Title

Cou

lom

b B

lock

ade

as a

Pro

be o

fIn

tera

ctio

ns in

Nan

ostr

uctu

res

Har

old

U. B

aran

ger,

Duk

e U

nive

rsity

Coh

eren

ce:

quan

tum

-mec

hani

cal

inte

rfer

ence

of e

lect

rons

Cor

rela

tion:

caus

ed b

y el

ectro

n-el

ectro

n in

tera

ctio

n

inte

rpla

y be

twee

n e-

e in

tera

ctio

n &

qua

ntum

inte

rfer

ence

[cla

ssic

con

text

: dis

orde

red

syst

ems a

nd m

etal

-insu

lato

r tra

nsiti

on]

extre

me:

cry

stal

mor

e ty

pica

l: liq

uid

Q.D

ots

Exam

ple

#1:

Qua

ntum

Dot

s

Com

p. T

ech.

[C. M

arcu

s gro

up, H

arva

rd]

conf

inem

ent

i

nter

fere

nce

dens

ity w

iggl

es

in

tera

ctio

ns→

→→

⇒in

terf

eren

ce c

ontri

butio

n to

the

inte

ract

ion

ener

gy

expe

rimen

t: se

nsiti

vity

to sm

all p

aram

eter

cha

nge

(B, V

g, …

)H

ow b

ig?

Doe

s sha

pe m

atte

r?

Exam

ple

#2:T

rans

port

Thr

ough

Sin

gle

Mol

ecul

es (1

)

Sing

le m

olec

ules

mol

ecul

ele

ad 1

lead

2

VI•C

ondu

ctan

ce?

•I-V

curv

e?•e

-e in

tera

ctio

ns?

•Vib

ratio

ns?

•Dev

ices

?

Tran

spor

t Thr

ough

Sin

gle

Mol

ecul

es (2

)

Mol

ec E

lect

r

Com

p. T

ech.

[D. R

alph

and

P.M

cEue

ngr

oups

, Cor

nell]

Exa

mpl

e #

3:N

ano

–m

agne

tism

, sup

erco

nduc

tivity

, … (1

)

Nan

o-m

ag.,

supe

r.,•E

lect

ron-

elec

tron

inte

ract

ions

cau

se c

erta

in c

olle

ctiv

eph

enom

ena

whe

n su

ffic

ient

ly st

rong

•Doe

s qua

ntum

inte

rfer

ence

influ

ence

thes

e co

llect

ive

effe

cts?

(mag

nitu

de?,

spat

ial d

istri

butio

n?, b

reak

dow

n?, …

)

M

idea

l

M

real

Doe

s in

terfe

renc

e af

fect

the

dire

ctio

n of

the

mag

netiz

atio

nin

sm

all f

erro

mag

nets

?

Nan

o –

mag

netis

m, s

uper

cond

uctiv

ity, …

(2)

Nan

o-m

ag.,

supe

r.,In

terp

lay

of s

uper

cond

uctiv

ity a

nd d

isor

der

Supe

rcon

duct

ivity

cha

ract

eriz

ed b

y a

loca

l ord

er p

aram

eter

, the

gap

:

↑↓

∆r

rc

cr

~)(

For w

eak

diso

rder

and

wea

k co

uplin

g, ∆

(r) i

s ind

epen

dent

of r

But

, if d

isor

der i

s stro

ng, i

nter

fere

nce

dest

roys

supe

rcon

duct

ivity

the

supe

rcon

duct

or –

insu

lato

r tra

nsiti

onca

lcul

atio

n of

∆(r

):sc

hem

atic

:

[Gho

sal,

Triv

edi,

and

Ran

dier

i]

OU

TLIN

E

Out

line

-mas

ter

Com

p. T

ech.

Intr

oduc

tion:

exam

ples

of i

nter

play

of i

nter

actio

ns a

nd in

terfe

renc

eW

e ne

ed a

tool

th

e C

oulo

mb

bloc

kade

and

how

it c

hang

es

Goa

ls: 1

. Sur

vey

the

exce

llent

phy

sics

in th

e ar

ea o

f CB

2. P

repa

re y

ou to

be

able

to u

se th

is to

ol in

you

r ow

n w

ork

1.B

asic

Cou

lom

b B

lock

ade

•con

duct

ance

thro

ugh

a ne

arly

isol

ated

sys

tem

2.M

esos

copi

c Ef

fect

s in

the

Cou

lom

b B

lock

ade

•flu

ctua

tions

of C

B p

eak

heig

hts

and

spac

ings

3.K

ondo

Effe

ct in

Qua

ntum

Dot

s

OU

TLIN

E

Out

line

-loc

alB

asic

Cou

lom

b B

lock

ade

Com

p. T

ech.

I. O

verv

iew

of t

he p

heno

men

a –

no q

uant

um in

terfe

renc

e ef

fect

s!

II. M

agni

tude

of t

he c

ondu

ctan

ce

a)G

eom

etry

, 1 ju

nctio

n, c

harg

e st

eps

b)2

junc

tions

c)C

harg

e pu

mp

d)N

onlin

ear t

rans

port

e)S

uper

cond

uctin

g gr

ains

a)Th

erm

ally

act

ivat

ed re

gim

e –

rate

equ

atio

nsb)

Inel

astic

cot

unne

ling

c)E

last

ic c

otun

nelin

g

III.

Qua

ntum

fluc

tuat

ion

of th

e ch

arge

on

a gr

ain

Mai

n So

urce

s

Sour

ces

Com

p. T

ech.

•“El

ectro

n Tr

ansp

ort i

n Q

uant

um D

ots”

, L. P

. Kou

wen

hove

n, e

t al.

in

“Mes

osco

pic

Elec

tron

Tran

spor

t” b

y So

hn, K

ouw

enho

ven,

& S

chon

, 199

7.•“

Sing

le E

lect

ron

Phen

omen

a in

Met

allic

Nan

ostru

ctur

es”,

M. H

. Dev

oret

, D. E

stev

e, &

C

. Urb

ina,

in “

Mes

osco

pic

Qua

ntum

Phy

sics

”,A

kker

man

s, et

al.

Les H

ouch

es, 1

997.

•“Lo

w-T

empe

ratu

re T

rans

port

Thro

ugh

a Q

uant

um D

ot”,

L. I

. Gla

zman

& M

.Pus

tilni

k, c

ond-

mat

/05

•“Sp

ectro

scop

y of

Dis

cret

e En

ergy

Lev

els i

n U

ltras

mal

l Met

allic

Gra

ins”

, J. v

on D

elft

and

D. C

. Ral

ph, P

hys.

Rep

. 345

, 61-

173

(200

1).

•“Fe

w-e

lect

ron

Qua

ntum

Dot

s”, L

. P. K

ouw

enho

ven,

D. G

. Aus

ting,

and

S. T

aruc

ha,

Rep

. Pro

g. P

hys.

64, 7

01-7

36 (2

001)

.•“

Mes

osco

pic

Fluc

tuat

ions

of C

otun

nelin

g an

d K

ondo

Eff

ect i

n Q

uant

um D

ots”

, L.

I. G

lazm

an, i

n N

ATO

ASI

boo

k ed

ited

by K

ulik

, et a

l., 2

001.

•“Q

uant

um S

mea

ring

of th

e C

oulo

mb

Blo

ckad

e”, K

. A. M

atve

ev, i

n N

ATO

ASI

boo

k ed

ited

byK

ulik

, et a

l., 2

001.

•“Th

e St

atis

tical

The

ory

of Q

uant

um D

ots”

, Y. A

lhas

sid,

Rev

Mod

Phy

s. 72

, 895

-968

(200

0)•“

Theo

ry o

f Cou

lom

b-B

lock

ade

Osc

illat

ions

in th

e C

ondu

ctan

ce o

f a Q

uant

um D

ot”,

C

. W. J

. Bee

nakk

er, P

RB

44,

164

6 (1

991)

.•“

Qua

ntum

Eff

ects

in th

e C

oulo

mb

Blo

ckad

e”, I

. L. A

lein

er, P

. W. B

rouw

er, a

nd

L. I.

Gla

zman

, Phy

s. R

ep. 3

58, 3

09-4

40 (2

002)

.•M

esos

copi

c ef

fect

s in

the

Cou

lom

b B

lock

ade,

G. U

saj,

D. U

llmo,

HU

B;

Phys

. Rev

. B 6

4, 2

0131

9(R

); 64

, 245

324;

66,

155

333;

67,

121

308(

R) (

2001

-200

3).

•“Se

mic

lass

ical

Den

sity

Fun

ctio

nal T

heor

y: S

trutin

sky…

”, D

. Ullm

o &

HU

B, P

RB

63,

125

339

(01)

.

Geo

met

ry a

nd C

harg

ing

Geo

met

ry

Com

p. T

ech.

2ne

utra

l

2

)(

2Q

QCe

E−

=

eU

CQ

/g

neut

ral=

1-ju

nctio

n gr

ain

Isla

nd w

ith 1

junc

tion

Com

p. T

ech.

Gra

in c

onne

cted

to o

ne tu

nnel

junc

tion

and

one

“gat

e” c

apac

itor

Ave

rage

cha

rge

incr

ease

s in

st

eps

as a

func

tion

of v

olta

ge

appl

ied

to th

e ca

paci

tor

How

acc

urat

ely

is th

e ch

arge

qu

antiz

ed?

Is c

ondu

ctio

n th

roug

h th

e gr

ain

poss

ible

?

[Fro

m D

evor

et, E

stev

e, &

Urb

ina,

95]

2 ju

nctio

ns (1

)2

Junc

tions

: Con

duct

ion!

Com

p. T

ech.

V d

rives

cur

rent

U s

ets

the

neut

ralit

y po

int

(gat

e vo

ltage

)

Set

U s

o th

at s

tate

s (1

) and

(0)

have

the

sam

e en

ergy

cu

rrent

flow

s!

[Fro

m D

evor

et, E

stev

e, &

Urb

ina,

95]

Con

duct

ion

Thro

ugh

a C

B Is

land

(con

tinue

d)

2 ju

nctio

ns (2

)

Com

p. T

ech.

2ne

utra

l

2

)(

2Q

QCe

E−

=

•wea

kly

coup

led

lead

s: Q

jum

ps sh

arpl

y•Q

jum

ps a

nd G

has p

eak

whe

n:

Sem

icon

duct

or C

oulo

mb

Blo

ckad

e St

ruct

ures

Dat

a 1-

sc st

ruct

Com

p. T

ech.

Dat

a: L

ater

al Q

uant

um D

ot

Dat

a 2

-lat

eral

Com

p. T

ech. [C

. Mar

cus g

roup

, Har

vard

]

Dat

a: V

ertic

al Q

uant

um D

ot

Dat

a 3

-ver

tical

[Tar

ucha

& K

ouw

enho

ven

grou

ps, N

TT &

Del

ft]

Tran

sfer

ring

Elec

tron

s 1-

by-1

: A P

ump

Cha

rge

pum

p

Com

p. T

ech.

2 is

land

s (3

junc

tions

) req

uire

d:Tr

ansf

er s

eque

nce:

Rel

atio

n of

cha

rge

stat

es fo

r re

gula

r con

duct

ion

[Fro

m D

evor

et, E

stev

e, &

Urb

ina,

95]

Cha

rge

Pum

ping

: Dat

a

Pum

p da

ta

Com

p. T

ech.

[Sac

lay

grou

p]

Plat

eau

trans

ferri

ng e

lect

rons

1-b

y-1

Not

e lin

ear d

epen

denc

e of

pla

teau

cur

rent

on

frequ

ency

!

True

pum

ping

at V

=0C

urre

nt w

ith n

o dr

ivin

g vo

ltage

(but

mod

ulat

ion

volta

ges

doin

g w

ork…

)[F

rom

Dev

oret

, Est

eve,

& U

rbin

a, 9

5]

Cou

lom

b B

lock

ade:

Non

-line

ar T

rans

port

Non

-line

ar: s

chem

.A

pplie

d so

urce

-dra

in

volta

ge o

verc

omes

ch

argi

ng e

nerg

ycu

rrent

Com

p. T

ech.

(a) G

rain

in w

hich

R1,

C1

dom

inat

e. N

ote

step

s in

Ias

mor

e el

ectro

ns c

an b

e lo

caliz

ed o

n th

e gr

ain.

Cou

lom

b St

airc

ase

(b) G

ranu

lar f

ilm –

man

y gr

ains

with

ave

rage

C a

nd R

Firs

t obs

erva

tions

and

dis

ucss

ion

of C

B in

50’

s an

d 60

’s (G

iave

r & J

akle

vic)

[Fro

m G

lazm

an, 0

0]

Non

-line

ar C

B: D

ata

in m

etal

gra

in

Non

-lin:

met

al

Com

p. T

ech.

[D. R

alph

, et a

l. ’9

7]

Cha

ract

eris

tic d

iam

ond-

shap

ed re

gion

of C

B

Non

-line

ar C

B: D

ata

in s

emic

ondu

ctor

qua

ntum

dot

Non

-lin:

sem

ic.

Com

p. T

ech.dI/d

V p

lotte

d vs

. Vg

and

Vsd

[Tar

ucha

and

Kou

wen

hove

n gr

oups

, ‘01

]

Supe

rcon

duct

ing

Gra

in: R

ole

of th

e G

ap

Supe

rcon

d: sc

hem

.

Com

p. T

ech.

Sup

erco

nduc

tivity

rem

inde

rs:

•BC

S g

roun

d st

ate:

ele

ctro

ns p

aire

din

sin

glet

s•I

f the

re a

re a

n od

d nu

mbe

r of

elec

trons

ther

e m

ust b

e at

leas

ton

e qu

asi-p

artic

le o

utsi

de th

e co

nden

sate

•The

min

imum

ene

rgy

for c

reat

ing

aqu

asi-p

artic

le is

the

gap,

∆•∆

can

be

chan

ged

with

by

appl

ying

a m

agne

tic fi

eld

For N

odd

, one

mus

t add

∆

to th

e C

B e

nerg

y pa

rabo

lapa

ss d

irect

ly fr

om

N to

N+2

(fo

r N e

ven

and ∆

>Ec)

[Fro

m D

evor

et, E

stev

e, &

Urb

ina,

95]

Cha

rge

in a

Sup

erco

nduc

ting

Gra

in

Supe

rcon

d: d

ata

Com

p. T

ech.

3rdca

se: ∆

< E

c

N(U

) mea

sure

d w

ith a

CB

ele

ctro

met

er

for d

iffer

ent m

agne

tic fi

elds

--B

eaut

iful v

erifi

catio

n th

at s

uper

cond

ucto

rs re

ally

are

in a

com

plet

ely

paire

d st

ate

–a

mac

rosc

opic

sin

glet

.--

Can

tell

the

parit

y of

a m

etal

gra

in c

onta

inin

g 10

12el

ectro

ns!

[Fro

m D

evor

et, E

stev

e, &

Urb

ina,

95]

A M

odel

Dot

Mod

el H

amTi

me

to b

ecom

e qu

antit

ativ

e!! s

o w

e ne

ed a

mod

el…

Com

p. T

ech.

nla

bels

stat

es in

the

dot (

disc

rete

)k

labe

ls st

ates

in th

e le

ads (

cont

inuo

us)

α=

L, R

(ie

. Lef

t or R

ight

)sl

abel

s spi

n

is a

n op

erat

or

t αn

is in

depe

nden

t of k

–

poin

t con

tact

geo

met

ry

Wha

t is

the

Mag

nitu

de o

f the

Con

duct

ance

?

Mag

G 1

Com

p. T

ech.

Sta

rt at

hig

h T

and

cons

ider

su

r-T

regi

mes

.cc

essi

vely

low

e

∞≡

+=

GG

GG

GG

RL

RL

Hig

hest

T:

CET>>

Cha

rgin

g en

ergy

is n

eglig

ible

com

pare

d to

ther

mal

exc

itatio

ndi

scre

tene

ss o

f ele

ctro

n ch

arge

irre

leva

ntA

dd re

sist

ance

of t

he

two

junc

tions

in s

erie

s:

CET

1stIn

term

edia

te re

gim

e:

≤≈

In a

dditi

on, a

ssum

e:1.

No

quan

tum

mec

hani

cal i

nter

fere

nce

effe

cts

2.R

apid

ther

mal

izat

ion

of e

lect

rons

in th

e do

t –in

cohe

rent

trans

port

(ie. i

nela

stic

rela

xatio

n ra

te >

esc

ape

rate

)3.

Res

ista

nce

of e

ach

junc

tion

is >

e2 /h

Seq

uent

ial t

unne

ling

regi

me,

trea

t with

rate

equ

atio

ns[F

ollo

win

g G

lazm

an`0

0 an

d Pu

stiln

ik&

Gla

zman

`04]

Com

men

ts o

n po

ints

1 a

nd 3

Nee

d G

<e2/

h1.

Neg

lect

qua

ntum

inte

rfere

nce

negl

ect s

ingl

e pa

rticl

e qu

antiz

atio

n(d

iscu

ssed

tom

orro

w!)

T sm

ears

den

sity

of s

tate

s tre

at s

pect

rum

as

cont

inuo

usw

ith D

OS

ν

3. N

eed

for G

L,GR

<< e

2 /h : C

omp.

Tec

h.Tu

nnel

ing

thro

ugh

barr

iers

Qua

ntum

fluc

tuat

ions

of t

he n

umbe

r of e

lect

rons

Sho

uld

be s

mal

l ove

r tim

e of

mea

sure

men

t

RL,

/GC

RCt

==

∆

hG

CC

et

E>

=∆

∆)

/)(

/(

2U

se ti

me-

ener

gyun

certa

inty

rela

tion:

he

G/

2<

Sup

porte

d by

det

aile

d ca

lcul

atio

n as

wel

l…

Rat

e Eq

uatio

ns

Rat

e eq

sFo

r sim

plic

ity, c

onsi

der N

0cl

ose

to th

e de

gene

racy

poi

nt N

0*=1

/2co

nsid

er o

nly

the

char

ge s

tate

s 0

and

1E

lect

rost

atic

ene

rgy

diffe

renc

e:

Cur

rent

from

lead

αin

to th

e do

t via

Fer

mi’s

gol

den

rule

:

Com

p. T

ech.

P i: p

roba

bilit

y fo

r the

dot

to b

e in

cha

rge

stat

e i

f( ε) :

Fer

mi f

unct

ion

(not

per

turb

ed –

rem

embe

r poi

nt 2

abo

ve!)

Con

vert

sum

s to

inte

gral

s ov

er c

ontin

uous

den

sity

of s

tate

s:

2do

t

22

4t

heG

νν

πα

α=

Rat

e Eq

uatio

ns (c

ont.)

Rat

e eq

s (2)

Two

addi

tiona

l con

ditio

ns:

RL

II

−=

11

0=

+P

P

Sol

utio

n yi

elds

for t

he li

near

con

duct

ance

:

Com

p. T

ech.N

ote:

1.A

ll pe

aks

have

sam

e he

ight

!2.

Asy

mpt

ote

0.5

at lo

w

tem

pera

ture

–be

caus

e of

cor

rela

tion

(Can

car

ry o

ut fo

r mut

liple

cha

rge

stat

e an

d no

n-lin

ear b

ehav

ior.)

[Fol

low

ing

Gla

zman

`00,

and

Pus

tilni

k&

Gla

zman

`04]

Bey

ond

Firs

t Ord

er: I

nela

stic

Cot

unne

ling

Inel

astic

cot

unn

Abo

ve: l

owes

t ord

er in

|t|2 ,

invo

lved

tunn

elin

g vi

a a

real

sta

tein

the

dot

Res

ult:

expo

nent

ial s

upre

ssio

n of

Gin

the

CB

val

ley

beca

use

prob

abili

ty

of a

ther

mal

fluc

tuat

ion

to a

llow

occ

upan

cy is

tiny

, exp

(-E C

/T)

But

, hav

e to

con

side

r hig

her-

orde

r act

ivat

ionl

ess

proc

esse

stu

nnel

ing

via

virt

ual s

tate

sin

the

dot

Com

p. T

ech.

Initi

al S

tate

Fina

l Sta

te

*Lnt

RmtM

any

proc

esse

s!

[Fol

low

ing

Gla

zman

`00]

Inel

astic

Cot

unne

ling

(con

t.)

Inel

. cot

unn

(2)

Com

p. T

ech.

Fina

l sta

tes

for t

he m

any

proc

esse

s ar

e di

ffere

nt

sum

pro

babi

litie

s!

∑≈

allo

wed

,

2,

2

in|

|m

nm

nR

LA

heG

νν

1.C

onve

rt su

m o

n n,

m to

inte

gral

in

trodu

ces ν d

ot2

GLG

R

2.H

ow m

any

pairs

of s

tate

s in

the

dot c

ontri

bute

??P

hase

spa

ce a

rgum

ent f

amili

ar fr

om F

erm

i liq

uid

theo

ry:

Inco

min

g en

ergy

is ~

TTh

ere

are

~T 2

way

s to

div

ide

that

ene

rgy

amon

g th

e do

t sta

tes

By

com

parin

g w

ith a

bove

, ine

last

ic c

otun

nelin

g do

min

ates

ther

mal

ly a

ctiv

ated

tra

nspo

rt w

hen

2

/32

2in

⎟⎟ ⎠⎞⎜⎜ ⎝⎛

=C

RL

ETh

eG

GG

[Fol

low

ing

Gla

zman

`00]

Elas

tic C

otun

nelin

g

Elas

tic c

otun

n

Com

p. T

ech.

Wha

t if t

empe

ratu

re is

real

ly v

ery

smal

l, ie

. ver

y fe

w in

elas

tic p

roce

sses

?C

onsi

der e

last

ic p

roce

sses

:E

lect

ron-

like

Hol

e-lik

e

*Lnt

RntFi

nal s

tate

s fo

r the

man

y pr

oces

ses

are

the

sam

e su

m a

mpl

itude

s!

Not

e: e

lect

ron-

like

and

hole

-like

pr

oces

ses

have

diff

eren

t sig

n

[Fol

low

ing

Gla

zman

`00,

and

Pus

tilni

k&

Gla

zman

`04]

Elas

tic C

otun

nelin

g (c

ont.)

Elas

. cot

unn

(2)2 |

|)

(si

gn*

2

el∑

+≈

nn

C

nRn

LnR

LE

tt

heG

εεν

ν

1.K

eep

only

term

s in

whi

ch p

hase

can

cels

exa

ctly

(ie. d

iago

nal a

ppro

xim

atio

n –

we’

ll co

me

back

to th

is to

mor

row

!)2.

Den

omin

ator

cut

s of

f sum

whe

n ε n

~EC

Com

p. T

ech.

∑ <C

nE

nC

RnLn

RL

Et

the

G

||,

22

22

in1

~

ε

νν

a)O

ne s

um

only

one

fact

or o

f νdo

tm

ultip

ly b

yν d

ot/ν

doti

n or

der t

o ar

rive

at fo

rm G

LGR

b)N

umbe

r of t

erm

s in

sum

intro

duce

s fa

ctor

of E

C

⎟⎟ ⎠⎞⎜⎜ ⎝⎛

−+

+−

=0

* 0* 0

0do

t2

2el

111

1/

41N

NN

NE

he

GG

GC

RL

νπ

Elas

.cot

unn

(3)

Elas

ticC

otun

nelin

g(c

ont.)

Com

p. T

ech.

Com

pare

mag

nitu

des

of e

last

ic a

nd in

elas

tic c

otun

nelin

g:

Ela

stic

cot

unne

ling

dom

inat

es a

lread

y w

hen

C

C EE

TT

dot

elν

≡<≈

∆≈

CE

We’

ve s

tudi

ed th

ree

regi

mes

of t

rans

port

so fa

r:

whe

re ∆

is th

e si

ngle

-pa

rticl

e le

vel s

paci

ng

CEC

C EE do

tν

Tin

elas

ticco

tunn

elin

gin

val

ley

elas

ticco

tunn

elin

gin

val

ley

sequ

entia

ltu

nnel

ing:

disc

rete

cha

rges

cont

inuo

usflo

w

[Fol

low

ing

Gla

zman

`00,

and

Pus

tilni

k&

Gla

zman

`04]

Qua

ntum

Tun

nelin

g of

the

Cha

rge

Cha

rge

smea

r 1

Com

p. T

ech.

N0

N)

clas

sica

l

quan

tum

Find

us

ing

our m

odel

Ham

.(o

nly

one

lead

now

!)N)

[Fol

low

ing

Mat

veev

`00]

Qu.

Flu

ctua

tions

of C

harg

e: P

ertu

rbat

ion

Theo

ry

Cha

rge

smea

r 2Fi

nd c

ontri

butio

n of

virt

ual e

xcita

tions

into

and

out

of t

he d

otw

ave

func

tion

and

henc

e <N

>.to

the

grou

nd s

tate

1. E

lect

ron

from

ktu

nnel

s to

n, c

ausi

ng N

N+1

)]0()1(

[E

EtA

nk

nk−

−−

=ε

εn

kε

ε<

<0

Com

p. T

ech.

2. E

lect

ron

from

ntu

nnel

s to

k, c

ausi

ng N

N-1 )]0(

)1(

[E

EtA

kn

kn−

−−

−=

εε

kn

εε

<<

0

Con

vert

both

sum

s to

inte

gral

s:

Loga

rithm

ic d

iver

genc

e at

the

char

ge d

egen

erac

y po

int!!

Qua

ntum

Sm

earin

g of

the

Cha

rge

Step

s

Cha

rge

smea

r 3

Com

p. T

ech.

Div

erge

nce

have

to k

eep

mor

e te

rms!

Mat

veev

acc

ompl

ishe

d by

map

ping

ont

o th

eK

ondo

pro

blem

:

pseu

dosp

infli

ps th

roug

h in

tera

ctio

n w

ith re

serv

oir

real

spi

n st

ill p

rese

nt

2 ch

anne

l Kon

do

char

ge s

tate

0

spin

dow

nch

arge

sta

te 1

sp

in u

p

[Fol

low

ing

Mat

veev

`00]

OU

TLIN

E

Out

line

-loc

alB

asic

Cou

lom

b B

lock

ade

Com

p. T

ech.

I. O

verv

iew

of t

he p

heno

men

a –

no q

uant

um in

terfe

renc

e ef

fect

s!

II. M

agni

tude

of t

he c

ondu

ctan

ce

a)G

eom

etry

, 1 ju

nctio

n, c

harg

e st

eps

b)2

junc

tions

c)C

harg

e pu

mp

d)N

onlin

ear t

rans

port

e)S

uper

cond

uctin

g gr

ains

a)Th

erm

ally

act

ivat

ed re

gim

e –

rate

equ

atio

nsb)

Inel

astic

cot

unne

ling

c)E

last

ic c

otun

nelin

g

III.

Qua

ntum

fluc

tuat

ion

of th

e ch

arge

on

a gr

ain

THE

END

Title

Tem

plat

e

Com

p. T

ech.