Higher Order Processes in Field Emission Spectroscopy from Superconductors

Introdu tion 2

1 Field Emission from Metals 5

1.1 Field emission: the tunneling limit . . . 8

1.2 Current spe trum operator . . . 11

1.3 Perturbation theory . . . 13

1.4 Current spe trum fornormal metals. . . 16

1.5 Emission fromNiobium. . . 19

2 Field Emission from Super ondu tors: Single Parti le Pro- esses 21 2.1 BCS theory of super ondu tivity . . . 21

2.2 Super ondu tor's urrent spe trum . . . 24

2.3 Experimentalresults . . . 29

3 Field Emission from Super ondu tors: Andreev Pro esses 33 3.1 Andreev ree tion. . . 33

3.2 Spe trum of Andreev pro esses . . . 35

Con lusions 46 A Transfer Matrix 48 A.1 Bardeen's formula. . . 48

A.2 Harrison's semi lassi alapproximation . . . 50

Topi of this thesis is the study of the emission of ele trons from a material

under the appli ation of an intense ele tri al eld, known as eld emission.

Under the ee t of a very high eld, up to millions of voltsper entimetre,

appliedtoasampleofagivenmaterial,theele tronstunneloutofthesample

and travelto adete tor, likea uores ent s reen, pla ed ontheir traje tory.

The whole apparatus - sample, anode and dete tor - is kept in va uum, to

preserve the properties of the emitted beam. Metals are most ommonly

used for eld emission experiments, sin e the emission takes pla e even at

room temperature.

The analysys of the emitted ele trons an reveal many properties both

of the surfa e and the bulk of the sample. Usinga uores ent s reen or any

other method of dete ting the ele tron presen e, one an obtain the spatial

distribution of the ele tron ows, whi h turns out to be a proje tion of the

surfa e shape and impurities. Another property that is importantto study

is the energy spe trum of the emitted ele trons: if a spe trometer is pla ed

on the path of the emitted ele trons, one an measure their distribution in

energy. Thisdistribution anbeusedtoinferbulkproperties,sin eitree ts

the energy distributionof the ele trons insidethe sample. Su h experiments

were one of the rst onrmations of the Sommerfeldtheory of metals.

Numerousexperimental and theoreti al studies dealt with eld emission

from metals. It was only in 1998 that Nagaoka, Yamashita,U hiyama,

Ya-mada, Fujii and Oshima su eeded to perform a eld emission spe tros opy

experiment onsuper ondu ting Niobium,showing for the rst time the

sin-gle parti le spe trum of the super ondu ting state [NYU

+

98℄ in the energy

distributionof theemittedele trons. Theee t theyobserved hadbeen

pre-di tedbyGadzukin1969[Gad69℄andisadire t onsequen eofthemodied

density of states in super ondu tors. In this way, they showed that it was

possibleto dire tly inspe t the propertiesof the bulkof a super ondu tor.

Theee tpredi tedbyGadzukwasa urrentin reaseduetothe oherent

ele troni states in the super ondu tor, forming two sharp peaks near the

measure of the gap, but improvementsinthe te niques (su h asthe use ofa

nanotip asa sour e of ele trons [NFM

+

01℄)may allow itin the future.

Sin e most of the relevant properties of a super ondu tor are related to

the fa t that it has a oherent ground state, experiments, like those that

havebeen realizedby Oshima'sgrouponother non-super ondu ting

materi-als[OMK

+

02℄[CISO04℄usingaHanbury-Brown-Twiss interferometer,would

beanother importanttopi of future resear h.

Theroleofsuper ondu tivity orrelationsontunneling phenomenaisnot

limited to the presen e of a modied density of states. It has been known

sin e the seminal work of Andreev [And64a℄ [And64b℄ that other pro esses,

that involvethe onversion of(normal)parti lesand holesintoCooperpairs

are possible. These pro esses have been named Andreev pro esses.

The subgap urrent was derived by Andreev in the ase of a

metal-super ondu tor interfa e. He found that a parti ular kind of phenomena

showedup,inwhi hanele tronenteredthesuper ondu torandaholeexited,

whi his alled Andreev ree tion. In parti ular he showed that Andreev

re-e tiontookpla eeven atenergies lowerthanthegap. Thisree tion omes

fromthe fa tthat the ele tron tunnelingintothesuper ondu tormustform

a Cooperpair, and thus absorbs another ele tron from the jun tion to pair

with, leaving ahole in the metal [BTK82℄.

Inthisthesiswewillstudytheee tofAndreevpro essesineldemission

showing that it leads to importantmodi ations atenergies belowthe gap.

ToderiveAndreev ontributiontoeldemissionwewillneedtousethe

many-body treatment originally proposed by Gadzuk [Gad69℄. In this treatment

the emission pro ess is des ribed by a tunneling hamiltonian representing a

single ele tron jump, whi h is treatedperturbatively. The lastperturbation

order onsidered isthe maximumnumberof oherenttunnelings thatwe an

des ribe. Gadzuk, limiting his treatment to the rst order did not a ount

for Andreev tunneling and obtained a vanishing subgap spe trum. We will

pro eed by al ulating the spe truminthis energy range. The resultis that

su hpro essesarestillbeyondthepossibilityofanexperimental onrmation

due to their extremely low intensities. Moreover, the experiment ondu ted

showed some still unexplained spe trum distortion in the relevant energy

range, whi h made impossible to experimentally see the gap. In ase su h

issues ould be over ome in the future, the possibility of dire tly revealing

the emission of an entire Cooper pair would be of great help in studying

the a tualbound state that formsthose pairs, and in generalthe properties

of the super ondu ting bulk. Also, the possibility of obtaining maximally

entangled ele trons in this way ould be very interesting.

the Fowler-Nordheim theory is reprodu ed in the many-body pi ture and

how some of the resolution limiting pro esses an be taken into a ount.

In the se ond hapter we apply the same al ulations to the ase where a

super ondu ting material is used instead of simple metal. Then we explain

howthe1998experimentwas arriedoutanditsresults. Inthethird hapter

we explain the al ulations that we have done to derive the spe trum of

Andreev pro esses in eld emission. We briey omment on the possibility

ofmeasuringsu hspe trumwiththepresent-dayapparati. Intheappendi es

weshowinmoredetailthefundamental al ulationsthatareusedinthetext.

In the rst one, we treat the approximation ofthe transfermatrix elements,

whileinthese ondone weexplainallthestepsneededtoderivethe Andreev

Field Emission from Metals

Field emission experiment are ondu ted pla inga metalli sample. A high

eld(

1



100 Vnm

−1

)isapplied,possiblyusingasmalltiptoenhan etheeld

strength. Ele trons that tunnelout ofthe metal are dete tedand measured.

Spatialimaging,spe tros opyandinterferometryarethemost ommon

mea-surements whi h are arried out on the ele trons.

Fieldemissionexperimentsinthe twenties were oneof therst

onrma-tions of the tunneling theory based on the newly born quantum me hani s.

The experimentswere arriedmeasuring the total emitted urrent and were

aimed at probing the properties of the interfa e. As su h they fo used on

the urrent dependen e on the experimental parameters, su h as eld and

temperature. S hottky [S h23 ℄ attempted to explain eld emission at low

temperatures with the vanishing ofthe eldbarrier of ele trons. His

predi -tions, however, were not inagreement with experiments.

Partially inspired by Oppenheimer's work on emission from hydrogen

atoms [Opp28℄, in 1928, Fowler and Nordheim [FN28℄ found an expression

fortheemission urrentfrommetalstowhi hHendersonandBadgley[Phy31 ℄

gave rst experimental onrmations in 1931, and whi h be ame the basis

for all subsequent treatments of the eld emission problem. By using

Som-merfeld theory of metals, Fowler and Nordheim explained both thermioni

emission and eld emission using the same treatment, parti ularly showing

that ele trons that were emitted ame, in both phenomena, from the

on-du tion band.

Their model was a very simple one, redu ed to a one-dimensional

sys-tem. It showed that the most relevant emission properties ame from the

assumptionthatthe energydistributionofele tronsinsidethemetalwasthe

Fermi-Dira one, and used the al ulation of the tunneling probability done

by Nordheimin the same year [Nor28℄.

applying a strong repulsive eld to ele trons in a metalli tip ould proje t

them onto a dete ting surfa e. The image formed on the surfa e provided

useful informationonthe surfa e properties of the tip.

Inthepresentthesiswefo us onspe tros opyinstead,whi h orresponds

to measuring the energy distribution of emitted ele trons. Sin e the energy

gain of the ele trons due to the voltage dieren e is xed, the resulting

distributiondependsonthepropertiesofboththemetalandoftheinterfa e.

To properly link the energies of the ele trons insidethe metal tothe

en-ergiesof emittedones,one hastotakeintoa ountthetunnelingprobability

at a given energy, whi h is a property of the barrier. It has been estimated

in many ways, the most ee tive one being the use of WKB's semi lassi al

approximation,whi h ledtoabehaviournot sofarfromwhat wasshown by

Gamow for

α

de ay in the same years [Gam28℄ when Fowler and Norheim rst formulated their theory. In fa t a Gamow fa tor appears also in the

theory of eld emission, and its approximation by means of a Taylor

se-ries expansion isthe basis of most parametrizationsof the Fowler-Nordheim

formula.

In their original work, Fowler and Nordheim derived the total urrent

onvolutingtheele troni energydistributioninthemetalwiththetunneling

probability [FN28℄. This wasderived assumingasimpletriangularpotential

(see Figure1.1)

V (x) =



−φ

x ≤ 0,

−eF x x > 0,

(1.1)

in the emission dire tion, where

φ

is the work fun tion, i.e. the potential dieren e between an ele tron inside the bulk and one pla ed at aninnite

distan e while the ele tri eld

F

is zero. Mat hing the solution for

x < 0

and

x > 0

andapproximatingthe energydependen e ofthe probabilitythey obtained the following expression for the transmission probability

D(W )

D(W ) ≈ e

−c+W/d

,

(1.2)

where the

c

and

d

are the interfa e parameters following the notation by Young [You59℄, and depend on the experimental parameters

φ

and

F

. The exponentialdependen y onthe energy wasdemonstrated to hold alsoin the

many-body ase by Bardeen [Bar61℄.

Theotherimportant omponentoftheFowler-Nordheimtreatmentisthe

supplyfun tion

n(Ω, E)

,representingthenumberofele tronsinthesample whi hhavetotalenergy

E

anddire tion

Ω

. BySommerfeld'stheorywemust

n(Ω, E) =

1

4π

Z

f (~

2

k

2

_{/2m)δ(E − ~}

2

k

2

/2m)d

3

k,

(1.3)

assuming an isotropi distribution inside the sample. Then the ux of

ele -trons with normal energy

W

and total energy

E

onthe barrier is

N(W, E) =

Z

δ(W − mv

⊥

2

/2)n(Ω, E)v

⊥

dΩ,

(1.4) where

v

⊥

=

p

2E/m cos(θ)

is the normalvelo ity atenergy

E

and dire tion

Ω

. The urrent spe trumis then

J(E) =

Z

dW N(W, E)D(W ) =

4πmd

~

3

e

−c+E/d

_{f (E).}

(1.5)

This is the regarded as the standard Fowler-Nordheim result for eld

emis-sion spe trum from old metals. It was rst experimentally onrmed by

Herderson and Badgley[Phy31 ℄usingaretarded potentialspe trometer (see

Figure 1.2), however itwas ompared with the normal energy distribution.

The dieren ebetween the total energy and the normal energy

distribu-tion ofemittedele tronswas learedlateron,in1959,by Young[You59℄. He

dened the total energy distributionas the numberof ele trons whi h rea h

a xed potential with the same kineti energy. Together with Müller, he

demonstrated thatthetotalenergydistributionwastobe omparedwiththe

spe tra of the ele tron, insteadof the normal energy (i.e. the kineti energy

due tothe impulsenormaltothe surfa e)[YM59℄. Finally in1961,Harrison

pointedout the importantroleof thedensity ofstatesof ele tronsinside the

metal in the nal expression of the emitted urrent spe trum [Har61℄.

Based on the premises by Young and Harrison, and using a many-body

approa h, in 1969 Gadzuk showed that the energy distribution of ele trons

inside a super ondu tor should lead to a hara teristi deformation of the

emissionspe trum[Gad69℄. Inparti ularheshowedthattwospikesappeared

aroundtheFermilevelatadistan e oftheorderofthe super ondu ting gap.

This spikes have eluded experimental onrmation until 1998, when an

ex-perimentbyNagaoka,Yamashita,U hiyama,Yamada,FujiiandOshimahas

seen thepredi ted urrentpeakinasuper ondu tingniobiumtip[NYU

+

98℄.

Many improvements an be applied to the basi Fowler-Nordheim

the-ory, to take into a ount a more realisti shape of the potential, or of the

surfa e. For example we ould in lude the image harge potential in the

des ription, possibly smoothed to onsider the ele tron penetration inside

−φ

0

Sample(

x < 0

)

0

Va uum (

x > 0

)

V

(x

)

(a) (b) ( )

Figure 1.1: Various barrier potentials are shown. The sample lls the left

part of the drawing until x=0, and has a potential of

−φ

orresponding to the Fermi levelrelative to the potential atgreat distan es in absen e of the

eld. It represents inthe ase of (a) onlythe applied onstant eld, (b) the

appliedand surfa e harge density (i.e. image harge) eldand ( ) the wave

pa ketee t whi h smoothes out the image harge ee t.

method of evaluating the ele tron redistribution inside the metal has been

developed by Watanabe's group [GNWW00℄. However su h modi ations

would only hange the formulas used to derive

c

and

d

fromthe experimen-tal parameters

φ

and

F

, leaving the energy distribution un hanged. Yuasa et al. have re ently studied modi ations to the Fowler-Nordheim formula

due toa non-planaremitter [YSOO02℄.

1.1 Field emission: the tunneling limit

The model we hose is the one proposed by Gadzuk [Gad69℄, sin e we are

on erned with many body ee ts. Moreover, it also presents a lear link

between the numberof ele trons that tunneloutiside the metaland the last

perturbation order onsidered. It will be used to derive the standard result

third ele trode is pla ed just behind the anode, with a voltage above it. It

is alled olle tor and only ele trons with energy above it voltage an rea h

it. Changingthe voltageand dierentiatingthe result givesthe total energy

distribution [GP73℄.

of super ondu tors, followingGadzuk's treatment, and nallywewillderive

the double parti le urrent, orresponding to Andreev pro esses.

In this model the system is omposed of the sample and the va uum

separated by a thin interfa e. Theinterfa eis lo ated at

x = 0

. We will all

L

the region at

x < 0

and

R

the other one, at

x > 0

. The

R

medium is va uum. We limit our des ription only to ele trons in the two subsystems.

Whilethisassumptionisrealisti fordes ribingthe dynami sof

R

,ele trons in

L

ouldbesubje t tomanyee ts, su hastheintera tionswith phonons, whi hwewillnegle t. Furthermorewedonot onsidertheCoulombrepulsion

between eletrons, assuming that either its ee ts are toosmall or are taken

into a ount renormalizingthe ele tron masses.

If the two sides were dis onne ted, the system would be des ribed by a

hamiltonian

H

0

whi h isthe sum of two hamiltonians

H

0

= H

L

+ H

R

,

(1.6)

H

L

and

H

R

being the two separate hamiltonians of the two subsystems

L

and

R

respe tively. The basi step to the tranfer hamiltonian approa h is that we now ouple the subsystems, so that the system is des ribed by the

new hamiltonian

where

H

T

is the hamiltonian for the ele trons in the interfa e

Z

b

a

d

3

x

−~

2

_∇

2

2m

ψ(x) + V (x)ψ(x),

(1.8)

where the interfa e isa smallplanein luded between

x = a

and

x = b

. Wewillassumethattheeigenstatesofthefullhamiltonianarestilldivided

in leftand rightstates, whi h willbeindexed by quantum numbers

α

and

β

respe tively, andannihilationoperators

c

α

and

d

β

. Rearrangingsomematrix elements we an ensure that diagonalelements of the followingform vanish

c

_α

H

T

c

†

α



=

D

d

β

H

T

d

†

β

E

=

*

c

_α

. . . c

_ω

|

{z

}

n

H

T

c

†

ω

. . . c

†

α

|

{z

}

n

+

= 0,

(1.9)

so that we an write

E

α

=

c

_α

Hc

†

_α



=

c

_α

H

0

c

†

α



, E

β

=

D

d

_β

Hd

†

_β

E

=

D

d

_β

H

0

d

†

β

E

,

(1.10) where the averages are over the void state. This rearranging an always

bedone asdemonstrated by Prange[Pra63℄. He alsodemonstrated that the

ommutatorsofeldoperatorsin

R

and

L

arevanishingintheweak oupling limit. Assumingmoreover that the interfa e is onservative (i.e. it does not

absorb nor emit ele trons), and that no intera tion between ele trons takes

pla e inside the interfa e, we see that the tunneling hamiltonian must be of

the form

H

T

=

X

α,β

T

α,β

d

†

β

c

α

+ T

α,β

∗

c

†

α

d

β

.

(1.11)

Itwillbene essarytoknowthetransfermatrixelements

T

αβ

later. Before this, wehavetode idewhi hparti ularset ofstates

α

and

β

weare goingto use. Forour purposemomentumandspinisthenatural hoi e. Furthermore

wedonot areaboutspineither: assumingthattherearenopro essesmixing

spinup anddown,

H

T

willsimply onservethespin. The urrentsoobtained is due to either spin-up only orspin-down only el trons, so the real urrent

is obtained multiplying itby 2.

Bardeen [Bar61℄ shows a method to derive the transfer matrix elements

in dierent systems, whi h is basi ally an appli ation of a one ele tron

ap-proximation. The followingformulaisderived intherstappendix, together

with all the onditionsin whi h itwillbe valid, but for now wewilltakefor

granted that it works inthe ase weare onsidering.

|T

kq

|

2

=



_~

2

2m



2

k

_⊥

q

_⊥

δ

k

k

,q

k

e

„

−c+d

−1

~2

k

2

⊥

2m

«

,

(1.12)

where

c

and

d

are the parameters depending onthe interfa e shape [You59℄ and will be spe ied later.

c

is dimensionless while

d

is dimensionally an energy.

k

and

q

are the impulses in

L

and

R

respe tively,

k

⊥

and

q

⊥

are their omponentsnormaltothe surfa eofthe interfa e,and

k

k

and

q

k

are the parallelones. The exponentialdependen e ofthetunnelingelementfromthe

energy normal to the surfa e is intuitive [RGM56℄, whereas the appearen e

ofthe normaldensityofstateswasrstshown byHarrison[Har61℄. This an

beexpressed as

|T

kq

|

2

= |T

0

|

2

k

⊥

q

⊥

δ

k

k

,q

k

e

d

−1

_ξ

k⊥

_,

(1.13) having put

|T

0

|

2

₌

~

4

4m

2

e

−c

and

ξ

k

=

~

2

_k

2

2m

.

Some observations are in order. First of all we observe that the

depen-den y on kyneti energies isnot symmetri albetween the left and the right.

The reason forthis willbe lear when we willderive this formula, but is

ba-si ally due tothe fa tthat kyneti energies in

L

and

R

of a parti leowing through the interfa e are not independent. Thus we have hosen to express

the transfermatrixasdependingon

ξ

k

be ausetheparti lesareowingfrom

L

to

R

in our ase. If we hose toexpress

T

interms of

ξ

q

instead,the oef- ients

c

and

d

should have been hanged. Moreover, as anti ipatedabove, there is a dimension mismat h, sin e the formula a tually gives the transfer

matrix perunit area of interfa e,instead ofthe total one. This isdue tothe

fa t that momentum paralleltothe surfa e is onserved atea h intera tion.

Whi h means that

T

is itself proportional to

δ

k

k

,q

k

and, in order to avoid divergen ies in the square of

T

, we had to divide by the total area of the interfa e.

1.2 Current spe trum operator

We must now des ribe the urrent distribution operator, i.e. the operator

whose average orresponds to the a tual measured value, inthe many-body

approa h. Wederivethe urrentowingfromthe

L

to

R

fromthe ontinuity equation

j = −e

D

N

˙

L

E

,

(1.14)

where

N

L

isthe operator ounting ele tronsontheleftofthe barrier. Asthe total numberof ele trons is onserved, it follows that

j = e

D

N

˙

R

E

N

L

=

X

α

c

†

_α

c

_α

,

N

R

=

X

β

d

†

_β

d

_β

.

(1.16)

The urrent is the hange in the number of ele trons on the right. An

experiment ountingonlyele trons omingatadeniteenergygivesa urrent

distribution of

∂j(E)

∂E

= e

*

∂

∂t

"

X

β

d

†

_β

d

_β

δ(E

β

− E)

#+

.

(1.17)

In the previous equation

E

β

is the energy of an ele tron in the state

β

. The expression for time derivative iseasily writtensin e by denition

∂

t

hOi =

 [O, H(t)]

i~



,

(1.18)

the average being taken over some, in prin iple time-dependent, state. We

want to knowthe values of

N

R

in the time evolution

|ψ(t)i

of some thermal state

|ψi

of the free hamiltonian

H

0

. This means we have to al ulate

∂j(E)

∂E

=

e

i~

hψ(t)|

"

X

β

d

†

_β

d

_β

δ(E

β

− E), H(t)

#

|ψ(t)i .

(1.19)

The thermal state is hosen to be the ground state of

H

0

at time

−∞

:

|ψ(−∞i = |ψ

0

i

. This orrespondstoassumethattheintera tionhasstarted somenitetimebeforethetime

t

ofthe tunneling. Wewillforgetthe

δ(E

α

−

E

β

)

fromnowon, forbrevity,itwillnotbedi ulttorestore itagain atthe end of the al ulation.

Sin e

H

0

onservesthe numbers ofele trons in

L

andin

R

separatelythe urrent be omes

j =

e

i~

hψ

0

| U

†

_{(t, −∞) [N}

R

, H

T

(t)] U (t, −∞) |ψ

0

i .

(1.20) having alsoexpli ited the time evolution

U

fromtime

−∞

totime

t

.

It must be noted that we have not assumed any parti ular form of

H

L

and

H

R

,and thiswillholdformanyoftheformulasinthefollowingse tions. Thus most of the results will still be valid when we will treat the ase of

super ondu ting materials.

The following notationfor

H

T

willease al ulations

where

A

and

A

†

are dened as

A =

X

α,β

T

α,β

d

†

β

c

α

,

(1.22)

A

†

=

X

α,β

T

_α,β

∗

d

_β

c

†

_α

,

(1.23)

so that

N

˙

R

an bewritten:

˙

N

R

(t) =

[N

R

(t), H

T

(t)]

i~

=

A(t) − A

†

_(t)

i~

.

(1.24)

Finallythe urrent an be written

j(t) = −

2e

_~

Im

A

†

(t)



.

(1.25)

This expression is general and will be our starting point for all parti ular

systems we willanalyze.

1.3 Perturbation theory

As afundamentalassumption of thetunneling hamiltonianapproa h isthat

the oupling between ele trons in

L

and

R

is very small. This translates into the requirement that the matrix elements of

H

T

are small ompared to typi alenergiesofele trons ontheleftandontherightoftheinterfa e. This

is realisti even assuminga highpotentialenergy inthe interfa e. Infa t an

in rease inthe barrierstrength wouldexponentiallylowerthewavefun tions

of the ele trons while linearly raising the energy, thus providing an overall

de rease inthe total energy dueto thepresen e of ele trons inthe interfa e.

In pra ti e the above assumption allows us to use perturbation theory,

up tosome nite order. Cal ulationsare done in the intera tion pi ture

|ψ

I

(t)i = e

iH

0

t

|ψ(t)i ,

(1.26)

and

O

I

(t) = e

iH

0

t

O(t)e

−iH

0

t

,

(1.27) for any operator

O

. We willomitthe subs ript

I

in the following equations. The evolutionoperator is given by the series

U(t, t

0

) =

+∞

X

n=0

Z

t

t

0

dt

1

H

T

(t

1

)

i~

·

Z

t

1

t

0

dt

2

H

T

(t

2

)

i~

· . . . ·

Z

t

n−1

t

0

dt

n

H

T

(t

n

)

i~

.

(1.28)

Calling

U

n

the terms ofthe sum inDyson's series

U

n

(t, t

0

) =

Z

t

t

0

dt

1

H

T

(t

1

)

i~

·

Z

t

1

t

0

dt

2

H

T

(t

2

)

i~

· . . . ·

Z

t

n−1

t

0

dt

n

H

T

(t

n

)

i~

,

(1.29) we obtain that

U

n

is of order

n

in

H

T

, and holds

j =

e

i~

+∞

X

n=0

+∞

X

m=0

hψ(−∞)| U

n

†

(t, −∞)[N

R

, H

T

(t)]U

m

(t, −∞) |ψ(−∞)i ,

(1.30)

in whi h the terms are of order

n + m

in

H

T

.

Atpresent, wetakeonlythe se ondorderapproximationin(1.25),i.e. of

the probability distribution, orrespondingtothe rst orderof the

probabil-ity amplitude. Sin e ea h appearen e of

H

T

des ribes one ele tron moving a ross the interfa e, this means sele ting the urrent ontribution due to

single parti le tunnelling between the two sides, i.e.

j(t) ≈ −

2e

_~

Im

D

U

₁

†

_{(t, −T )A}

†

(t) + A

†

(t)U

1

(t, −T )

E

=

= −

2e

_~

Z

t

−T

dt

′

Im

* 

A

†

_{(t), A}

†

_(t

′

_{) + A(t}

′

₎



i~

+

.

(1.31)

It an be seen that the terms ontaining

A

†

_(t

1

)A

†

(t

2

)

or

A(t

1

)A(t

2

)

are to be negle ted whenever one of the two regions is not super ondu ting. In

fa t separating the eld operators in

L

and

R

, su h terms would result in produ t of averageterms like

c

†

α

1

(t

1

)c

†

α

2

(t

2

)



hd

β

1

(t

1

)d

β

2(t

2

)i

,whi hare zero in normal metals or void. Sin e in our ase

R

is always void, we will not onsider these terms. They are however the main ontributions to ee ts

of oheren e between two points of a super ondu tor and give raise to the

Josephson urrent.

We express the above formulausing the eld operators,to obtain

j(t) ≈ −

2e

_~

X

α,β

X

α

′

_,β

′

Z

t

−T

dt

′

Im

 T

∗

αβ

T

α

′

_β

′

i~

Dh

c

†

_α

(t)d

β

(t), d

†

β

′

(t

′

)c

α

′

(t

′

)

iE

,

(1.32)

in whi h the ommutatoris easilyfound tobe

h[. . .]i =

D

d

β

(t)d

†

β

′

(t

′

)

E D

c

†

_α

(t)c

α

′

(t

′

)

E

−

D

c

α

′

(t

′

)c

†

_α

(t)

E D

d

†

_β

′(t

′

)d

β

(t)

E

.

(1.33)

for the ele trons in the two mediums, and therefore require knowledge of

ele tron dynami sto be al ulated.

By invarian e under time translation we an always de ompose Green

fun tions likethe above ones in the followingform

c

α

1

(t

1

)c

†

α

2

(t

2

)



=

Z

_+∞

−∞

dω

2π

A

α

1

α

2

(ω)f

+

_(ω)e

−iω(t

1

−t

2

)

_,

(1.34)

c

†

_α

₁

(t

1

)c

α

2

(t

2

)



=

Z

_+∞

−∞

dω

2π

A

∗

α

1

α

2

(ω)f

−

_(ω)e

+iω(t

1

−t

2

)

_,

(1.35)

where the

A(ω)

are spe tral weight fun tions and the

f

±

_(ω)

represent the

energy distribution of ele trons. Spe tral weight fun tions arry all

dynam-i al informationand will be dierent depending onthe material. The same

is doneforGreen fun tionsin

R

. We distinguishspe tralweights and distri-butions with subs ripts

L

and

R

.

Makinguse of the Greenfun tions (1.34)(1.35)the urrent be omes

j(t) ≈ −

2e

_~

X

k,q

X

k

′

_,q

′

Z

t

−T

dt

′

Im

T

∗

kq

T

k

′

_q

′

i~

·

·

Z

_+∞

−∞

dω

2π

A

R,qq

′

(ω)f

+

R

(ω)e

−iω(t−t

′

₎

Z

+∞

−∞

dω

′

2π

A

∗

L,kk

′

(ω

′

)f

_L

−

(ω

′

)e

+iω

′

_(t−t

′

₎

−

−

Z

_+∞

−∞

dω

2π

A

∗

R,qq

′

(ω)f

_R

−

(ω)e

+iω(t−t

′

₎

Z

+∞

−∞

dω

′

2π

A

L,kk

′

(ω

′

_)f

+

L

(ω

′

)e

−iω

′

_(t−t

′

₎



(1.36)

This expression is quite general and an be used to examine various ases

of dierent materials. For example, as in the previous simple model, ohmi

urrents an be obtained if

T

is weakly depentdent on the momenta. F ur-thermore it must be noted that we have retained orrelation properties

be-tween owing ele trons. This is apparent fromthe fa t that phases are still

present, and separating ele trons based on surfa e position just requires to

insert spatial

δ

fun tions in the expression and swit hing to spatial repre-sentation. Anyway weare not interested inthese phenomena, and willsoon

lose those phases from the expression. Intereferen e patterns were observed

by Oshimaetal. in2001 [OMK

+

02℄; alater experiment[CISO04 ℄ measured

the oheren e lenght basedon eld emissionpatterns.

In the ase of eld emission, the right region is va uum. Using this fa t

it an be seen that

j(t) ≈ −

2e

_~

X

k,q

X

k

′

_,q

′

Z

t

−T

dt

′

Im

T

∗

kq

T

k

′

_q

′

i~

·

·

Z

_+∞

−∞

dω

2π

Z

_+∞

−∞

dω

′

2π

A

R,qq

′

(ω)A

∗

L,kk

′

(ω

′

)f

_L

−

(ω

′

)e

+iω

′

_(t−t

′

₎

e

−iω(t−t

′

)

.

(1.38) Extending the

t

′

integration to

(−∞, +∞)

and performingit shows that

j ≈ −

2e

_~

X

k,q

X

k

′

_,q

′

Im

T

∗

kq

T

k

′

_q

′

i~

Z

_+∞

−∞

dω

2π

A

R,qq

′

(ω)A

∗

L,kk

′(ω)f

L

−

(ω),

(1.39)

whi h isthe generalizedformal resultfor eld emission's singleparti le

ur-rent. In the next se tion we will take

A

R

to be the spe tral fun tion of a free parti le. This approximation isfully justied by the shapesand sizes of

the apparati ommonlyused in eld emission experiments. However a more

pre ise des ription of the parti le dynami s ould be interesting, sin e the

ele tri eld extends over several entimeters before the ele tron rea hes a

onstant potential region. Moreover the possibility to redene

A

R

an be used to take into a ount spe tra distortiondue, for example, tothe

resolu-tion of the experimentalapparatus orthermalbroadening.

1.4 Current spe trum for normal metals

We an nowderive the Fowler-Nordheimspe trum withinthe tunneling

ap-proa h. The rst observation isthat both in

L

and in

R

we have

A

p,p

′

= δ

_p,p

′

A

_p

,

(1.40)

for every impulse

p, p

′

, sin e the free hamiltonian

H

0

onserves impulse. A tually,this property is true for any

α

and

β

set of eigenstates of

H

0

. The reason for whi h we have not assumed it earlier is that we still use ele tron

wave ve tor and spin as quantum number forthe super ondu ting ase, but

then they will not be onserved, sin e pure ele trons are not eigenstates of

the hamiltonian.

Furthermore,for the same reason,

A

L/R

fun tions are real, then

j =

2e

~

2

X

k,q

|T

kq

|

2

Z

_+∞

−∞

dω

2π

A

R,q

(ω)A

L,k

(ω)f

−

L

(ω).

(1.41)

j =

2e

~

2

X

k,q

⊥

|T

0

|

2

k

⊥

q

⊥

e

d

−1

_ξ

k⊥

Z dω

2π

A

R,q

(ω)A

L,k

(ω)f

−

L

(ω) =

=

2e

~

2

|T

0

|

2

(2π)

4

m

2

~

4

Z

dφ

k

dk

2

k

2

dξ

k

⊥

dξ

q

⊥

dω

2π

A

R,q

(ω)A

L,k

(ω)f

−

L

(ω)e

ξk

_⊥

d

=

=

2e

~

2

|T

0

|

2

(2π)

4

2πm

3

~

6

Z

dξ

k

dξ

q

dω

2π

A

R,q

(ω)A

L,k

(ω)f

−

L

(ω)

Z

ξ

k

0

dξ

k

⊥

e

ξk

_⊥

d

.

(1.42)

Nowwe restorethe

δ

fun tion of (1.17)that weleft impli it. Thismeans removingthe integralover

ξ

q

. Theenergy distributionofthe urrentdensity be omes

∂j

∂ξ

q

=

2e

~

2

|T

0

|

2

(2π)

4

2πm

3

_d

~

6

Z

dξ

k

dω

2π

A

R,q

(ω)A

L,k

(ω)f

−

L

(ω)



e

ξk

d

− 1



,

(1.43)

the

−1

being usually negle ted.

Asweanti ipated, nowweassumethatin

R

the statesare thoseofafree parti le in onstant potential

V

. This means that the spe tral fun tions

A

R

are

A

R,q

(ω) = 2π~ · δ (~ω − ξ

q

− eV ) ,

(1.44) where

V

is the potentialoutside the interfa e. The urrent density is then

∂j

∂ξ

q

=

2e

~

2

|T

0

|

2

(2π)

4

2πm

3

_d

~

6

Z

dξ

k

A

L,k

 ξ

q

+ eV

~



f

_L

−

 ξ

q

+ eV

~



e

ξk

d

.

(1.45)

Wenallyuse theknowledgethat

L

isametal, anduse the sameformof the spe tral desity fun tion for

A

L

A

L,k

(ω) = 2π~ · δ (~ω − ξ

k

) ,

(1.46) where

ǫ

F

is the Fermi energy in the metal. The nal form for the urrent spe trum is

∂j

∂ξ

q

=

2e

~

2

|T

0

|

2

(2π)

4

2πm

3

_d

~

6

Z

dξ

k

· 2π~ · δ (ξ

q

+ eV − ξ

k

) f

L

−

 ξ

q

+ eV

~



e

ξk

d

=

=

2e

~

2

|T

0

|

2

(2π)

4

2πm

3

_d

~

6

· 2π~ · f

−

L

 ξ

q

+ eV

~



e

ξq +eV

d

=

=

2e

h

3

2πmd

4

· e

−c

_{· f}

−

L

 ξ

q

+ eV

~



e

ξq +eV

d

(1.47)

write

f

−

L

as

f

_L

−

(ω) =

1

e

~

ω−ǫF

kT

+ 1

,

(1.48)

where

ǫ

F

isthe Fermienergy. Then we get

∂j

∂ξ

q

=

2πmed

h

3

· e

−c

·

e

ξq +eV

d

e

ξq +eV −ǫF

kT

+ 1

.

(1.49)

Wehave multipliedby 2to a ount fo spin degenera y.

Dening the unit of urrent per surfa e area

j

0

as

j

0

=

2πmed

2

h

3

· e

−c+

ǫF

d

,

(1.50) we an write

∂j

∂ξ

q

=

j

0

d

f (ξ

q

+ eV )e

ξq+eV −ǫF

d

,

(1.51)

f

being the Fermi-Dira fun tion relative to the Fermi level. This form of the urrent spe trumwas found by Fowler and Nordheim[FN28℄.

Firstof allweimmediatelynoti eone ondition forthe above formulato

be valid, whi h is that the dierential urrent is integrable, whi h, for this

parti ular formula, equivalent torequiring that itvanishes athigh energies.

This givesthe onstraint that

d > kT

.

The spe tral urrent isplottedhere inunits of

j

0

d

;the energies are repre-sented inunits of

d

.

Realisti values of

T

are around

d/300

, while others are provided for illustratethebehaviourofthe urrent. Lowertemperaturesarealmost

indis-tinguishable from the

kT = d/300

ase in the s ale used. A tually,

d

values estimated for ommonmetalsvaryinthe range

0.1eV . d . 0.25eV

,sothat the Tvalues we will onsider (

4

◦

_K

-

10

◦

_K

) are approximately

d/300k

. One annoti ethathavingassumeda

δ

-likebehaviourforallthespe tral densities, we anobtainother asesofdierent

A

L/R

exe utinga onvolution of the required fun tion with the above result. This simple fa t allows us to

take are of experimental limits in resolution and spe trum distortion and

broadening due to ollisionee ts.

On a side note, aslightly more pre ise formula for the urrent spe trum

would be

∂j

∂ξ

q

=

j

0

e

−

ǫF

d

d

e

ξq +eV

d

− 1

e

ξq +eV −ǫF

kT

+ 1

θ (ξ

q

+ eV ) ,

(1.52)

0

0.2

0.4

0.6

0.8

1

−4d

−2d

0

2d

4d

∂

j

∂

ξ

q

·

d

_j

0

ξ

q

+ eV − ǫ

f

kT = d/300

kT = d/20

kT = d/10

kT = d/5

Figure1.3: Theeldemissionspe trumfornormalmetalforvarious

temper-atures. Otherparametersare hosenaroundreasonable experimentalvalues.

where the

θ

a ountsfor keepingthe pro esses overthe bottom ofthe band. Missing su htermwould reateaninversionpointforthe urrentundervery

high elds, a behaviour of no physi al meaning. The need for this theta is

usually overlooked, solving the inversion by negle ting the negative term in

the urrent and givinga slightly overestimated result.

1.5 Emission from Niobium

The1998experimentineldemissionfromsuper ondu torshasbeen arried

on niobium, sin e it has the highest riti al temperature for the

super on-du ting transition. Fowler-Nordheim total energy distribution was veried

in a preparatory experiment [NYY

+

98℄ on Nb at high temperatures, using

the same experimentalsetup of subsequent low temperature experiments.

Theparametersoftheinterfa e

c

and

d

,dependingonthe appliedeld

F

andworkfun tion

φ

,werettedfromrawdata. Fittedvalueswere[YSOO02℄

c ≈ 15.2

and

d ≈ 0.19eV

, whi h gives

j

0

≈ 8.4 · 10

−13

A

/

nm

2

. These willbe

the values used in the next hapter. If we onsider their form to be the

one derived by Fowler and Nordheim [FN28℄ (see equations (A.18) (A.19)),

we get

F ≈ 3.5

Vnm

−1

and

φ ≈ 4.36

eV. A simple estimate tells that we must expe t typi al urrents to be of the order of

µ

A for tip radius around

100

nm likethe one that was used in the a tual experiment. The theoreti al spe trum resulting from these values in the ase of normal metal is plotted

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

−5.2

−5

−4.8 −4.6 −4.4 −4.2

−4

−3.8 −3.6

∂

j

∂

ξ

q

·

e

ǫ

F

/

d

d

ξ

q

(eV)

Figure 1.4: The eld emission spe trum for Nb in normal phase, using the

values of the relevant quantities as spe ied in the text. The temperature

used is

15

0

K.

in Figure1.4.

Deviations fromthe predi ted distribution were found slightly above the

Fermi level. The best explaination was found to be that of hot-hole

as- ade, by Gadzuk and Plummer [GP71℄, although later experiments seemed

to require further resear h. Another deviation wasfound fromthe standard

Fowler-Nordheimresult, atlowenergies. This waslater demonstrated tobe

Field Emission from

Super ondu tors: Single Parti le

Pro esses

Beforeweapplythemethodshownintheprevious haptertothe aseofeld

emission from super ondu tors, we introdu e the BCS theory of

super on-du tivitydeveloped by Bardeen, CooperandS hrieer in1957[BCS57℄. We

willuseittodes ribethedynami softheparti lesinsidethesuper ondu tor.

2.1 BCS theory of super ondu tivity

Wewilldes ribethesuper ondu tingmaterialwiththestandardBCStheory.

As in the previous hapter, we do not are about border ee ts due to the

fa t that the material does not ll the entire spa e, and use the full BCS

hamiltonian. The BCShamiltonian is [BCS57 ℄

H

BCS

=

X

k,σ

ǫ

k,σ

c

†

k,σ

c

k,σ

+

X

k,k

′

V

k,k

′

c

†

k,↑

c

†

−k,↓

c

−k

′

_,↓

c

_k

′

_,↑

.

(2.1)

where

c

k,σ

istheannihilationoperatorof anele tronofimpulse

k

andspin

σ

, and

ǫ

k,σ

= ξ

k

− ǫ

F

isthefreeenergy ofanele tron inthe samestate, relative to the Fermi energy.

We apply the mean eld approximationusing as the order parameter

∆

k

=

X

k

′

V

k,k

′

D

c

_−k,↓

c

_k,↑

E

,

(2.2)

H

BCS

=

X

k



c

†

_k,↑

, c

_−k,↓

  ǫ

k

∆

k

∆

∗

k

−ǫ

k

 c

_k,↑

c

†

_−k,↓

!

,

(2.3)

apartfromglobaladditional onstants. TheBCShamiltonianisdiagonalized

with the help of aBogoliubov rotation

c

_k,↑

c

†

_−k,↓

!

=



u

k

v

k

−v

∗

k

u

k

 γ

_k,↑

γ

_−k,↓

†

!

,

(2.4)

where

γ

operator destroy the resonant ex itationof the super ondi tor, and arrythesamequantumnumbersastheele trons. Thissamerotation anbe

written for both spin up and spin down ele trons. Leavingthe spin number

σ

undened we an write

c

_k,σ

c

†

_−k,−σ

!

=



u

k,σ

v

k,σ

−v

∗

k,σ

u

k,σ



γ

_k,σ

γ

_−k,−σ

†

!

,

(2.5)

wherewehaveposed learly

↑= − ↓

. Inthis ase,however, the oe ientsof the twomatri es(

σ =↑

and

σ =↓

)arenot indipendent. Thedependen y an beexpli itedapplying

σ

x

tobothsidesand takingthehermitean onjugated, whi hgives

u

_−k,−σ

= u

k,σ

,

v

−k,−σ

= v

k,σ

∗

.

(2.6)

The hamiltonian is nowdiagonalof the form

H =

X

k



γ

_k,↑

†

, γ

_−k,↓

  E

k

0

0

_−E

k

 γ

_k,↑

γ

_−k,↓

†

!

.

(2.7) where

E

k

=

q

ǫ

2

k

+ |∆

k

|

2

,

(2.8) and

|u

k,↑

|

2

=

1

2



1 +

ǫ

k

E

k



(2.9)

|v

k,↑

|

2

=

1

2



1 −

ǫ

k

E

k



,

(2.10)

u

_k,↑

v

_k,↑

|u

k,↑

v

k,↑

|

=

∆

k

|∆

k

|

.

(2.11)

Westillhavetodenethethermalstatebeforewestartanalyzingthemodel.

It is taken tobethe standard density matrix

ρ

T

= e

−

H

BCS

kT

,

(2.12)

at temperature

T

.

As we havepreviously seen we willneed toknow the Green fun tions in

the super ondu tor. There are two indipendent Greenfun tions, whi h are

D

c

†

_k

′

_,σ

′

(t

′

)c

_k

′′

_,σ

′′

(t

′′

)

E

,

D

c

_k

′

_,σ

′

(t

′

)c

_k

′′

_,σ

′′(t

′′

)

E

.

(2.13)

The rst typeof Green fun tions we willneed is

D

c

†

_k

′

_,σ

′

(t

′

)c

_k

′′

_,σ

′′(t

′′

)

E

= u

∗

_k

′

_,σ

′

u

k

′′

_,σ

′′

e

iE

k′

t

′

e

−iE

−k′′

t

′′

D

γ

_k

†

′

_,σ

′

γ

_k

′′

_,σ

′′

E

+

+ v

_k

∗

′′

_,σ

′′v

k

′

_,σ

′

e

iE

k′′

t

′′

e

−iE

−k′

t

′

D

γ

_−k

′

_,−σ

′

γ

_−k

†

′′

_,−σ

′′

E

.

(2.14) The

γ

ex itations are normal fermions, and as su h follow normal Fermi statisti . Hen e

D

γ

_k

†

′

_,σ

′

γ

_k

′′

_,σ

′′

E

= δ

k

′

_,k

′′

δ

_σ

′

_,σ

′′

f (E

_k

′′

),

(2.15)

D

γ

_k

′

_,σ

′

γ

_k

†

′′

_,σ

′′

E

= δ

k

′

_,k

′′

δ

_σ

′

_,σ

′′

f (−E

_k

′′

),

(2.16)

where

f

is the fermi distribution in the medium, and depends on tempera-ture and Fermi energy of the state. Usingthe previous expressions and the

symmetry properties of

u

,

v

and

E

,one an obtain

D

c

†

_k

′

_,σ

′

(t

′

)c

_k

′′

_,σ

′′(t

′′

)

E

=

Z

dω A

S

(k

′

, k

′′

, σ

′

, σ

′′

, ω)e

iω(t

′

_−t

′′

₎

,

(2.17)

where

A

S

isthe spe tral densityfun tion ofele trons inthe super ondu tor, representing a superposition of parti le-likeand hole-likequasiparti le.

A

S

(k

′

, k

′′

, σ

′

, σ

′′

, ω) = δ

k

′

_,k

′′

δ

_σ

′

_,σ

′′

f (~ω)·

·



|u

k

′

_,σ

′

|

2

δ



ω −

E

_~

k

′



+ |v

k

′

_,σ

′

|

2

δ



ω +

E

k

′

~



.

(2.18)

Fa toring the

δ

fun tions, the spe tral density is found depending only on the quantity

E

k

′

,

∆

k

′

A

S

(k

′

, k

′′

, σ

′

, σ

′′

, ω) = δ

k

′

_,k

′′

δ

_σ

′

_,σ

′′

A

_S

(∆

_k

′

, E

_k

′

, ω),

(2.19) having posed

A

S

(∆

k

′

, E

_k

′

, ω) = 2πf (~ω)



|u

k

′

|

2

δ



ω −

E

_~

k

′



+ |v

k

′

|

2

δ



ω +

E

k

′

~



.

(2.20)

In the same way wend the anomalous Greenfun tions

c

k

′

_,σ

′

(t

′

)c

_k

′′

_,σ

′′

(t

′′

)



= u

_k

′

_,σ

′

v

_k

′′

_,σ

′′

e

−iE

k′

t

′

e

iE

−k′′

t

′′

D

γ

_k

′

_,σ

′

γ

_−k

†

′′

_,−σ

′′

E

+

+ u

k

′′

_,σ

′′

v

_k

′

_,σ

′

e

−iE

k′′

t

′′

e

iE

−k′

t

′

D

γ

_−k

†

′

_,−σ

′

γ

_k

′′

_,σ

′′

E

.

(2.21) Usingagain Fermistatiti sagain for

γ

ex itationsand

u

,

v

,

E

propertieswe have

c

k

′

_,σ

′

(t

′

)c

_k

′′

_,σ

′′

(t

′′

)



=

Z

dω C(k

′

, k

′′

, σ

′

, σ

′′

, ω)e

iω(t

′

−t

′′

)

,

(2.22)

having dened the Cooperspe tral density fun tion

C(k

′

, k

′′

, σ

′

, σ

′′

, ω) = u

k

′

_,σ

′

v

_k

′

_,σ

′

δ

_k

′

_,−k

′′

δ

_σ

′

_,−σ

′′

f (~ω) [δ(ω − E

_k

) − δ(ω + E

_k

)] .

(2.23)

We anagainfa torizethe

δ

fun tions,withtheresultingfun tiondepending, this time,also onthe sign of the spin

σ

′

:

C(k

′

, k

′′

, σ

′

, σ

′′

, ω) = δ

k

′

_,−k

′′

δ

_σ

′

_,−σ

′′

C(∆

_k

′

, E

_k

′

, σ

′

, ω),

(2.24)

moreover, itis odd asa fun tion of

σ

′

C(∆

k

′

, E

_k

′

, σ

′

, ω) = σ

′

C(∆

_k

′

, E

_k

′

, ω).

(2.25)

2.2 Super ondu tor's urrent spe trum

With the above denitions we are now able to derive the urrent spe trum

in eld emission from a super ondu tor in the rst order approximation in

|T |

2

. To do this we use the form of the spe trum derived in the previous

level

A

L,k

(ω) = A

S



∆

k

,

p

(ξ

k

− ǫ

F

)

2

+ |∆

k

|

2

, ω − ω

F



,

(2.26) by posing

~

ω

F

= ǫ

F

we an obtain

∂j

∂ξ

q

=

πemd

~

4

e

−c

Z

_+∞

0

dξ

k

f (ξ

q

+ eV − ǫ

F

) e

ξk

d

·

· A

S



∆

k

,

p

(ξ

k

− ǫ

F

)

2

+ |∆

k

|

2

,

ξ

q

+ eV − ǫ

F

~



,

(2.27) where by hangingthe integration variableot

ǫ

k

= ξ

k

− ǫ

F

and substituting the formula(2.20) for

A

S

,wehave

∂j

∂ξ

q

=

πemd

~

4

e

−c+

ǫF

d

Z

+∞

−ǫ

F

dǫ

k

2π~f (ξ

q

+ eV − ǫ

F

) ·

·



|u

k

|

2

δ (ξ

q

+ eV − ǫ

F

− E

k

) + |v

k

|

2

δ (ξ

q

+ eV − ǫ

F

+ E

k

)



e

ǫk

d

.

(2.28) After substitution of the expressions for

u

k

and

v

k

, we get

∂j

∂ξ

q

=

2π

2

_emd

~

3

e

−c+

ǫF

d

Z

+∞

−ǫ

F

dǫ

k

1

2

f (ξ

q

+ eV − ǫ

F

) e

ǫk

d

·

·



1 +

ǫ

k

E

k



δ (ξ

q

+ eV − ǫ

F

− E

k

) +



1 −

_E

ǫ

k

k



δ (ξ

q

+ eV − ǫ

F

+ E

k

)



,

(2.29)

then we let

ǫ

F

→ ∞

. This an be done bea ause in the previous equationit represents the Fermi energy relative tothe bottom of the band and there is

no invarian e by redening the zero of the energy. Then

ǫ

F

≈ 11.2 ≫ d, kT

(in Niobium),and we have

∂j

∂ξ

q

=

j

0

4d

Z

+∞

0

dǫ

k

f (ξ

q

+ eV − ǫ

F

) ·

·



1 +

ǫ

k

E

k



e

ǫk

d

+



1 −

_E

ǫ

k

k



e

−

ǫk

d



δ (ξ

q

+ eV − ǫ

F

− E

k

) +

+



1 −

_E

ǫ

k

k



e

ǫk

d

+



1 +

ǫ

k

E

k



e

−

ǫk

d



δ (ξ

q

+ eV − ǫ

F

+ E

k

)



.

(2.30) This is simpliedgreatly with the help of the usual

δ

property

δ[f (x)] =

X

x

i

|f(x

i

)=0

δ(x

i

)

∂j

∂ξ

q

=

j

0

2d

Z

_+∞

0

|E

k

|dǫ

k

|ǫ

k

|

f (ξ

q

+ eV − ǫ

F

) ·

·



1 +

ǫ

k

E

k



e

ǫk

d

+



1 −

ǫ

k

E

k



e

−

ǫk

d



δ



ǫ

k

−

q

(ξ

q

+ eV − ǫ

F

)

2

− ∆

2



,

(2.32) whi h, alling

ξ

′

_{= ξ}

q

+ eV − ǫ

F

, is integrated to give the nal result

∂j

∂ξ

q

=

j

0

d

f (ξ

′

₎











ξ

′

p

ξ

′2

_{− ∆}

2











·

·

"

cosh

p

ξ

′2

_{− ∆}

2

d

!

+

p

ξ

′2

_{− ∆}

2

ξ

′

sinh

p

ξ

′2

_{− ∆}

2

d

!#

,

(2.33)

whi histheexpressionderivedbyGadzuk. Therequirementthattheenergies

areoverthebottomofthebandhasbeenleftout,but anberestoredtoavoid

the non-signi ativedivergen e atenergies

ξ

′

_{→ −∞}

. The urrentspe trum

is plotted in Figure 2.1, where we an see that near the Fermi level there

is an in rease of the urrent per unit of energy, above the maximum value

rea hed by the spe trum of tha normal metal ase (dashed). The spe trum

divergesunlessone onsidersee tswhi hwouldsmooththeresultingenergy

distribution, su h as relaxation ee ts for the quasiparti les. In Figure 2.2

we plotted the spe trum of the super ondu ting state in the range near the

Fermi level, ompared to the normalmetal ase.

Themainfeaturesofthisresultarethesuppressionof urrentintherange

ofenergies

(−∆, ∆)

andthepresen eoftwospikesaroundthesuppresionarea with heightproportionalto

∆

. Thephysi almeaningis easilyrelatedtothe intrinsi hara teristi softhe super ondu tingstate. Infa twe ansee that

the rangeof energies forwhi hthe spe trumvanishesis entered aroundthe

Fermi level withwidth equaltothe gapin theenergy-momentumdispersion

relation of quasiparti les in the super ondu tor. This range is the one in

whi h no fermioni ex itation an live inside

R

. The orresponding states are moved tothebordersofthe suppressionrange providingthe (integrable)

spikes.

As we have remarked before, the transfer matrix elements an be

ex-pressed asfun tionsof theparti leafter thetunneling. Thetwoenergies are

not independentsin eanimportantassumptionwhilederivingthe matrix