Villain approximation for the Coulomb blockade problem in small tunnel junctions

PHYSICA /i, Villain approximation small tunnel junctions A.

for the Coulomb

In small tunnel junctions with capacitances lesh than a few femtofarads. charging energy EL ~~ ~‘!2(’ associated with a single electron can cscced the thermal fluctuation temperature 7‘. In this cast the Coulomb interaction of the electrons blockades tunnclling. In particular. the Coulomb blockade effect has been observed I-Cin

;i linear

array

of

junctions

and

in

;I

[I-J] The charging effect4 occurred due to discreteness of the electron charge. and they reveal themselves through the observation of single electron tunnclling oscillation\; (SET) and the formation of the Coulo~nh gap on I-V characteristic of normal junctions. ‘Fhe problem of electron tunnelling in such small structures W;lS considered by Averin and Likharev [S]. They obtained the master equation for the charge probability function CJ( Q. t) in ;I normal current-biased junction and calculated the I-V characteristics of this junction in various limits for temperature and current. The complications occasioned by shunt resistance [6] as well single

problem

in

Goluh

1. Introduction

cently

blockade

junction

iis I,> ;I resistor III 4t’rics in :I voltage driven junction have been considered in thcorctical works [7-l?] (:\cc’ review) article [ 1.3) for li\t ot rcfercnces). In

thi4

article

formation

to

lhc

like to anal~\c the tunnelling in normal anti Villain tran\1~) applying

k\zc‘ would

problem of clcclron Josephson junction\ part

of

the

action

txyonsihle

single electron lunnelling proccs3. The rcahon for this i\ that such an approxim;ition compared to the semi-classical approach [ 51 providc4 2 simple rcprescntation for gcner-sting functional (Z) or ;I partition function which includes quantum fluctuations in the theory 01 for

the

charging

effects.

The

form

of

% is ximilar

1’01

and Josephson clemcnts. as arc the I--L’ characteristics for small currents. We have reproduced somt known results and obtained ncu OIlC\ a\ hell. concerning the intcrfercnce bytwccn (‘ooper pair and quasiparticle curl-cnt\. We have also applied Villain transformation to the problem of ;I scluarc array of normal tunnel junctions in order to study the phase diagram of this \ystcm. In this case. we considered the model with random tunnclling on differ-ent \ite\. Finall> . although it has some specific prcq~crticx. out- mcthotl for calculating the average \oltagc normal

A. Goluh

/ Viiloin trunsformution

exploits the energy band picture in “quasicharge” space [ 141. Some complications originate from the current term in the effective action, which breaks the cp+ q + 27r symmetry of the partition function. So to preserve the symmetry, we consider I( pie), ( p = 1 /T), as an integer, leaving the analytical continuation to real current to the last step of the calculations.

2. The normal

tunnel junction

Single electron tunnelling is the main dynamic process in normal structures. We restrict ourselves to the current-biased junction and investigate the limit of small currents and large shunt resistance. It has been shown [13, 151 that tunnel elements can be described in terms of a single phase variable by the action on the imaginary time contour

I I dr

0

0

x [cos(cp(r)

-- (p(T’)) - l] ,

(1)

where y(7) = (/Ii! sin(nr//?)))‘. The first term in eq. (1) represents the changing energy due to the junction’s capacitance C, the second term accounts for the interaction with biased current I, and the last term describes the single electron tunnelling. Its strength is characterised by dimensionless tunnelling conductance (Y = hi4e’R = R,,IR. It is convenient to introduce a generating functional. For this purpose we add a source term to s: B

s,(cP>4) = f Here

q(T)

1dT dT)‘P(T) is an

imaginary

(2) time

dependent

95

= q(0).

As a result

Z,( 4(T)) = 1 Dq exp]-S(cp)

~ &(cp,

q)l

(3)

If I = 0 then, obviously, Z( q(T)) possesses the symmetry cp-+ cp + 2nn. When I # 0 the simplest way to preserve this symmetry is to change I+ il and to consider I( pie) as an integer. In our final results we will return to the real current in the same way as in Matsubara’s theory of linear response where we get different susceptibilities. As a next step it is useful to substitute the genuine cosine interaction (cq. (1)) by the periodic Gaussian one:

/dr_/

CXp~a

0

dr’

y(T

-

T’)[COS((p(T)

I)

‘P(T’))

+ i d7’ r(r - T’)

junctions

and q(p)

charge

B

L’

-ff

itI smull tunnd

~

11)

c m,,.(P(T) 77

I

(P(T’)) >

(4)

where mTT. are integer numbers, and C1,,,7,,, denotes the product of sums ll,7. C’,,li7 L’ B,(x) is the Villain function. This transformation is known as the Villain approximation. In the Villain model the parameterization [ 161

(p(T)

=

4(T)

+

y

.

4(B) = dd())

(5)

opens the way for decoupling the continuous and discrete degrees of freedom. The former can be integrated rigorously. Equation (4) needs some comment. Due to two time integrations, after discretezation, we have a double sum s,t_’dr J”,t_’ dr’ . = X,, E? . . (E is the elementary length on the time lattice). Therefore the integer numbers mrT. in the right hand side of eq. (4) depend on two time (T, T’) variables. Such a dependence does not introduce difficulties. However, as we will see later, the

constraints-continuity 9

integration

equations

will

that appear after

rather

ass~inic‘

an

un~15uai

term. .l‘he

remark

next

when

Ilsually

a term have

like

H,

=

Here,

/3.

rate

of the former

another

approximation

a detailed

Let ate

[~B,(.Y)] q.

that eqs. ~21.

we

(3).

then

first the

CY-

the part

I‘hen z,;,

bc

(eq5.

substitute (7,.

ttic

solution

ref.

term

charge

x f

arc appropri-

0.

cvcn

example.

let

in

the

.Y ~=-F.

1.3.

(5 ) into

of Z( y(r))

involve

it as

may

and (4)

over

which

the periodic

he rewritten

variable

C/J(T)

haa the form

Z,:,)

as an integral

Q(T) [I-i].

Villain fi.

tunction

/I\( \-I i5 small

Thcrct’or-c.

the first

( 12) singles out ontl, value\ cnablinp the s~inl OVCI- I?I.. formed. poncntially

right

hand

in cq. (7)

the following

1nt0

[ 171

I‘hc

term

(‘J)

have

and large

The

ot cq.

(S)),

exact

we

(see

;I

I,arameterizatioii

integration

(let us denote

The over

H,(X)

WC of

the

Therefore.

all

For

introduce

the functional field

(6)

for

’ = 7. Then

After

is

to

then

explanation):

us note when

x.

the case

I which

limit. for

approximations

case

and if ,!3 +

(Y <

/I, (_I 1.

is Lrpplictl

we are studying

opposite for

the function

transformation

cxp( /3 cos q)

tunnelling

small

concerns

Villain

side

of cq.

(8)

and we obtain continuity

replaces after

equation

the tirst

d-integration

Neglecting small.

the wc

ha~c

0.

/?I;~

terms

to

loi-

exponent

be

0

1 t



which

.-

in ctt

;IW

lli~15

pci. LX\

i

A. Golub

! Villain transformation

in small tunnel junctions

97

P

Q) = -

F(P,

F j-dx Y(X)

(15)

0

where (I

the function

f,(y)

has the form

7--x/? (14)

where the overbar means, as in eq. (lo), the time average and Q(T) is defined in (10). The expression (13) diverges when the function in the exponent is positive. For such values of current I and charge 4 formally divergent, F can be defined by an analytical continuation procedure. This situation is similar to that described in ref. [18]. The Z-V characteristic depends on the imaginary part (or the real part after transformation I+ -iZ) of F(p, Q). Im F(p, Q) is a well-defined quantity entirely expressed in terms of the physical parameters of the tunnel junction and current. Let us consider quasicharge q(7) as time independent: q(7) = q, and denote QP = p - q. In the case Z = 0, Z,,,,(q) (eq. (13)) represents the sum over the band spectrum. Then energy bands are functions of the quasicharge q [14] (see fig. 1) (Numbers 0, 1 indicate the first two energy zones.) This form of the spectrum reflects cp-+ C,C + 21rn invariance of the partition function. The energy bands E,, are e-periodic. For finite F. the real part of F leads to normalization of the energy spectrum, which if (Ye 1, is a small correlation to E,,. The Im F defines the decay rate of the charge state and brings the main contribution to Z-V dependence of the junction. If Z # 0 and q(r) = q, at the low temperature limit, eq. (14) becomes

IQ,, + 17) - Ecx] .

f%(y)=jdrexp[F

We rewrite

eq. (15) as B

F(Z’, Q) = - F

1 dx Y(X) 0

x [(f,(P) -f-i(%)) +L(dl

Y -e/2

z,(q) =

5 ev[ -y (Q,, + PO]

p=-x

x e(i- IQ&Q, where

Fig. 1. The dependence quasicharge q.

F,

=

+m

(i

-

Qp - PI)]

the last expression

2e(u (+(;-Q,-ar)G n2E,Z

of

e/j first

two

energy

bands

on

here

(18)

7

a( Q,, + pZ> = exp F,;

Integrating

0

(17)

Here T,, = (ie - Q) /I is the zero of the expression in exponent of eq. (16). The last integral in eq. (17) is well defined and gives no contribution into Im F. After analytical continuation, the first term (proportional to as an imaginary part which (f,(P) -f,(r,,))) h determines Im F(p, Q). For small current, Im F #O if ]Q,l < ie and we maintain in eq. (13) only terms with Im F( p, Q) f 0.

x exp[ --F

-e

(16)

W is the cutting

frequency.

.

we have

(19)

Using the xrics representation for the MacDonald function K,, which include on In-type term. we easily find the imaginary part ot F,:

01. after l-ends

replacing

I--i/.

foi-

IOM

1.

eq.

(7(l)

a4

llerc

T!,

instanton

(31 J

second

15 ttlc

LX~IICCII~C

(77).

xolution and

third

term\

coordinate

(,I,, III

Sf:‘,fC,.

ccl.

01

‘l’licri

( I ) (MC dcnotc

the

th<, iI\

\L1111 iI> .S>) l~ecomc~

Let us consider C//C, belonging to the intcr\,al [ ~~1. I]. This corresponds to the sum in eq. ( IS) over 1) = 0, + 1:

1X voltage can be obtained using cqs. (27)-(2-l) and integrating the following formula over q [ 141 (SW also ref. (131):

notations arc the ‘ranic ;I\ in ccl. 1 l(l) In ccl. (3’). the main integration over T IICY II! the region ._ I ‘-I,,: here J,, i\ the handuidth (XC below). We consider the c71\c 0,, -I A,, \\h~~r-c the. approximation (I;( 7-i I?!, Ihc ii-function / I .:j I\ po5sihle: $( 7) Tf( T T i .losephsc>n co\inc, ;IL’ tion prevents the qaration of continuouy ,trl~i discrete degrees ot freedom Therefore. to pc~form this separation we exploit the semi-cl:rGcal approximation for generating ;I tunctional which represents Z, as ;I win over IL’. instanton ;ind .‘L’ anti-instantons tr;ijectories [ 13. 141. Thcl \u111 Other

(‘5,

3. The Josephson

tunnel

The small Josephson described by the same tional term

,s, =

-E,

.I

d7 cos &C(T)

junction

tunnel junction may he action ( I ) with an add-

(20)

of

Lx.

all

phnsc

dependent

( I. 26.X))

terms

in the

action

i< (WC

A. Golub I Villain transformation in small tunnel junctions

the I-V .

dr $(7)5(r)

characteristic

99

has the form [14]

(31)

0 So after integration nate T,, we obtain

Z,,,(n)=

over

(35) the collective

coordi-

J04 exd-SdJ)l

The action S(q) + S, (eqs. (1) and (26)) describes the Josephson tunnel junction with a finite subgap conductance 1 lR # 0. More elaborate analysis [ 151 reveals additional corrections. They may be incorporated in an effective renormalized dynamic capacitance [ 151, C,=c+6c(l-+cos2&

x

(N+!N_!)-’ (\

dr exp(2rri[(r))N*

which

3R,, SC = R,16A

for 6C G C can be approximated

’

as

0 2;, = C,( 1 - & cos 2q) , .

X (1 dr exp(--2+((r)))*-

(32)

1 4, &=C,KIhd.

1 (36)

0 Here

A,, =

Ih(~)“z($1’4 exp(-s,,)

, (33)

s

= rx

8E,

If1

( E, 1

Collecting obtain

all expressions

(28),

(31),

we finally

Here C, = C + 6C; A is the superconducting energy gap. 6C and F depend on the conductance l/R, of the junction in the normal state. There is some controversy about the sign and value of E [20,21]. We will follow ref. [15] because there the cos 2~ or quasiparticle Cooper pair interference current is settled in the classical Josephson approach [22,23]. The resulting I-V characteristic has the form (35), where the bandwidth A,(E) is now the function of the parameter F. To find this dependence, we need the semiclassical solution for phase difference &(T). If E e 1 the instanton solution has the form $()(T) = k-2 arctg exp(-we

+

A” j dT cost4 5 -

P + t(~Nl]

(34)

0 It is possible to replace the cosine function in the second term by a periodic parabola (which is an appropriate approximation for low temperature and current limit. cos()

=$

+i?((-p

+ 5(T))‘.

As a result the same formula for Z, as in the normal metal junction eq. (12, 13) follows. Only substitution has to be performed: ~T~A,-+ E,. So

w = w()(l + &)

+ ~F~WT)

, (37)

Using solution (37) together with a modified form due to (37) of the action, we obtain

s,, = (g”‘(l+ ;)

(38)

L

Considering only this correction to the exponent in (33), we obtain the main contribution in A,(E) dependence. So the band splitting energy A,(E) is decreased.

4. The arrays of normal tunnel junctions The the

Villain

arrays

The

of

phase

tunnel

normal

was limit

C‘,,+

type

transition

phase. the hours

0)

make

the action

label the (39)

formula

applied

regular

1231.

(4)

has

the

same

to

clearer,

way

term

ing

in

and

( 13) were obtained.

as

Wc

choose

,I model hcrc

for

of \n1;1ll

and occurring

ortlcl-

unc~orrelatcd

according

0i1

tunnci

\vith random a)n\~c‘nicncc ii

;I\ the random \ar~able.

* 6 ‘(: 1

on difl~~--

to ;i prob;ibilir\

of the

(when

the

form

nearest

g~vcn ci( f i\ dcfineti

) configuration.

the partittoil

tuiic.

ncigh-

on the lattice. we may

transfer

gas model.

on the 2D in eq. (30).

when

l-or tlon

Foi-

lattice

is

Proceed-

expressions

( 17)

we have

The

phase

described ~mc

diagram

in

ref.

results.

In

the

ckd bv the curve

According from (SW

fig.

J’. All \‘ = Y,, exp( - : /3Ek ). ;I prime

on the

sum

(fig.

2)

(WC fig.

3).

Indeed.

show

I (the

fixed

the lies

phac

increasing

Ho\+. tli;l

point\

main in

h)u11-

its

clittercncL, re-entrant

at low temperature

let us analyse above

( 4.; I W;I\ I-c’pracnt

71

picture

to a metallic

al

WC only

\‘- 7’ plane

tunnelling

trajectories

the system Here

of stable

to this

coherent

transition

(40)

for

[?‘I.

it is a regioil

gram).

where

in the limit

(I and in ~concl

(39)

the Coulomb

to the cosine

ling.

follows

neigh-

of junctions

(islands)

generalized

114consider

Let

cnt xitcs

that the

ii.

of [24]

concluct;incc

distribution

to conducting

through

rcfcrs

the sites into

the

tunnelling

can be generalized.

array

problem

(‘,,

In

Kosterlitz-Thouless

tunnelling

q5,, = 4, -- 4, i. ;.

this.

a

to the ground SchGn

Supposing

picture

coherent

on the

(1 ~--COS(r#?,(T)~~d,,(T’)))

Here hours.

for

with

‘1%~ r-csult the

the renormalization

randomly this

with

C‘ located

insulating

caused

occurs

action

J

To

the

the

temperature.

junctions.

(‘,,~+

found

Tunnelling

transition

The

and

from

investigated.

junctions

Fazio

when

two-dimensional

capacitance

by

0 they

tunnelling

art:

the

capacitance

and with

obtained

for tunnel

ncarcst-neighbour junction

(4) is useful

junctions

diagram

of

array

transformation

tig. 1 for wi~ill

the separatice J’ (or

decreasing

1I,,, I’ ] ot the

A. Golub

I Villain transformation in small tunnel junctions

101

ture and small current. Using Villain transformation the same approach has been developed for both types of tunnel junctions. The influence of the interference between Cooper pair and quasiparticle currents on the average voltage was also considered. It was shown that this influence amounts to depressing the voltage. We have analysed the phase diagram of an array of normal tunnel junctions with random tunnelling between different sites and have found the reentrant behaviour of the system in this case.

t

Y

To

T-

T+

-I-

Fig. 2. Hamiltonian flows in the (T, y) plane. There is a line of fixed points at J = 0. The heavy trajectory which starts at T,, and ends at T, bounds a region of insulting behaviour.

Acknowledgements I am grateful to B. Horowitz and E. BenJacob for their most valuable discussions, and also to Mrs. Rivka Shaanan for technical assistance.

References Ill P. Delsing,

Fig. 3. Re-entrant

transition

to a metallic

phase

effective chemical potential of charges) in the limit of the large scale length. This corresponds to the metallic (free charges) phase. When u = n/8 the separatrice is shrunk to a point, so for v > n/8 we have only the metallic phase. If (T < n/S then there are three regions T < T,,, we have a T,, < T < T,, and T >. T,. Therefore domain (under the separatice) where scaling results in the insulating behaviour of the system.

5. Summary In conclusion, we have calculated the Z-V characteristic of the normal and Josephson current-biased junction in the limit of low tempera-

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