECTS

mi rosJOSEPHSONopi des ripti.owever, qualitativ ly ey are the sam(the at all temperatures T < T , hen e

The quantita iv

n of Josep

son

at T T

usual ase in appli ations) needs a

Considertreatmenvery thin weak linkwithinbet eene woe tshalfs

ofLandausuper ondu tor (Fig. 27).

qualitatively they may be treated

Ginsburg-

theory.

= 1

+ x

(x2)

2 = j

FIG. 27: A weak link between

halfs of a super

du tor.

link is small, and may be

ally small

x = 0: Hen e,therman superodynamiurren

through the w

The order p

has its

value

sides x < x ; x > x ; but is expo-

neno

dered onstan

in both bulks of

. In the w

lineak, not only j j is small, also its

Without the righrameterhalf, the boundary ondition (58) would hold at x :

phase may hange rapidly (e.g. from super= ondu tor=

+ bothyeakvery small perturbation).

2ie

x +

= 0:

In the presen e of the right half, this ondition must be modi ed to slightly depending on the value

x +

2ie

(90)

~ Ax

1B. D. Josephson, Phys. Lett. 1, 251 (1962)

jei1 2

small

umber depending on

properties of

eak link. Time

version

that (90) remains valid for

; A ! A;

henthe

must be real as long as the phase of

wheremandsdo not depend on A. For the moment thew hoose a gauge in whi h A

= 0: Thein, the supersymmetryurren

density at x1

js;x(x1)

ie~

1 x

i x

jm sin 2 1

(91)

( 2

ei( 1

We generalize the argument of the sine fun tion by a general gauge transformation (22):

= 0 ! A

;

A +

~ ;

2e Z

dxAx;

1 !

= 2 1

(92)3

tential

by a voltmeter.

Re all that is the ele tro

Now, the general Josephsonhemiquational

readsmeasured

= j

sin :

(94)

6.1

The d. .

quantum

interferen e

and a ordi

g to (93)Josephsonoten

di eren e

is zero

that ase.

A ording

(94), a d. . su er urrene eof anyt,

value between j

and j

may ow through the jun tion,

Now, on ider a jun ion in the y z-plane with a magneti eld applied in z-dire tion. There are

super urrentos s reeningthehe eldtialway from the bulks of the super ondu ting halfs.

(y)

Figure 28: A Josephson

jun tion in an external magneti eld B.

Let us onsider (y); and let (0) = 0

at the edge

= 0: We have

+ Z

1 dyAy:

Byd = Z

1234

ds A = Z

2 dxAx

+ Z

3 dyAy

+ Z

4 dxAx

In the jun tion we hoose the gauge A

= By; A

= 0: Then, the y-integrals vanish, and from (92),

(0) = 2

= 0

(y) =

dxBy = 0

2 Bd

;

The d. . Josephson urrent density through the

os illates with y a ording to

2 Bd

(y) = j

sinjun tion+

= j

; hen e

= =2; and

In experiment, at B = 0 one always starts from a biased situation with j

j (y) = jm os

2 Bd

(95)

The total urrent through the jun tion is

dy os

20 Bd

= jm

where is the thi kness in z-dire tion. F =

is the area of the jun tion. With

i b

sin( b)

dy os( y) = <

dyei y = <

urrent at a given eld B is

we nd that, depending on the phase

; the

0)j

(96)

Is;max

= Fjmaximal

An ven simpler situati

sin(2Bbd=0

ds A =

if one splits the jun tion into a double jun tion: Now,

is the magneti ux through the ut-out, and

ds A = 2

appears,

0 b

Figure 29: A simple SQUID geometry.

Hen e, the

d. . Josephson urrent is

a +

2 #

2 =

Is;max = F jmaximalsin a + sin

max

Is;m x

(97)

Figure 30: Phase relations in Eq.(97).

= 2F j os

This is the

experimentally oun

6.2

The a. . Josephson e e t

a w thbasisdevi e alled

super ondu ting

quantum interferometer (SQUID).

= (2e=~)V t:

We return to (93) and (94), and apply a voltage 2

= V to the jun tion, that is,

An a. . Josephson urrent

(t) = j

sin !

= 2eV=~

(98)

resul s,

a onstant v

f 10 V a frequen y !J =2 = 4.8 GHz

is applied. For a v

is obtained: the a. . Josephson

e t is in the

mi row

vltageregion.

If onealthoughverlays a radio frequoltagen y voltage over the onstant voltage,

(99)

= V + V

os(! t);

one obtains a frequen y modulation of the a. . Josephson

sin

t +

2eVr

sin(! t)

~!r

urren

2eV

Jjnj

sin(!J + n!r)t:

(100)

= 1

~!rr

Jj j

is the Bessel fun tion of integer

to a ount that

non-zero voltage a ross the jun tion

interpret the

ts one m

take

auses also a

experimennormal urrenustI

V=R; where R is

of the jun tion for normal

ele trons.

index

in of the radio frequen y.

If the impedan e of

The experimental issue de

ends on the

radio sour e small ompared to that of the jun tion, wthehav

voltage-sour e situation, and

the total urrendissipativthrough the jun tion averagedouplingver radio

resistan e

(101)

I = I

+ V=R;

= j F:

1S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).

frequen ies

d. . Josephson spike, Im

~!r=2e

V V=R

Shapiro spikes for

Figure 31:

urrent vs. voltage in

+ n!

= 0

voltage-sour e situation.

as is usually the ase,

wtheJosephsoav a nurrent-sour e

ation, where the fed-in total urrent determines

If the impedan e of

r dio

is large omp red tothe

impedan e of the Josephson jun t on,

2e dt

situ

the voltage a ross the jun tion:

= R(I I ) = R(I I sin ):

(102)

V =

Shapiro steps

V =R

Figure 32: Josephson urren

vs. voltage in the urrent-sour e situation. V

The a. . Josephson e e t yields a possibility of pre ise measurements of h=e.

7 MICROSCOPIC THEORY: THE FOCK SPACE

;

d of the

The S hr•odinger w v

of an ( sola ed)

of its po

spin variable,

: = ( ;

): Si

there are only

wo funindepetionden

spin states for an ele tron

| the spin

efunwithtionrespe t to any

ele trhosen axis may be ei her up (") or down (# |,

distakreteon onlyomponentwvalues, + and , hen e may be thought as onsisting

of twition,fun tions

(103)

(r;

(single)=

(r; +)

lo al) one-parti le operator

a spinor fun tion of r: The expe tation value of any

formingA( ; s; r ; s ) = Æ(r r )A(r; s; s ) isX Z

(104)

(r; s)A(r; s; s

) (usuallyr; ):

We will often use a short-hand notation x r( ; s).

The S hr•

wavefun tion

s;s

: : : x

) of

:):

many-parti le quantum state must be

(: : : x : : : x

: : :) = (fermioni: : : x

(105)

totally antisymmetodingeri with respe t to parti le ex hange

(Pauli

pie e of

solid, N

prin iple):

, this fun tion is

in omprehensible a

-fun tions.

oordinatesumerisystematial

manner

fun tionaloordinatebas s in the fun tional spa e of those horrible

not

ll of those 2

brough

independen

| with the symme ry propertpra ti (105)ally

the +- and

-v lues

an always

to an order thattotally-values pre ede

+-valu s,

en e winahavessible:only

F a

y set of N valu s, + or for ea

s ; it is

fun tion of 3N position

oordinates. Although

disti

guish 0, 1, 2, . . . , N -v

that

is (N + 1) ases | and

h ugh ea h of

those

(N + 1)

funhe symmetrytions

property (105)

ea h

need on y be

givmensionfor

rtaionorder of the par

for all s

= + parti les |, it is lear that even if wspaweuld be onwithten

= fundtionsforalues,all

need

ly be

ertain

se tor of

3N-d

1023

again

givea

all posit

with 10 grid

points al

axis w

would need 10

grid

ts f r

very

7.1

representation of that

-fun tion. Nevertheless, for formal manipulatiopoins, we

introdui le

Slater determinan

hosen) omplete

set of one-parti le spinor

Cons der some (for the moments

fun tions

(x);

arbitrarily

orthonormalll

0 ;

d r

( j

(r; s) 0 ( ; s) = Æ

already1 2 )orbitalsN

Æ(

0):

(106)

X (x) (x0) = Æ(x x0) = Æss0

T ey are ommonly all d (spinor-

lThe quantum

umber

ref rs

to both the spat al and

for a plane wave), and we agree upon

erta

andinstanf eve

given

near order of thoseorbital-

di es.

Choose

N of those orbitals,

; ; : : : ;

;indexas ending order of the l

and form the determinant

the spin state and is usu

lly

multi-

(for

(nlm ) for an atomi

or (k )

L = (l

: : : l

is a new (hyper-)multi-

whi h labels an orbital on guration. This determinan of

L(x1

: : : xN ) =

det k

(xk)k:

(107)

a mat ix a

= (x

) for every pointindex(: : : x ) in the

-position spa e has the proper

are di erent, and it would be identisymmetryall

property (105). In view of (106) it is normalized, if all l

spin

zero, if at least tw

f the

ould be equal

t with

equal raws): Two fermions annot

Now, given a o

plete set of

(106),(determinanw mention

without

proof that all possible orbital

on g

ra sameof N orbitals(107) orbitalsform omplete set of N-fermionspew

vefunial

tions (105), that is, any

be in the

spin

. This is the ompared to (105) very

ase of the Pauli pr n iple

h is the mmo

know

ase).

(whiavefun

tion

(105) manly be represented as

X CL L(x1

: : : xN

(108)

(x1

: : : xN ) =

with ert in oeÆ ients CL;

(r ) +

era0 ( tion'); ;

(109)

P jCLj2

= 1 (` on Lguration in

For

y operator whi

an be expanded into its one-parti le,

w -parti leand so on parts as for

instan ean Hamiltonian with

-dependent)

(possibly spin

i=6 j

s s ;sjtera tions,j

pair

the matrix with Slater

ts is

jhjl

) ( 1)

= hdeterminanjHj 0

jPji

lPklk

;

(110)

1 X X

w l0 l0) ( 1)jPj

P is any permutation of the subs ripts

i; j; k; and

jPj

ts order. The matrix elements are

j(=6 i)

k(=6 i;j)

wherede n d as

X Z

=6 j

jhjl

) =

d r

(r)

(r; s );

(111)

(r; s)h

ss0

and l l jwjl0 l0) =

Z d3

0 (

; s0 ) 0 (r ;

0):

(112)

(r ; )

(r ; s ) w

(r ; r )

(Note our

vention

the order of indi es whi

may di er

from that

other

but

and

elemen . The se ond line is non-zero

ly if the

s s ;sjsj

di er

most intorxtbooksw

i j

j i

s s0

;sjs0

to a ertain

anoni al way of w iting of formulas later

The rst lin

of (110)leads

perm

P has

only

on-zero term

this

ase, determi

ing the

sign f

that matrix

the sumutationsver all permutati

ns hasationsw non-zero terms in

ase: if P is a perturbation with

=all

alled dir t intera tion

and the se ond one ex hange onin eragurationstion.

non-zero

only if the tw

on gu

L and L

di er

most

one orb tal,

he sum

ver

for

ing ontribution is (1=2)[(

orbitals,j

k =6 i; j, then the onerrespo

jwjl0 l0) ((l

jwj

0 0) ( 1)jPj.

For L = L

and P = identity (a

sometimes

also in

general

ase)

the

rst matrix element is

7.2

The Fo k spa e

of quantum me hani s by wav

with the

Up to here we onsid red repr

partiingle

number N of the system

xed. If

thistationsumber

large,

efunnnottions

xed at

of any s le.

(In

des ription any partimale ros opibe allyreated or

ed, possibly together

de ite value in

. Zero mass bosons as

.g. photons

y be emitted or absorbed in sys

temsof

with its

tipartiexperl ,lativistimen a uum region just by apply ng

Fromannihilamere te hni al

ensemble with varying

parti le

umber, than with

the

anoni al oneulate. H n e

there

are many good

view,

quantum statisti s

denti al parti les is mu

simpler

to form

with

grand anopoini al

reasons to onsider quantum dynami s with hanges in partienergyle umber.

In order to do so, w

start with buil

ing the Hilbert

f quantum

of this wider

ntro

symmetry with onsideredspe t to par i le ex hange will b

y H

. In the

last

u ed

g in H . Instead of

thspamultie -

L asstatesw of N indi esframe:l w

may denotespa basise

state by spe ifying the o upation numbersdenoted(being either 0 or 1) subseof all orbtiontals

the Fo k

. The

o now Hilbert spa e of all N-parti le

es having the appro-

priate:

index

spe ifying

= N:

(113)

: : :

: : : ;

Our previous determinantal state (107) is now represented as

0 : : :i:

ear spa e

= j0 : : : 01

0 : : : 01

by the basis v

(113), . . the st

es of HN are either linear ombinations

states (113) not oin iding

all

num ers

are

. H

is the omple

ity)

limits of Cau

msequen

e oftorssu thelinear

ombinations. (AorthogonalCau y sequen e is a

ig with limm;n!1h

umo upation= 0.

The in lusion of all limits of su h

sequen es

importan

in all

of limits. This ompleteness of the spa e

onfused with

P j

of states (1

3) (with the

squared absolute values of the oeÆ ients C

al to

to H meanspannedrealizing the topologi

ompleteness property of the Hilbert spa e, being extremely

the

ofnsiderationsbasis set f g.

ompleted by orresponding Cau h

with ompletenessabov given

rthogonality retained,

nned by allpartistatle ve tors (113) for all N

de ned as the ompleted dire t sum of all H . It is sp

seq

The extended Hilbert spa e F (Fo k sp e) of all states w

the

notumber N not xed is

en es, just as the real line is obtained from the rationandl

line

by ompleting it with the help of

ber are allowed too. (For bosoni elds as e.g. laser light those quantu ombinationsu tua ions an be tiomle

Cau they

of rationalde nitionumbers.

are linear

with v

ying

Note

hat F now on

not only q antum states w

so thasequendoese

not havtainsde nite valu

in the quantum state

(o upation

umber u tuations),

also

linear

ombinations with varying N so that now quantum

ations of

he total par

important

experimentally even for ma os opi N.)

7.3

whi

umber representation

r as

provide abandonjust transiti

between basis states (113) whi h are as lose to eawith

TheOde nitiupationof these r

the

operators for

mustsimplesthav regardoperatorsthe an-

ompl tely

wful w vefun tions (105) and will ex lusively work

o -

tisymmetry of the quantum states

to Pauli's

ex lusion1 : : : n

( 1)

follo

;

from this antisymmetry.

upation

umber eigen

^ j : : : and: : :annihi= j : : :

are

ates (113) and m trix elements between t

em. The

pos

ible: those whi h di er in one o upat on

umber only.

ation

prin iple

^ j :ation: : : : :

= j : : : n

+ 1 : : : (1

fermionsP

(114)5

The u

will

lear below. By

j<iwingthe

elements with

They are de ned as

j<i

umbersefulnessdi erenthefrom fa0 tors1 do

otbeappomear: appli ation of ^

= 0 gives zero,

tothat state with n

eeded

do p

rti ularly

not

reate

non-fermion

states (tha

states with o upa

possible

signumber

= 1

(113), it

eas ly seen

hese operators ha all the

and appli ationo upatiof ^

to a state with

es zero as well). The ^

and is,^ arematrixutually

Hermitiona

onsiderj

properties,

eigenstates+ ij

onjugate, obey the key

relations

giv

and

^ j : : :

: : :i = j : : :

: : :

: : : ^ ^ j : : : n

[^ ; ^

= Æ

;

[^ ; ^

= 0 = [^

; ^

(117)

d ned in

way.

Reversely

the anoni al

wi h the anti mmutator [^ ; ^

= ^ ^

+ ^ ^

anti ommu atio

elatio

(117)

all the

lgebrai

properties of the ^-op ators and moreover

lasses of

representati

ns of those algebrai relations withstandarddi erent stru ture

nd not unitary

The

(113) of the Fo k spa e

generated out of a singleare,b sis ve tor, the

however, vast

de ne up to unitary equiv len e the Fo k-spa e representation (114, 115). (There

equivalen furtherto

Fo k-spa e represde netation.)

ally

va uum basistate ji j0 : : : 0i (with N=0) by apply ng ^ -

in any order agrethe

sign fa tors of (114, 115). Sin pr

du ts

systematif signset

of N ^ -operat

again the order

jn1

: : : : : :

= : : :operators:^ ^ ji

(117) in

(118)

de ning

fa tor

in view

ent with

with ea other up to possibly

sig

, all possible

(118) do not

s agree

more di erent

Observbasis e ors than those

(107) with the ongiventionexpressionsthe order

the

upon there.

With the help of the

^-operators,

inear operator in the Fo k

may be

expressed. It is

Hen forth, by using (118) w

written

need not bo her any more about the given

of the orbital

linegenerateord

indi es.

atrix e ements with

Hamiltonianumber eigenstates (113) as

Hamiltonian (109))

the same

H =

^ ( jhjj)^

^ ^ (ijjwjkl)^ spa^

(119)

not diÆ ult to demonstrate that the

X y

e ause both span the Fo k spa e,

tal states (107) and the o up o upationumber eigenstates and

has wi h determinantal st

in (110). Be ause of the one-to-o

orresponden e between the deter-

ijkl

prin iple

may be realized withmiltoniaboso ihanger at

ger

operators

minanof y

tor

en in the S hr•

tation

eviden

(119).

f the equivalen

y the

tes

(109)

(119) are equ valen .

The building

The S hr•odinger w giv

n tion

fand

-parti le

tum statefromust be totally symmetri

belinearitresp ed

y sy

efutrized prod

bosot (permanrepresens), withquanlightly more

volved normalizatiothen

fa tor. The orbitals

ow be

(omis

with

arbitrary many

= 0; 1; 2; : : : : This ase

with

t tooper

rti le

on of the

minus sign is

(105)). The det rminants re

^ j : : :o :upied: : b b j : : :

: : :i = j : : : n

(123)

[b ; b

= Æ

;

[b ; b

= 0 = [b ;pbarti les:

with the anoni al ommutation

relations

ijannihilatijon

: : : = j : : :

1 : : i n ;

b j : : :

: : : = j : : :

+ 1 : : :i ni + 1;

bi j : :

^ ^

The basis states of the Fo k spa e are reated out of the va uum a ording to

theorder.h.s. of (120, 121) not only ensures that

(122)

holds but also ensure the mutual Hermitian

: : :

: : :i =

ji:

(124)

The

of these operators in the produ t does not makei

any di eren e.

The hoi e of fa ors

onjugation of bi

and b

7.4 Field operators

eld operators

A spatial representation may be introdu ed in the Fo k spa e by

(125)

^(x) =

(x)^ai;

40^

(x) =

i de(x)^aning;

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