PHYSICAL REVIEW B VOLUME 49, NUMBER 19 15 MAY 1994-I

Zero-frequency shot noise for tunneling through a system with internal scattering

J. Carlos Egues, Selman Hershfield, * and John %.%ilkinsDepartment ofPhysics, The Ohio State University, Columbus, Ohio 4321011-68

(Received 22 November 1993)

Within the sequential tunneling approach we calculate the zero-frequency shot noise for electrons tun-

neling through a mesoscopic system with internal scattering. This scattering is included by coupling theresonant level to another set of "internal" states. Our results show that the internal scattering has no

egect on the noise provided that the internal states are not coupled to the collector. On the other hand,when electrons can hop from the internal states to the collector, the noise is affected. In some cases thezero-frequency noise can be suppressed below one-half of full shot noise, 2eI. All limiting cases for theratio of noise to the current, S/2eI, are presented and discussed.

I. INTRODUCTION

Over the past few years there has been a flicker of in-terest in noise in mesoscopic systems. Noise here refersto the Fourier transform of the current-current correla-tion function, usually taken in the zero-frequency limit.%hile the noise in equilibrium is related to the linear-response resistance by the fluctuation-dissipationtheorem, ' the noise out of equilibrium is not directlyobtainable from the resistance or the current. At largevoltage bias it is often proportional to the current andcalled shot noise, indicating that this noise is due to thediscrete nature of the charge transport. Most of the newinterest has focused on the reduction of the shot noisebelow the value for completely random current fluctua-tions, 2eI. This value is often referred to as the classicalshot noise or full shot noise.

The reduction in the shot noise has been the subject ofboth theory and experiment. One set of theoretical pa-pers calculated the noise exactly for noninteracting elec-trons in mesoscopic systems. These papers found thatthere was a suppression of the shot noise when thetransmission probability approached unity. Thus, for ex-ample, for a resonant-tunneling structure the noise issuppressed below 2eI near resonance. Another set oftheoretical papers used master equations to calculate thecurrent fluctuations in a variety of mesoscopic devices,including resonant-tunneling structures and the Coulombblockade. ' ' In the case of the double-barrierresonant-tunneling structures the results of the nonin-teracting quantum calculations and the master-equationcalculations agree exactly in the large bias limit. In thesecalculations the suppression of the noise was at mostone half of the full-shot noise These t.heoretical resultsare supported by an experiment on double-barrierresonant-tunneling structures, which preceded most ofthe theoretical work. '

At this point a natUral question is what does thesuppression of the shot noise depend on. For example,the dependence on different statistics, Fermi and Bose,has recently been studied. ' In contrast to the suppres-sion of noise for ferrnions, the noise for bosons isenhanced at low temperatures. Another interesting aspect

is how the shot noise suppression for fermions dependson the amount of scattering. The fact that thenoninteracting-quantum coherent and the master-equation (sequential-incoherent approach} calculationsgive exactly the same result for double-barrier, resonant-tunneling structures would seem to indicate that the shotnoise does not depend on the amount of scattering.

In this paper we consider a model for "internal"scattering in a resonant-tunneling system in order toanswer the question of whether scattering within the res-onant level can change the result for the suppression ofthe zero-frequency shot noise. ' By coupling the reso-nant level (e.g., island 1 in Fig. 1) to a set of internalstates (e.g., island 2 in Fig. 1), we include internal scatter-ing for electrons while on the resonant site. Our model,shown in Fig. 1, consists of an emitter, a central region,and a collector. The central region is composed of twoset of states (island 1 and 2), one of which is accessiblefrom the emitter and the other which is only accessible byscattering from the first set. Figure 1 also illustrates allthe possible paths an electron can follow. Because weconsider only the high bias regime, there is no back flowfrom the central region to the emitter or from the collec-tor to the central region.

Our results for the effect of scattering internal to theresonant site can be summarized as follows: (a} when theinternal states (island 2} are not coupled to the collector,internal scattering has no effect on the noise, (b) when theinternal states are coupled to the collector, noise can de-pend on the internal scattering rates, but their effect canbe included in the case (a) result by renorrnalizing theexternal rates, and (c) in a limited parameter range —i.e.,when all the rates are similar and the internal states areconnected to the collector —the noise can be suppressedbelow one-half (i.e., to 0.45) of the classical value. Thus,there is only limited parameter range in which the noisecan be affected by internal scattering.

The rest of the paper is organized as follows. In Sec. IIwe define our sequential model for resonant tunneling interms of a master equation and the corresponding ratesequations. In Sec. III we define both the current and shotnoise in our system; the shot noise is then calculated interms of "hop-hop correlation functions. " Section IV

0163-1829/94/49(19)/13517(11)/$06.00 49 13 517 1994 The American Physical Society

13 518 EGUES, HERSHFIELD, AND WILKINS 49

Nl- nl0'e=

Emitter ~

N2

Il lrcl-~cl

~ Collector

l n2{N&- n, )il g =&i

I12

~c2

The characteristic times ~„~,1, v.,2, and ~; can in princi-ple be determined by Fermi's golden rule. Note that,even though the internal rates g; ( n, , n z ) and r, ( n, , n z ) asdefined in Fig. 1 have a nonlinear term, n1n2, the rateequations turn out to be linear because the nonlinearterms cancel themselves out. In addition, since the bias isassumed to be large, electrons are neither allowed to hopinto the central region from the collector nor to hop backto the emitter from island 1.

By setting Eqs. (1) and (2) to zero we obtain thesteady-state solution, ( n, ) and ( n ~ ):

Central Region

FIG. 1. Schematic of the model for the tunneling systemsolved in this paper. The internal degrees of freedom are charac-terized by the two coupled set of states (two islands labeled 1

and 2). The islands 1 and 2 can hold at most N& and N2 elec-trons, respectively, while the actual number of electrons isdenoted by n

&and n&. Electrons hopping in island 1, from the

emitter, can either hop to the collector or (due to internalscattering) to island 2. Electrons in island 2 can again eitherhop to the collector or hop back to island 1 (due again to inter-nal scattering). No hops from the islands back to the emitterare allowed because we consider only the high bias limit. All ofthe hopping rates obey by construction the Pauli exclusion prin-ciple: if an island is full, no electron can hop into it. In addi-tion, if an island is empty no electron can hop out of it.

N1~, 1

(n, &=rq) +1e + (N27e1 q) IN) rq2+1 ( )

N1N2 Vc1+c2(n, )=

(N, r,2+1; )(rq)+re)+N27er, )(4)

where 5n, (t)=n&(t) —(n, ), 5n2(t)=n2(t) —(n2), and

1 1 Nz

Equations (1) and (2) can be put in the following matrixform:

5n, (t) 5n, (t)

5n2(t) 5n2(t)

contains some plots of the shot noise (as a function of thecharacteristic times of the system) as well as several of itslimiting cases. We summarize our results in Sec. V.

A. Rate equations

From Fig. 1 we can immediately write down the rateequations describing the average number of electrons inthe two islands (central region}, n, (t) and n2(t}:

dn&(t) N& n&(t)—dt

n, (t) n2(t)[N, —n, (t)]++cl

n, (t)[N, —n, (t)]

dn2(t) n, (t)[ zN—n2(t)] n2(t)[N, n,(t)]-

dt I l

n2(t)

C2

II. SEQUENTIAL MODEL

As pointed out by Luryi, ' transport in resonant-tunneling systems can be described within a classicalsequential picture in which the tunneling event takesplace in a sequence of hops. This alternate view to thequantum mechanical (coherent) one assumes that theelectrons are incoherent due to some phase-breaking in-teraction, e.g. , electron-phonon coupling in the resonantsite. The sequential model is formulated in terms of amaster equation for the number of electrons in the severalsites (two in our model) and its corresponding rate equa-tions.

C

Nq

and

A, +fn &(t) = ( n, ) +(a+ N+b+M+c+ )e +

+(a N b+M+c —)e

+ P11n (t)2=(n )+z(a+N+b+M+c+ )e +

P12—P11+ (a N b+M+—c )e

Piz

where

P„+P22 Q(P„+P22 } —4 detpA+= +

2 2

are the two eigenvalues of P. Since detP & 0 andP»+P» &0, it can be verified that A.~(0. This showsthat, after a random fluctuation in the number of elec-trons in any of the two islands, the system always tendsto relax towards the steady-state regime. The quantitiesa+, b+, and c+ are defined as

A, + —P»a+—

P12+ g g 7

C2

Solving Eq. (5) subject to the initial conditions n, (0)=Nand n2(0) =M, we obtain,

49 ZERO-FREQUENCY SHOT NOISE FOR TUNNELING THROUGH. . . 13 519

and

(n, ),1=a +a+,

(12)

(13)

master equation' is the simplest way to account for Auc-

tuations.

B. Master equation

N, —(n, ) (n, ) (n, )

+c1 c2

I1 I2+, (15)e e

where e is the absolute value of the electronic charge and(I&):I& and—(I2 ) =I& are the partial average currentsvia island 1 and 2, respectively. The rate equations donot describe the random fluctuations in the current; a

I

—(n, )=c +c+ .

Equations (7) and (8) are used in the calculation of thehop-hop correlation functions which are the key in-gredients in calculating the shot noise (Sec. III).

From the steady-state solution we can easily obtain theaverage current (I ) =I thr—ough the system,

The master equation we consider is consistent withboth the Pauli exclusion principle and the rate equations,Eqs. (1) and (2). It describes the time evolution of theprobability, p(n„nz, t), of finding n& and nz electrons inislands 1 and 2, respectively, at a time t. The numbers n,and n2 are taken to be integers. By looking at Fig. 1 wecan see that electrons can hop in island 1 from both theemitter and island 2 at "generation" rates g, (n, ) and

g, (n „nz ), respectively. Also, electrons can hop off island1 to both the collector and island 2 at "recombination"rates r, &( n

&) and r; ( n ~, n 2 ), respectively. On the other

hand, while electrons can hop off island 2 to both the col-lector and island 1 at rates r,2(n„n2) and g, (n&, n2), re-

spectively, they can hop into island 2 only from island 1.Taking into account all of these processes, we can write

dp (n„n2, t)=g, (n) —1)p(n) —l, n2, t)+r, )(n)+1)p(n)+1, n2, t)

+g;(n& —l, n2+1)p(n& —l, n2+1, t)+r;(nt+ l, n2 —1)p(n &+ 1,n2 —l, t)

+r„(n, +1)p(n, , n2+1) —[ g( n, )+g;(n&, n 2) +r, (n„n 2)+r, &(n&) +r,z(n 2)]p(n&, n 2t) . (16)

Note that in our model master equation the number ofparticles in the central region (Fig. 1) is conserved underinternal scattering. The following end-point conditionsare satisfied by construction:

r„(0)=0, r,2(0)=0, r;(O, n2) =O,g, (n;(n „0)=0

(17)

and

I

the current i (t):

NT NT1

i(t)=ae g f(t t,')+pe g—f(t —t,")j=1 j=1

NT2

+pe g f(t —t,"), (19)

r;(n„N2)=0, g;(N„n2)=0, g, (N, )=0 . (18)

These constraints ensure that unphysical processes of ei-ther hops from an empty island, Eq. (17), or hops into afull island, Eq. (18), do not occur.

The steady-state distribution function, po(n, , n 2 )

which is obtained by setting Eq. (16) to zero, is used todefine the variances, ((5n, ) ), ((5n, ) ), and (5n, 5n2),which are computed in Appendix A. In the next sectionwe show how the shot noise depends on these quantities.

III. ZERO-FREQUENCY SHOT NOISE

In this section, we calculate the zero-frequency shotnoise. We first define the time dependent current i(t)through our system in terms of the emitter-to-island-1,island- l-to-collector, and island-2-to-collector hops.Next we define the noise and relate it to the several (i.e.,nine) hop-hop correlation functions in our model. Atzero-frequency we then calculate the correlation func-tions in order to fully obtain the zero-frequency shotnoise.

As an extension of what was done in Ref. 11, we define

I f (t)dt =1 . (20)

The Auctuation in the current, about its average value,at time t, is 5i (t) =i (t)—(I ). The Fourier transform ofthe current

fluctuation

autocorr elation function,(5i(t+t')5i(t')), , is defined as the noise, S(co), in thecurrent

where a and p are device-dependent (geometric) factors'and tJg, tJ", and tJ" are the times associated with theemitter-to-island-1, island-l-to-collector, and island-2-to-collector hops, respectively. Due to charge conservationthe sum of a and p is one. Note that there are, respec-tively, XT, NT, and XT electron hops for each one ofthese three hop processes within the observation periodT. Equation (19) shows that i (t) is a fluctuating quantitydue to the discrete nature of the charge, i.e., whenever a(random) hop takes place, it generates a pulse in theexternal circuit. The function f (t) (pulse-shape function)describes the shape of the pulses produced in the externalcircuit' by the individual electron hops. We will assumethat the only relevant property of f (t) is its normaliza-tion

13 520 EGUES, HERSHFIELD, AND WILKINS 49

S(w)=2 f e' '(5i(t +t')5i(t'));dt

=2 e'"' i t+t' i t', .—I dt . (21)

where F (cu) is the Fourier transform of f (t) and satisfiesF (0)= 1 [due to Eq. (20)], via

It is related to the cosine Fourier transform of the current S( )= —4 (I)'5( ).T

(23)

I(co)=eF(w) . age '+Pg e ' +Pg eJ J J

(22)From Eq. (22) we have

[I(co)~ =e ~F(co)~ a ge ' +P ge ' " +P pek,j k,j k,j

i N(tg —t ) (tg-t'2) (i"-i )

k,j k,j k,j(tc1 tc2) (tc2 t ) (tc2 tc1)

+p2y j k +p y ' j E +p2y ' j k

k,j k,j k,j(24)

By defining appropriate "hop-hop correlation functions" we can express the double sums in Eq. (24) in an integralform. ' For instance, let us define the quantity h, i,2(t) to be the ensemble-averaged rate of r, ~ hops (island-2-to-collector hops) given that there was one r„hop (island-1-to-collector hop) at t =0 (the time origin is arbitrary). Forh, z, i(t) an analogous definition holds. In terms of h„,z(t) and h, z„(t) the average of the seventh and the ninth doublesums in Eq. (24) can be rewritten as

( g e ' " + g e ' " =2N& f cos(cot)h„,z(t)dt +2Nz f cos(cot)h, 2„(t)dtk,j k,j

=2NrH, 2, i(co)+2NrH, i,q(co) ) (25)

h (t)N, —(n, &

lim ~ h, i(t) =h =taboo T e

'

h„,(t}(26)

where H, , i(m2) and H, 2(co) are the cosine Fouriertransform of h„,z(t) and h, 2„(t), respectively. We candefine hop-hop correlation functions for all of the otherpairs of hop rates that can be formed out of g, r, &, and r,2.Note that there are nine different hop-hop correlationfunctions in our system: h (t), h „(t), h, 2(t), h„(t),h„„(t),h„„(t),h„s(t), h„„(t),and h„„(t). For t~ ~(no correlation between the hops) the hop-hop correlationfunctions have a trivial asymptotic behavior, i.e.,

h„,(t)lim h, ~„(t) .=h~=

t —+ oo

h„„(t)

&n2}

c2

NT I2T e

(28)

g„s(t)=h, i (t) —h, . (29)

where h, hi, and h2 denote the average values of theemitter-to-island-1, island- l-to-collector, and island-2-to-collector hop rates, respectively. Note that h =h, +h2(NT=NT+NT ). It is convenient, as it will soon becomeevident, to define "reduced" correlation functions g bysubtracting off the long-time limit of the hop-hop correla-tion functions, e.g.,

and

h, ig(t)

lim h, i„(t) =h i=

taboo

h„,2(t)+Cl

(n ) NT I,T e

(27)

The cosine Fourier transform of Eq. (29) is

G„s(co)=H„~(co) m(I } 5(co) . — (30)

Ultimately, all of the double sums in Eq. (24) can be writ-ten in terms of these reduced hop-hop correlation func-tions. Hence,

(~I(co)~ }=2m(I} 5(co)T+e ~F(co)~ [Nz [a (1+26ss(co)}+2aPG,i (co)+2aP6, 2 (co)]

+NT[P (1 +26„„(co)) +2 aPG„(co)+2P 6,2„(co)]

+NT[P (1+26,2,z(co))+2aPG, z(co)+2P 6„,2(co)]] . (31)

49 ZERO-FREQUENCY SHOT NOISE FOR TUNNELING THROUGH. . . 13 521

+2 [P G„„(0)+aPGg„(0}+PG,z„(0)]T

N+2 [P G,2,z(0)+aPG, 2(0)+P G, i,2(0)] .

T(32)

Note that the singular terms in Eqs. (23) and (31}dropout of Eq. (32). In Appendix B we calculate, as an exam-ple, one of the nine hop-hop correlation functions, name-

ly, h, i,2(t}. In addition, we present (see Table I) the ninezero-frequency components of the cosine Fourier trans-form of the reduced correlation functions that appear inEq. (32}. Note that the hop-hop correlation functions de-pend explicitly on the variances of po(n „n2 } (AppendixA}. Thus, using the results from Table I we obtain

&n, &s=1—2Ni Ni

& n, &(5n, 5n & 1/r;+2

Ni(1 —&n, &/N, } 1/r, 2+Ni/r;

(Ni —1)/Nir;

1/r, +1/r, i+$2/r;

(33)

The zero-frequency shot noise can be readily obtained bysubstituting Eq. (31) into Eq. (23} and then setting co=0.The noise ratio s =S(0)/2eI is given by

s =a +P +2a Ggg (0)+2aPG„(0)+2aPG, 2 (0)

It should be noted that s does not depend on either a orP. In particular, s~ i & o=s~ o& i=s holds, i.e., thezero-frequency noise is the same regardless of where thecurrent is "measured. " This is a consequence of chargeconservation and continuity equation as pointed out inRefs. 11 and 22. In the next section we discuss some as-pects of Eq. (33) as well as some of its limiting cases.

IV. DISCUSSION

Equation (33) describes the zero-frequency shot-noiseratio, s=S(0)/2eI. It generalizes the results in Refs. 11[cf. Eqs. (4.38} and (3.15)] and 12 [cf. Eq. (25)] to includeinternal scattering. Because the cross variance (5n, 5n2 &

is negative semidefinite [see Appendix A, Eqs. (A7) and(A10) and also Fig. 2(d)], it always decreases the noise.This is not surprising since the shot noise must be de-creased by any correlation (e.g., (5n, 5n2&) among theelectrons in the islands.

In Figs. 2(a}—2(c) we present some plots ofs(r„/r;, r,2/r, )~n iv =, for diff'erent ratios r, /r; The.

1 2

plane logio(r, i/r;) =0 roughly determines when internalscattering is significant, i.e., when the island-1-to-island-2hop rate competes with the island-1-to-collector hop rate.Note that this plane also determines where the cross vari-ance term (5ni5nz& first deviates from zero [see Fig.2(d)]. In Figs. 2(a) —2(c), we denote region I as lying onthe right-hand side of this plane and as region II and IIIthe two Qat portions lying on the left-hand side of theplane.

In the following paragraphs we explain some of the

TABLE I. Zero-frequency components of the cosine Fourier transform of the reduced hop-hopcorrelation functions (Appendix B). The variances are defined in Appendix A. The mean values (ni )and ( n2 ) and the matrix elements of P are defined in Eqs. (3), (4), and (6), respectively.

7 r, (n, &

Geiei(0) = [&(5ni ) ) —(ni ) ]+ (5n, 5n, )t, i%i N2w, i ni )

7g r, (n, )N

[((5n, )'&-(n, )]+ '(5n, 5n, )

&ciN2 N, r„n,G, i,2(0)=

(n,), (n, )G, «(0)=—,', [((5n, )') + (n, ) N, ]+ (5n, 5nz )—

(N~ —yn~ g)w, & N] 2

(n, )r, &n, &r,G,~„(0)=—, , 1+Pi, [((5n, ) ) —(ni)]+Pii (5ni5ng)

s n ) zw, 2N) 1 2

G 2 2(0)&n, )r„N,

(n, &r, &n, &~,1+P„' &5ni5ni)+Pii'

[&(5n, )'& —&n, &]1 2

&n, )r, &n, &r,1+P„[((5ni)')+(n, ) N, ]+P» (—5n, 5n, )

Gg, I(0)=—

Gg, 2(0)=

1 &n, )[&(5n, )'& —(ni &]— &5ni5ni &

I N2 ni

[((5n ) ) —(n ) ]— (5n, 5n )&n, &

2 N, n~

Ggg(0)= ~ [((5ni) )+(ni) Ni]+ ( n, 5n 5)—z

&n, &, (n, &

(Nl —nI ) N2

13 522 EGUES, HERSHFIELD, AND WILKINS

10. 9-

~ 0. 8-

0.7.ll 0. 6-

'~ 0. 5

&oglO(&'~/&

10.9.0.8.0.7-

tl 0. 6-

0. 5.

8 8

(c)

og lO(&cl /& j

1.+ 0. 9-

0. 8.W 0. 7.

II

«~ 0. 5-

&og 1O(~c2/~i

01

-2.-3.

4Q4-

o —5.

(b)

1O(&cl /& )

FIG. 2. Ratio of the zero-frequency shotnoise (calculated within a sequential tunnelingapproach) to its classical value, s =S/2ei[(a)—(c)], and cross variance (5n, 5n2 ) (d), forthe tunneling through the system shown in

Fig. 1, plotted as a function of the ratios ~, I/7. ;and ~,2/~; for the particular case N l

=X2 = l.Figures (a) —(c) show s for three values of~, /~;: 0.1 in (a), 1 in {b), and 10 in {c). Allthree plots clearly show suppression of theshot noise with respect to its classical value. Asurprising result is that as ~,2~ ~ (i.e., no hopfrom island 2 to the collector) the internalscattering has no effect on the calculated noise;the system behaves exactly as if it were thesingle-island system. Figure (b) displays anoth-er interesting feature: the noise can besuppressed to values below one-half. The crossvariance in Fig. (d) is negative semidefinite. Itplays an important role in suppressing thenoise s below one-half.

og 1 O ( Tcd /ri 8 8 oglo(&. l /& ) loglO(rc2/ i lO(~cl/~i I

features in Figs. 2(a) —2(c) in terms of "effective single-island models. " The basic idea is to think of electronstraversing the system in two hops just as in a single-islandmodel. This picture holds as long as one of the hop rates(as defined in Fig. 1) is much bigger than any other in thesystem. Figure 3 shows several "effective single-islandpictures" which hold in different regimes.

In region I, because there is little internal scattering(~„&r, ~ r„&r; ), transport occurs mostly via island 1

regardless of ~,z', our system behaves as the single-islandsystem treated in Refs. 11 and 12. In addition, since inthis region transport occurs basically in two hops, onefrom the emitter to island 1 and the other from island 1

to the collector [see Fig. 3(I)j, the relevant characteristictimes are w, and '7 I. In the absence of internal scattering(i.e., r;~ ac ), we obtain the noise ratio s given in Ref. 11,which is shown in the first row of Table II.

On the other hand, in region II (r„&)r; »r, 2) trans-

port involves both islands. Electrons hop to island 1

from the emitter, then to island 2, and finally to the col-lector. Since r„«r,2 and r;, we can think of island 2and the collector as constituting an "effective collector. "We regain the single-island model picture: electrons hopto island 1 from the emitter and from there, due to inter-nal scattering, to the "effective collector" [see Fig. 3(II)].The relevant characteristic times in region II are ~, and

~,-, which is now the "collector time. " The values as-sumed by s in region II correspond to those obtainedwithin the single-island model of Ref. 11 with ~, replacedby r;/Nz (for r, l~ ao and r,2~0o).

Transport within region III always involves both is-

lands even though the final hop to the collector mayoccur either from island 1 (r„&r,2), or from island 2

(r„&~,2). In either case, we have r, &&r„,r, z. There-fore, we can think of island 1 and island 2 as a single"effective island. " We have again a single-island picture

in which electrons hop from the emitter to the effectiveisland and from there to the collector [see Fig. 3(III)j.The relevant characteristic times in this case are ~, andsome "effective collector time" v,*. The third and fourthrow of Table II illustrate two possibilities for thiseffective collector time. Note that with

r;=r„N, /(N, +N2) in entry (3) and r,"=r,iN, /Ni in

entry (4) the standard single-island result (see first row ofTable II) is regained.

So far, based on the single-island model of Refs. 11 and12 which holds in region I of Figs. 2(a) —2(c), we have in-

troduced two "effective" single-island models one ofwhich describes transport in region II ("effective collec-tor" picture) and the other in region III (effective islandpicture). A natural question is: do these single-islandmodels also describe the "dips" in s in Figs. 2(a) —2(c)?The answer is: any dip is explained by the model holdingwhere the dip is. For example, the dip in Fig. 2(a), whichlies entirely within region I, can be described by thesingle-island model of Refs. 11 and 12, i.e., the minimumof s should occur at v, =~,I. On the other hand, the dipin Fig. 2(c), between regions II and III, actually lies in aregion where the effective island picture for transport (in

region III) holds. The limiting case on the fourth row ofTable II, where the effective island picture holds, can beused to verify that the dip in Fig. 2(c) occurs atr, =r, =r,2 (for Ni =N2 = 1).

In Fig. 2(b), the dip in between regions II and III is en-

tirely due to the cross variance term in Eq. (33). To seethis we note that in region II of Fig. 2(b) s equals —,

' and

(5n, 5n2 ) =0 [see Fig. 2(d)]. For the particular set of pa-rameters of Figs. 2(b) and 2(d) (5n &5n& ) has a minimum

below zero at r,2=r;, hence, according to Eq. (33), thenoise is suppressed to values smaller than one-half of itsclassical value. This result shows that the inclusion ofinternal scattering in resonant-tunneling systems can

49 ZERO-FREQUENCY SHOT NOISE FOR TUNNELING THROUGH. . . 13 523

cause a further suppression of the shot noise. Unlike Fig.2(c), we cannot describe this dip in terms of any"effective" single-island model since it occurs within a re-gion where 7 7 2

The dip in s between regions II and III in Fig. 2(a) canbe understood in terms of another "effective" single-island picture. The dip occurs at v,2=~; &&~„which,because r, =0. Ir; in Fig. 2(a), implies that g,))r;,r,2))r, &. Therefore, we can think of the emitterand island 1 as an "effective emitter. " The single-islandpicture then describes an electron hopping to island 2from this effective emitter and from there to the collector[see Fig. 3(IV}]. The relevant characteristic times in thispicture are ~,2 as the collector time and some "effectiveemitter time" ~,'. The fifth row of Table II illustrates thispicture in the limiting case ~, ~0 and ~„~00. Fromthis entry we can see that the efFective emitter time is

r;=r;/N, Not.e also that (n, )/N, =l in this case

meaning that the second term in Eq. (33} is zero. There-fore, the S /2eI contribution proportional to & 5n, 5n2 ) inEq. (33) is crucial in suppressing the noise.

A surprising result is that our system behaves as thesingle-island model of Refs. 11 and 12 irrespective of ~;provided that ~,2)~„. The second row of Table IIshows that the single-island result is actually regained inthe limit ~,2

—+ 00. This basically says that internalscattering, no matter how strong, has no ejfect on noisewhen the internal states (island 2) are not coupled to thecollector. This result explains why the dip in s along thelog, o(r„/r;) axis (for r,2&r„) in Figs. 2(a) —2(c) alsooccurs at r, =r,

&(as in the single-island model valid in

region I) regardless of r;Unlike the other rows in Table II, the first and the last

two rows of this table represent some "obvious" limitingcases. The first row just says that if an electron cannothop to island 2 (because r;~ 00 ) then the single-island

TABLE II. Limiting cases for the noise ratio S/2eI, Eq. (33). The first two rows show that when ei-ther ~; or ~,2 goes to infinity, the shot noise ratio is the same as the single-island model of Refs. 11 and12. While this result is obvious in the limit ~;~~, since electrons in island 1 cannot hop to island 2, itis surprising in the limit r,2~ ao. The latter case (r,z~ ao ) implies the internal scattering has no egecton the shot noise as long as the internal states (island 2) are not coupled to the collector. The third andfourth rows indicate that, for ~;—+0, when either ~„=~,1 or ~,1~00, the system behaves as an"effective" single-island system with the island-1-to-collector characteristic time ~, 1 being replaced byNl~, l/(Nl+N2) and N1~,2/N2, respectively. In the limit shown in the fifth row, our system alsobehaves itself as an "effective" single-island system; however, island 1 and the emitter constitute an"effective emitter" with characteristic time ~,*=~;/Nl. The collector characteristic time is still 1

The last two rows show that full shot-noise result is recovered when one of the two island-collectorrates is infinity. Note that in the sixth row both r, 2 and ~; must be zero in order to get full shot noise,while in the seventh row only ~, 1 must be zero. These two rows correspond to transport purely throughisland 2 and island 1, respectively.

Limits

7 ~00

&n, )Nl

+cl

(n, )N2

S/2eI

1 —2 1—(n, ) (n, )

1 1

(2) VC2~ 00&c1+&,

+cl

V, 1+W,

(n, ) (n, )1 —2 1—

Nl Nl

(3) 7 C2= 7 c 1 and 'T; ~0

Nl V'cl

N, +N2

Nl +N2+w,

(n, )Nl

(n, & &n, )1 —2 1—

N, Nl

(4) ~,1~00 and ~;~0

Nl ~C2

N2

Nl ~,2 +g

&n, )N,

&n, & &n, &

1 —2 1—Nl Nl

z, ~0 and ~, 1—+~ C2

T

C2 +1

&n, & &n, &

1 —2 1—N2 N2

~,2~0 and ~;~0

(7)

13 524 EGUES, HERSHFIELD, AND %'ILKINS

I: "single-island model"

Emitter ~&~ N l ~&~ Collector

N2

III: "effective island"

Emitter ~~~ N ) Collector

N&c l &~& i ~~ &c 2=:

IV: "effective emitter"

II: "effective collector" in a limited parameter range we found a suppressionbelow one-half (0.45) of the classical value. On the otherhand, when the internal states are not directly coupled tothe collector, we find that internal scattering has no effecton the noise. By analyzing several limiting cases of ourmodel, we are able to understand most of the features inthe noise plots in terms of "effective single-island mod-els." These models reduce the interacting problem withinternal scattering to a noninteracting ("renormalized")problem without internal scattering.

Emitter ~~~ N] Emitter ~&~ N& Collector

N2 N2

&i «&c 1'~c2 &e «&c 2'&i «&c l

FIG. 3. Effective single-island picture of our system in fourdifferent regimes. The single-island model (Refs. 11 and 12)

holds in region I. Note that this picture is valid irrespective ofinternal scattering provided that ~,2~ oc. %'ith this model we

can understand all of the "dips" along the loglo(~ 1/7.;) axis in

Figs. (a) —(c). The effective collector picture describes transportin region II. In this regime, because r, 2 is much bigger than anyother hop rates in the system, island 2 and the collector can be

thought of as an "effective collector"; hence, electrons traversethe system in two hops: from the emitter to island 1 and fromthere to the effective collector. The dotted line enclosing island

2 and the collector, in II, denotes the effective collector. Theeffective island picture holds in region III where internal

scattering is strong (v;~0). In this limit, island 1 and island 2

can be thought of as a single "effective island" which is denoted

by the dotted line enclosing both islands in III. The "dip" along

the log»(~, 2/v. ;) axis, in Fig. 2(c), can be described in terms ofthis effective island model since it lies in region III. On the oth-

er hand, when the emitter-to-island-1 hop rate is much biggerthan the other hop rates in the system, we have the "effectiveemitter" picture. The dotted line enclosing both the emitter and

island 1, in IV, represents the effective emitter. This regime

holds only in the region of Fig. 2(a) where the dip along the

loglo{ ~„/~; ) axis lies.

ACKNOWLEDGMENTS

This work was supported by the Brazilian FederalAgency for Higher Education —CAPES (J.C.E.), theU.S. Office of Naval Research (J.W.W.), and the NationalScience Foundation (S.H. , DMR-9357474). One of us

(J.C.E.) thanks Per Hyldgaard for valuable discussionsduring the early stages of this work.

APPENDIX A: VARIANCES

In this appendix we compute the variances ((5n, ) ),((5nz) ), and (5n&5nz) of the steady distribution,

p (on„n )z. Just as (n&) and (nz) were determined bywriting equations for n, (t) and nz(t}, the expectationvalues ( n f ), ( n z ), and ( n, n z ), can be determined fromtheir equations of motion. For example, multiplyingthe master equation, Eq. (16},by n, and summing overboth n

&and nz, the time derivative of (n f (t}) is

(&fd(tli [X,—n, (ti])

[I+2n, (t)]dt +e

n, (t)+ 1 —2n) t

n &(t)[Nz —nz(t)]

(+ [1 2n)(t)]—

model result should hold. The last two rows say that ifwe have just an emitter and an "effective collector, " with

no intermediate island, the full shot noise result should berecovered. In the sixth row the group island 1 + island 2+ the collector plays the role of' the effective collectorand in the seventh row, island 1 + the collector form theeffective collector.

where

nz N, —n, (t)+ 1+2n) t

NI, N2

(A 1)

V. CONCLUSION(n', (r)) =

n =On =01 ' 2

n ip(n„n zt) . (A2)

In summary, we have presented a simple classical mod-el for studying the effects of internal scattering on thezero-frequency shot noise in tunneling systems. Withinthe sequential tunneling approach, using a master equa-tion, we have included internal scattering by coupling theresonant site (island 1) to another set of internal states (is-

land 2). When compared to earlier results, without inter-nal scattering, our results indicate that internal scatteringcan contribute to a further suppression of the shot noise;

Expressions analogous to Eq. (Al) can be written ford ( n z(t) ) Idt and d ( n, (t)nz(r) ) Idt.

In the limit t~~ [steady-state regime: p(n„nz, t}~po(n, , nz)] the time derivative on the left-hand side ofEq. (Al) is zero, and (n, (t)) becomes (n, ) +((5n, ) ).In this limit the three equations for (n &(t)), (nz(t) ),and (n, (t)nz(t)) reduce to

ZERO-FREQUENCY SHOT NOISE FOR TUNNELING THROUGH. . . 13 525

Te +cl

N2

1 1 N

Nl++c2 +i

Nl

Nl —1

N2 —1

Nl 1 N2 1+ + + +

+e +cl +c2

((5n, )') 1

(5n, 5n, )

I +I1+I'

2

I2+I'2

—I'

(A3)

where I, I&, and I2 are defined in Eq. (15) and

(N2 —(n2) }(n& ) (N& —(n, ) )(n2)+e

(A4)

+ + + (5n, 5n ).1 1 1

e +C1 +C2(A5)

An interesting consequence of Eq. (A3} is that current isdirectly related to the variances via

+ ((5n, )') + ((5n, )')'Tc 1 'Te c2

&n, )

Nl

+Te TI

Tc 1 Te Ti

&n, &

N2 +c2

&n, &

Nl

side of Eq. (A3)

+1C2N1

(A6)

(A7)

Using the following relations to simplify the right-hand we obtain the variances

&n, )((5n, )2& =(~,—&., &) +

1

Nl —1

(5n, 5n, &,1 1 N2

Te

N2 —1

(A8)

((5n )')=(& —(n, )) +2 2 2 N&5n, 5n, ), (A9)

&5n, 5n, &=

1 1 1 1

c 1 TC2

c2

&n2& &n2)

N2c2

(n, )

Nl

1 + 1

+e +cl

1+ 1 N2

+cl

N —1+ 2

c2

c2

+

(A10)

Since ( n 2 ) /N2 ( ( n, ) /N, [from Eq. (A7)], the expres-sion for (5n &5n2 ) in Eq. (A10) is negative. It should be

emphasized that no explicit knowledge of po(n„n2) wasneeded in the above derivation.

I

correlation functions can be determined in the same way.By definition, h, &,2(t) is the ensemble-averaged rate of r„hops, at time t, given that there was one r„hop at t =0,

Nl, N2

APPENDIX B: CALCULATIONOF THE CORRELATION FUNCTIONS

h„,2(t) =nl = l, n&=0

r 1(n 1 }P 2(n 1 n2 (B1)

Our procedure to determine the hop-hop correlationfunctions follows that of Ref. 11. Here we determineh„,2(t) to illustrate the procedure. All of the other

where p,2(n„n2, t) is the probability distribution of theelectrons in islands 1 and 2, at time t, given that therewas a r, 2 hop at t =0. Note that the sum over nl ex-

13 526 EGUES, HERSHFIELD, AND WILKINS

eludes the n1=0 state because island 1 has to have atleast 1 electron in order for a r„hop to occur. The prob-ability distribution p, z(n, , n 2, t) is given by

p, 2(N, M) = —pp(N, M 11)r,z(M + I ) .1

0

The normalization factor D is

Nl, N2

(B3)

N N —1l' 2

p, z(n&, n&, t)= g p(n&, n 2tl»M 0)p, 2(N M),N=O, M=O

N=1, M=0pp(N M)r p(M):—h2: (B4)

(B2)

where p(n„n2, tlN, M, O) is the conditional probability offinding n, and n 2 at time t in islands 1 and 2, respective-ly, given that there were N electrons in island 1 and Melectrons in island 2 at t =0. Observe that the M =N2state is not included in Eq. (B2) since a r, 2 hop occurredat t =0. The probability distribution of electrons rightafter a r, 2 hop (at t =0) is denoted p,2(N, M). This quanti-ty is proportional to the product of the steady-state prob-ability of having N and M + 1 electrons in islands 1 and2, respectively, and the probability of occurrence of a r, 2

hop:

Nl N2

h„,2(t) =~2+c1%c2 N=O, M=O

n, (N, M, t)

Xpp(N, M + 1)(M + 1),(B5)

where n, (N, M, t) is given by Eq. (7). By using the end-point conditions, Eqs. (17) and (18), we obtain

Note that we are assuming that the system is in thesteady-state regime right before the r, 2 hop at t =0, i.e.,we are considering fluctuations about the steady-state re-gime. Substituting Eqs. (B2) and (B3), together with theexpressions for r„and r,2, into Eq. (B1), we have

l( +t

h„,~(t) —h ( =g„,~(t) =h 27 c1+c2

[a+ (5n, 5n& ) +b+ [((5nz) ) —(n2 ) ]]

+ [a (5n, 5n )2b+[((—5nz) ) —(n2)]] . (B6)

In the noise expression, Eq. (33), we need the zero-frequency component of the cosine Fourier transform of the reducedcorrelation functions. In Table I we present the zero-frequency component of all nine correlation functions needed tofully determine the noise.

'Present address: Department of Physics, University of Florida,Gainesville, Florida 32611.

1J. B.Johnson, Phys. Rev. 29, 367 (1927).2H. Nyquist, Phys. Rev. 32, 229 (1928).R. Landauer, Physica (Amsterdam) 38D, 226 (1989).

4G. B. Lesovik, Pis'ma Zh. Eksp. Teor. Fiz. 49, 513 (1989)[JETP Lett. 49, 592 (1989)].

~B. Yurke and G. P. Kochanski, Phys. Rev. B 41, 8184 (1990).M. Biittiker, Phys. Rev. Lett. 65, 2901 (1990).

7R. Landauer, and Th. Martin, Physica B 175, 167 (1991);Th.Martin and R. Landauer, Phys. Rev. B 45, 1742 (1992).

~C. W. J. Beenakker and H. van Houten, Phys. Rev. B 43,12 066 (1991)~

C. W. J. Beenakker and M. Biittiker, Phys. Rev. B 46, 1889(1992).L. Y. Chen and C. S. Ting, Phys. Rev. B 43, 4534 (1991).John H. Davies, Per Hyldgaard, Selman Hersh6eld, and JohnW. Wilkins, Phys. Rev. B 46, 9620 (1992).L. Y. Chen and C. S. Ting, Phys. Rev. B 46, 4714 (1992)~ Thiswork is equivalent to that in Ref. 11; however, in the presentpaper we follow the notation and procedures used in thelatter to calculate the shot noise.

' Yuan P. Li, A. Zaslavsky, D. C. Tsui, M. Santos, and, M.Shayegan, Phys. Rev. 8 41, 8388 (1990). For some experi-

mental results on microwave noise in double-barrierresonant-tunneling structures, see Refs. 25 and 26.M. Buttiker, Physica B 175, 199 (1991).For a study of the co&0 noise in alloy-based resonant-tunneling structures, see E. Runge, Phys. Rev. B 47, 2003(1993).S. Luryi, Appl. Phys. Lett. 47, 490 (1985).

' For a discussion in the context of tunneling see R. Landauer,J. Appl. Phys. 33, 2209 (1962).According to the Ramo-Shockley theorem, a g hop causes acharge ae to flow in the external circuit. See S. Ramo, Proc.IRE 27, 584 (1939) and W. Shockley, J. Appl. Phys. 9, 639(1938).

M. J. Buckingham, Noise in Electronic DeUices and Systems(Wiley, New York, 1983),pp. 34.

20See, for instance, Athanasios Papoulis, Probability, RandomVariables, and Stochastic Processes, 2nd ed. (McGraw-Hill,New York, 1984), pp. 270, 491, and 492.For an illustration of this procedure in the context of correla-tion and structure functions in disordered materials see J. M.Ziman, Models of Disorder (Cambridge University Press,Cambridge, 1979), pp. 124.Selman Hershfield, John H. Davies, Per Hyldgaard, Christo-pher J. Stanton, and John W. Wilkins, Phys. Rev. B 47, 1967

49 ZERO-FREQUENCY SHOT NOISE FOR TUNNELING THROUGH. . . 13 527

(1993).While ((5n, i ) and ((5nz) ) are by definition positive quan-tities, (5n &5n2 ) is necessarily negative (or zero) in our model.The physical reason for this is particle-number conservationunder internal scattering. Any relevant fluctuation that de-creases n, relative to (n, ) must increase n2 relative to (n2 ).Thus, the main contributions to (5n, 5n2) come from 5n&

and 5n2 having opposite signs. Mathematically this effectcomes from including particle-number conservation in themaster equation (16).

We should point out that the variances can also be determinedby following the general approach for multivariate processsystems proposed by K. M. van Vliet and J. R. Fasset, inFluctuations Phenomena in Solids, edited by R. E. Burgess(Academic, New York, 1965), p. 267.J. I. M. Demarteau, H. C. Heyker, J. J. M. Kwaspen, T. G.Van de Roer, and L. M. F. Kaufmann, Electron. Lett. 27, 7(1991).T. G. Van de Roer, H. C. Heyker, J. J. M. Kwaspen, H. P.Joosten, and M. Henini, Electron. Lett. 27, 2158 (1991).