Noise spectrum and the fluctuation-dissipation theorem in mesoscopic rings

ANNALS

OF PHYSICS

206, 68-89

(1991)

Noise Spectrum and the Fluctuation-Dissipation Theorem in Mesoscopic Rings YUVAL GEFEN

The Weizmann

Department Institute

of Nuclear of Science,

Physics, Rehovot 76100, Israel

AND

ORA ENTIN-WOHLMAN Raymond

School of Physics and Astronomy, and Beverley Sackler Faculry of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Received June 4, 1990

A disordered one-dimensional metallic ring threaded by a magnetic flux is considered. The ensemble-averaged noise spectrum of the thermodynamic, current fluctuations is calculated, and its dependence on flux, temperature, system size, and electron density is studied. Weak localization results are reproduced in the infinite-size limit and found not to contribute, to leading order, to the flux dependence. For a finite-size ring, the fluctuation-dissipation theorem is invoked to relate the noise spectrum to the real part of the ac conductance. The latter may be either a maximum or a minimum as a function of the flux, depending on the parity of the number of electrons. The results are discussed in view of previous studies of related problems. 0 1991 Academic Press, Inc.

I. INTRODUCTION The study of transport in submicron quantum systems has been proven to be an exciting area of activity for both theorists and experimentalists [l-3]. Concomitantly, efforts have been made in studying thermodynamics of mesoscopic systems, most notably, systems of Aharonov-Bohm geometries [3-lo]. Attempting to bridge together thermodynamics and transport, one is evidently allured by the fluctuation-dissipation theorem [ 111. Understanding the onset of dissipation in such quantum systems is of fundamental importance. In order to gain an insight into the nature of dissipative processes, it is convenient to define and analyze simple model systems. An example is a strictly one-dimensional (metallic) ring of circumference L, threaded by a magnetic flux, @. This system is depicted in Fig. 1. Once 0 = Q(t) varies with time, an electromotive force is generated around the loop, which, in turn, brings forth currents. One may 68

OOO3-4916/91 $7.50 Copyright 0 1991 by Academic Press, Inc. All rights of reproductmn in any form reserved.

FIG.

NOISE

SPECTRUM

1.

One-dimensional

IN MESOSCOPIC

Aharonov-Bohm

RINGS

69

geometry.

calculate the work done by the external field. If the system is to remain in a steady state (when averaged over long times), the energy added to it has be be extracted by some external agent (for example, a reservoir of harmonic degrees of freedom), i.e., it has to be dissipated [12]. Thus, by calculating the total work done on the system, one can estimate the power dissipated. Two distinct mechanisms have been proposed so far in this context: (1) One may invoke a relaxation mechanism [ 131 that governs the dynamical equations for the density matrix, p, in the basis of the adiabatic energy states, E(@(r)). This density matrix is assumed to be diagonal. A finite relaxation time causes a delay between the instantaneous p(@(t)) and the equilibrium p,,(@(t)), associated with the same value of the flux. This, in turn, gives rise to net work done on the system, hence to dissipation. This mechanism, in the context of periodic adiabatic spectra, has been put forward by Landauer and Biittiker [14], Buttiker [15], Imry and Shiren [16], Triveldi and Browne [9] and others [17]. (2) Varying @ with time may induce interlevel Zener transitions. Energy may be pumped into the system and will eventually by dissipated [l&22]. This channel of dissipation (as opposed to the previous one) survives in the zero temperature limit. We note that this mechanism goes beyond linear response since the probability for a Zener transition vanishes exponentially as the induced electromotive force (proportional to &D/tit) tends to zero. In this work we study the fluctuations in the current along a metallic loop, associated with the first machanism described above. In addition to the basic importance of studying such a phenomenon and relating it to the frequencydependent conductivity (associated with dissipation [ 11, 23]), our study has direct experimental relevance as far as one (or few)-channel samples are concerned [24]. It may also be viewed as an essential step in understanding the physics of multichannel systems. One expects three physical factors to be determinant in the noise spectrum of a mesoscopic sample: (i) The discreteness of the energy spectrum implied by the finite size of the system considered; (ii) The coherence of the electronic wave-functions around the loop; this gives rise to effects related to Ahoronov-Bohm oscillations [6, 251; (iii) Static disorder brings forth elastic scattering, leading to localization effects. There are several controllable physical parameters which affect the noise spectrum. These include temperature, magnetic flux, and the amount of disorder. It is the interplay among the above factors on which we focus our attention. We stress

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GEFENANDENTIN-WOHLMAN

that there are various significant aspects that will be ignored in the following analysis, most notably, inelastic scattering and coupling to an external reservoir. In the present paper we study the noise spectrum of the current in a finite, onedimensional conducting ring threaded by a magnetic flux. The noise spectrum is derived by Fourier transforming the currentcurrent time correlation and is related, through the fluctuation-dissipation theorem, to the ac conductance of an isolated ring. In Section II we define our model and construct the quantity to be studied (Eqs (11.11) and (11.12)). In Section III we evaluate the noise spectrum within the bare “bubble” diagram approximation (Eq. (111.13)) and simplify the expression for various limiting cases. In particular, we find the leading dependence on the magnetic field due to the Aharonov-Bohm effect. Localization corrections (more precisely, corrections due to ladder and maximally-crossed diagrams [26,27]) are evaluated in Section IV. We show that they depend weakly on the flux and these, in general, become substantial only in the thermodynamic limit. Section V contains a discussion of the physical significance of our results. Some of our expressions are related (through the fluctuation-dissipation theorem) to previously obtained results on transport in one dimension. There are two appendices to this paper. Appendix A discusses a toy model of two levels which depend periodically on a “flux.” The noise spectrum is calculated assuming a single relaxation time [13-151 and is shown to be compatible with the conductance of that system, as calculated earlier [ 13, 14,28,29]. In Appendix B we consider the contribution of other families of diagrams to the noise spectrum.

II. FORMULATION

OF THE PROBLEM

We consider the system depicted in Fig. 1, whose Hamiltonian

H=&(P-;A>2+v(e).

is (11.1)

The coordinate 13,as well as all vectors, are measured along the tangential direction. The magnitude of the vector potential, A, is (11.2)

A = @IL,

where L is the circumference. The scattering potential satisfying

V(O) is taken to be random,

(v(e) V(P)} = LV2 qe-

(V(Q> =o,

e’),

(11.3)

where S is a periodic delta-function (with period L), and { } denotes the ensemble average over the scattering potential configurations. The strength of the potential fluctuations is related to the elastic mean free time, z, by h/T

= 2?‘CN(Ep) v2,

(11.4)

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RINGS

where N(E~) is the density of states ( number of states per unit energy) at the Fermi energy. We are interested in the current-current time correlation, K(t), (11.5)

(K(t)} = { U(t) Z(O)) - (<0)*~~

where ( ) denotes a thermodynamic average. K(t) for a two-level system is calculated in Appendix A. The current operator, Z(t), is Z(l)=C

zyy,C~Cy’ei(~~~~~~)f’*,

(11.6)

YY’ where the subscripts refer to the single-electron eigenstates, 17). CT (c,) is a Fermion creation (annihilation) operator of the yth-state. ZyySare the $-matrix elements of the current operator in that basis and E,, are the respective energies, both quantities being functions of @. Employing standard identities of Fermi operators we obtain K(t)=~Z,,~Z,,~,e IT’

i’“-“,“‘hf(E,)(l

-f&J).

(11.7)

Hereafter f denotes the Fermi-Dirac distribution. We now express the matrix elements in Eq. (11.7) in the basis of the “pure states,” lp), i.e., the electronic eigenstates in the absence of the scattering potential (plane waves). A similar approach has been taken in Ref. [8]. The current operator is diagonal in that basis. Defining ayp = ( p 1y ), it follows that (11.8)

Zyys = C Zpa&ayp, where ZP= ZgP. Substituting

this in Eq. (II.7) and Fourier transforming

we obtain

(11.9)

x(1-f(E,.))d(E,-E,,+hw).

In order to carry out the ensemble average of K(o), it is convenient to write it in terms of advanced and retarded Green’s functions, GA and GR, (11.10) From Eqs. (11.9) and (11.10) we then find (ti = 1) &f(E

+ a) --f(E))

Y(E, 0, $1,

(11.11)

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ENTIN-WOHLMAN

where

Y(E, ~4) = c Z,Z,,((G’?p, P’,E+0)- GR(p, P’,E+0)) P. P’

x (G”(P’,

(11.12)

P, E) - GR(p’, P, &)I}

and /I = l/k,T. Equations (11.11) and (11.12) are our basic expressions for the noise spectrum. In the following we apply them to a finite-size system; in that case the summations are over a discrete set of momenta which depend upon the flux [8] (11.13)

P=$(n+o,

where 4 = @/GO (GO = he/e being the flux quantum) and n is an integer. The ensemble average implied in Eq. (11.12) will be evaluated diagrammatically; the main ingredients are the ensemble-averaged Green’s functions {Gb,p’,d}

=6,&p,

E),

where &p- p2/2m are the pure system single-electron energies (i.e., in the absence of disorder). The important contributions to the physical quantities at hand arise from levels at the vicinity of the Fermi energy, .sF. It is therefore convenient to linearize the energy spectrum [6, S] of the pure system, as depicted in Fig. 2. Using (11.13)

FIG.

2.

The original

spectrum

cp (solid

line) and the linearized

spectrum

(dashed

line).

NOISE

SPECTRUM

IN MESOSCOPIC

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73

to relate p and n, the discrete energies E,, on the right and left branches of the energy spectrum, are given by E,=E~+Ao+(nT(b)A~.

(11.15)

Here A, is twice the spacing between energy levels at the Fermi energy, A,=-

27Tv, L ’

(11.16)

where vF is the Fermi velocity and A, depends upon the parity ,of the number of electrons, N, in the conduction band A,=

0 4112

Accordingly,

we also approximate

for even N, for odd N.

(11.17)

ZP by its value at the Fermi level, ZPF, (11.18)

for the right and left branch, respectively. Furthermore, since the dependence on q5 is periodic with period 1, q5may be restricted to vary between, say, -f and 5. The &dependence of I,,, may thus be neglected, giving rise to errors of the order of l/N4 1. It follows that Z,, = T ev,lL,

(11.19)

where vF = rNJmL.

III.

MAGNETIC

FLUX

DEPENDENCE

In this section we calculate contributions to {K(o)} due to the bare “bubble” diagrams shown in Fig. 3. These consist of replacing the ensemble average of a product of Green’s functions by the product of their averages. In particular, we derive the flux dependence. In the next section we analyze the d-dependence due to other diagrams (ladder and maximally-crossed) and argue that it is negligible

FIG. 3. Bare bubble diagrams: a. G RG R; b. GAG*; c. GRGA (the solid lines represent ensemble averaged Green’s functions).

74

GEFEN

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ENTIN-WOHLMAN

compared with the dependence we find here. Thus, the expressions derived in this section (see Eq. (111.13) and the limiting cases (i)-(v)) yield the leading dependence of the noise spectrum on the gauge field. Using Eqs. (11.14) and (11.15), we find that the bare “bubble” contributions to Y (Eq. (11.12)), denoted Y,, take the form

(111.1)

+ {4- -41.

The first term here represents the summation over the right linear branch of the spectrum, where a=(&+m-Ado-i/2z)/A1, (111.2) b=(E-Ado-i/2T)/A,. The contribution of the left linear branch is obtained from that of the right one, by changing 4 into -4 (last terms in (111.1)). Using the identity [30]

71x, cn -=7ccot x-nl Equation

(111.3)

(III. 1) becomes i/T

YO(E,f-9 4) = -A,ow~+(~/T)*

C(w + i/T)k(a)

- g(b*)) (111.4)

- (w - i/zMa*) - g@))l, where we have introduced

the notation

g(x) = cot x(x + 4) + cot x(x - 4).

(111.5)

In order to simplify these results, we now assume that the difference between a and b (cf. Eq. (111.2)) is negligible. This is, for example, the case when or 4 1. Equation (111.4) then yields 47r1zF sinh r@A1

YO(E,w,d) = - - zA, CO’+ ( 1/r)2 x 1 coshn/rA,-cos2n(y+q+)+((+ [ i(

-,)I.

(111.6)

We note (cf. Eqs. (11.4) and (11.16)) that TC/ZA,= L/2zu, = L/21, where I = uFr is the elastic mean free path. Thus, in the L/lb 1 limit, Y, tends to the value -4e2(1/L)/(1 + (07)~) and is independent of 4, as expected. (Here we have used Eq. (11.19)).

NOISE

SPECTRUM

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RINGS

The contribution of the bare bubbles to the noise spectrum is thus given by inserting Eq. (111.6) into Eq. (11.11) and carrying out the energy integration. We denote this contribution by (K>,, and evaluate it in two limiting cases: (i) or < 1, /lo 4 1, PA, 9 1. In this case we may neglect the energy dependence of Y,. The energy integration in (II. 11) then yields a factor of -w, whereas the Bose factor becomes (exp( -/3w) - 1))’ - -k, T/o. The trigonometric functions in (111.6) reduce to cos 27r(4+ A,/A,) and hence yield fcos 2x4 (cf. Eq. 11.17)), depending upon the parity of the number of electrons. Consequently, {Z&=k,T=

sinh( L/21) 1 z L cosh(L/21) T cos 24 1 + (or)”

(111.7)

where the upper (lower) sign stands for even (odd) parity of the electron number. (ii) wz < 1, j?wS 1. To consider the large /?o limit, we note that f(e + u) --f(c) = (exp( --Pa) - l)f(E)(l --f(s + 0)). At low temperatures, the energy integration is thus limited to the range --w
x [tan-‘(sinh(L/21)

tan ~(4 + w/A,)/(cosh(L/21)

T 1)) + (# + -d)].

(111.8)

We note that for small enough frequencies such that w G A,, this result reduces to sinh( L/21) {K),=02eZi 71 L cosh(L/21) T cos 2nd This is the low temperature there replaced by o.

1

1 + (~r)~’

(111.9)

counterpart of the previous case (Eq. (111.7)), with k, T

In both cases (i) and (ii), {K},, is periodic in 4 with a unit period (which translates into @,-periodicity when considered as a function of the applied flux, a). The sensitivity to the flux decreases exponentially with L/21, i.e., as the system size or the amount of disorder increases. This is clearly seen in Eqs. (111.7) and (111.9)), where the leading behaviour of the first harmonic in 4 is attenuated by a factor of (cosh(L/21)))’ [7]. It is therefore desirable to obtain a systematic expansion of {K(o)}, in harmonics of 4. This will also enable us to address more general situations. Denoting (fw)},=~

{K(~))O

e-i2ny6,

v-integer,

(111.10)

76

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AND

ENTIN-WOHLMAN

and using Eq. (111.4) we obtain cos-+-sin y 1,.

YO(&, w, 4) = - 52

4 *(l+e[ L 1+(0X)

-y]

xe -vL’21( - 1 Y” cos(2nv53COS2~v~).

Next we carry out the energy integration [30] implied in Eq. (II.1 1 ), 2n 1 dC(I(E+w)-~(E)jCOs(2XV~) (2ni)’ e-j0 - 1 I 1 sin m~w/A, =- 1 28 e-B0 - 1 sinh 21r~v//3A, ’

(111.12)

As a result we obtain the noise spectrum in series of the &harmonics bubble approximation)

’

(

W

-+4n k,T -“L/2/(

f V=l _

1 )NV

cos [ sin

(in the bare

7Lvw 1 . 7cvw -I- I + -Wt sm I- 1 I

nv*lA

1

cos 27cv$ . (111.13) > This expression is amenable to experimental investigation. It describes the #-harmonic dependence upon various controllable parameters of the system, i.e., the frequency, temperature, disorder, and system size. In deriving it we have ignored the effects of inelastic and phase-smearing scattering. Otherwise, the range of validity of (111.13) is quite wide, as has been indicated throughout the calculation. One notes that a finite temperature and higher amount of disorder reduce the sensitivity to the flux, higher harmonics being more attenuated. Let us now consider several limiting cases of Eq. (111.13): xe

sinh 2n2v//3A,

(iii) L/Z9 1. In this limit only the lirst few harmonics survive. (iv) or Q 1, PA, G 1. In this case we approximate sinh(2rr2v//?A,) by its leading exponential term and perform the summation over v, keeping only the second term in the square brackets in (III. 13). The result is

x(

w : 7c [ e”Tcos2n# _ e”Ty2n(qS+w/A,) k, T 2wr cash CLT cos 27~4 cash a + cos 2n(p5 + w/A,)

+(4+

-4)

I> 7

(111.14)

77

NOISE SPECTRUM IN MESOSCOPIC RINGS

where we have denoted (111.15)

c+l+27ctk,Tr), and the f signs stand for an even (odd) parity of the electron number.

(v) wr b 1, fld i $1. Here we keep only the first term in the square brackets of (111.13) and sum over v as in the previous case, to obtain

sin 2rr(# + w/A,) +((b+cash CIr cos 27c(~,5 + w/A 1)

4

.

(111.16)

The expression above yields the limiting behaviours as a function of 4, within the bare bubble approximation. In the next section we discuss further contributions to the noise spectrum which, however, contain weaker dependences on 4. In the last section we put the above results in the context of related works.

IV. FURTHER IMPURITY CORRECTIONS The additional impurity corrections to the noise spectrum, derived in this section, are associated with two infinite series of diagrams: ladder and maximally-crossed graphs (Fig. 4). These are frequently discussed in the context of weak localization corrections to the conductance. Returning to Eq. (11.12), we separate Y as m, 094) = YRR(E, f&4) - YRA(E, w, 41,

(IV.1)

where

YRR(&, 0,4) = 2 Rec ZpZps{GR(p, P’,E+w)GR(p’, P,&)I, P-P’

YRA(GW,

$I=2

Re

c

ZJ+,{GR(p,

P',

E+W)

G*(p',p,~)}.

P.P’

FIG. 4.

(a) Ladder

and (b) maximally-crossed

diagrams.

(IV.2)

78

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We now consider in detail Y,,. In the case where this contribution is dominant, Y,, may be disregarded (see below). The contributions of the ladder diagrams to YRA, denoted Yk,, and the maximally-crossed ones, denoted Yzz (cf. Fig. 4), are given by

Yi, = 2 Re c l,Jp,GR( p, E+ w) G*(p, E) GR( p’, E+ w) G*( p’,

E)

P3 P’ u

(IV.3)

and YF: = 2Re c IpIp.GR(p,

E + o) G*(p, E) GR(p’, E + w) G*(p’, E)

P. P’

U”X( p + p’, E, E + Co) xz-ux(p+p’,c,E+w).

Here GR and GA are the ensemble-averaged U=

(IV.4) Green’s functions, (see Eqs. 11.14));

1 = Al/4712 27crN( EF)

is the matrix element squared of the scattering potential

(IV.5) (cf. Eq. (11.4)); (IV.6)

X(k,~,&+o)=~G~(p,~+a)G*(p-k,E)

with the momenta p and p’ given by Eq. (11.13) and the momentum 2m, /L, n i = integer.

transfer k being

As is usually the case, the contributions (IV.3) and (IV.4) become important when the denominators in those expressions tend to zero. In the present situation, however, the quantity X depends upon 4 and the size of the system and thus will not necessarily lead to the usual diffusion pole. Calculating X by the linearization procedure defined in Section II, we find X(k, E, &+co)=

-*

1 A, (o + i/z)‘-

(uFk)’

((0 + i/z) C, + MC,),

(IV.7)

where we have denoted c, = f (cot 7T(u* + 4) + cot x(a* - $b)-cot R(b + qb)- cot 7r(b - (b)), c2 = ; (cot 7T(d + 4) -cot n(a* - 4) -cot K(b + 4) + cot 7c(b- fs)),

(IV.8)

NOISE

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and a and b are defined in (111.2). Using (IVS), (IV.7), and (IV.8), the denominator of Yk* is 1 + C,/(z(o + i/z)) and will become small in the small OT limit when C, approaches -i. To examine this possibility, we note that the relevant range for the energy integration (cf. Eq. (11.11)) is 1~1
-2eC,/(o+i/z),

(IV.9)

P

where C2 is given in (IV.8); ~Zp~,ZpGR(p,&+~)GA(p,

4GR(k--p,

~+m)G*(k-p,~)

P

Al nl+2~f(w+i/r)2+A:(n,+2gl)2 c2

= --

1,

x((W+i/T)C1+Al(nl+2gl)C,) where we have denoted k = 2nn,/L

(IV.10)

and

C,=$(cot$a*+4)-cot~(a*-#)+cot~(b+d)-cotr(b-4)).

(IV.ll)

Using these results and the identity

u l-UX(O,E,E+W)

U2X(k.

E. E + 01

’ l-UX(k,s,s+w)

U2X(0, E, E+ Co) =l-UX(O,s,s+~)+l-UX(k,s,s+w)’ 595/206/l-6

u

(IV.2)

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we obtain

=2Re

4eZC2 U2X(0, E, &+ 0) (W+i/:)Z1-uX(O,&,E+W)

-2Re%f

’ u 7c (w + i/g2 cn, l--X(n,,&,&+O)

Al G n,+2~+(w+i/+A;(n,+2qh)2

((0 + i/z) C, + A,(q + 24) C,)

1 .

(IV.13)

Under the conditions for which 1 - UX is very small (see discussion following Eq. (IV.8)), both C2 and C3 are exponentially small (exp( -L/21)), whereas C, - -i. In that case Eq. (IV.13) becomes Yk,+

e2 Yr(‘,C=2Rez40,

(IV.14)

i.e., it is given solely by the contribution of the maximally-crossed graphs. To examine weak localization corrections to this quantity we add the corresponding contribution of the bare bubble (i.e., 2 Re{GR} {GA>). The latter reads Y@)=2Re-2u RA

e2 L

-C, Fm+i/r’

Note that this is identical to Eq. (111.6) for or 6 1, i.e., the contribution negligible. The results (IV.14) and (IV.15) may be written in the form YgA= YC+

Yk,+

YF’,=2Re-

2e2v, i L w+M(o)’

(IV.15)

of Yfi is

(IV.16)

where (IV.17)

This is identical to the expression obtained in [26], for an infinite system. In the present case, however, we are able to consider the effect of the finite size of the system. A straightforward calculation of the second term in (IV.17) yields 2u ck w + ir(+k)’

=iy&cotnG.

(IV.18)

NOISE

This is, however, derived term becomes comparable [26], replacing the bare arrives at a self-consistent

SPECTRUM

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81

with the bare diffusion coefficient, D,, = vir. When this to l/r one has to carry out a self-consistent calculation diffusion coefficient by D(o) = D,(i/zM(o)). One then equation for M(o), (IV.19)

The solution corresponding

to localization

has the form

i.e., it describes an insulator. Equation (IV.19) then yields o,, N l/27, i.e., the infinite system result in one dimension [26]. Examination of Eqs. (IV.17) - (IV.19) reveals that under the restrictions specified above (cf. discussion following Eq. (IV.8)) (which led to the form (IV.17)), Eq. (IV. 20) with O,,N l/22 is the only consistent solution. We therefore obtain (127 4 1) rg!&-to leading order. Introducing spectrum

4e2v, L

d/t

0; +

(IV. 21)

(o/zy

this expression into Eq. (II.1 l), we find for the noise

This yields

the first factor here being twice the conductance of the one-dimensional system. This calculation shows that weak localization corrections are reproduced (for small frequencies) in the L/l B 1 limit. In that regime, the flux dependence is exponentially small. When finite-size systems are considered, the denominators (1 - UX) in (IV.13) deviate from the usual diffusion pole form. For small frequencies we find that o + iz(~,k)~ is modified into co+:+

i7(v,k)2,

C=l-iC,=l+

sinh L/21 cash L/21 T cos 214’

(IV.24)

One notes that iC/r (in the regime where C > 0) formally plays the same role as the spin-flip scattering rate or the inelastic scattering rate [26]. Hence, just as in the

82

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case where those rates are included, the theory does not yield localization but rather corrections to i/r, of the order of l/8. We have included in this calculation the contributions of {GRGA), and disregarded the terms of the form {GRGR}. We have checked that in the small frequency limit (or 4 1) the latter contribution to the energy integration (cf. Eq. 11.11)) is exponentially small in L/2L. These terms contribute to the imaginary part of the conductivity (giving rise to diamagnetic persistent currents [7]). Since {K(o)} is real [ll], the {GRGR}-terms do not play a role in the noise spectrum, in that limit. Other possible contributions to {K(w)} are discussed in Appendix B.

V.

DISCUSSION

We have considered the noise spectrum of the current fluctuations, {K(w)}, in a one-dimensional mesoscopic ring threaded by a magnetic flux. Here, the term mesoscopic refers to the fact that the single-electron wave-functions are assumed to be coherent around the ring (Z4$ L, where I, is the phase-smearing length [l-3]). It has been shown previously that interesting flux-dependent effects, which may arise in thermodynamics as well as in transport quantities, are direct consequences of coherence and discreteness. These are also the origin of the sensitivity of (K(w)} to the flux, which is a manifestation of the Aharonon-Bohm [25] effect in such geometries. The analysis was performed within the independent electron picture. We have found the dependence of the noise spectrum of various parameters of the system, including flux, temperature, disorder (r), and the system size (A,). More technically, we have shown that the principal flux dependence arises from the bare “bubble” diagrams. Other contributions (i.e., localization corrections due to maximally-crossed graphs) may add important corrections to {K(o)} under certain conditions; these, however, depend on the flux very weakly. Particularly instructive is the expansion of the noise spectrum in harmonics of 4 (Eq. (111.13)). It is seen that these harmonics are attenuated exponentially with temperature and disorder [68]. The characteristic energy scale for both attenuation factors is A,. Higher harmonics are attenuated stronger. From our experience with other physical quantities [7, 81 we propose that the harmonics may be further attenuated when a finite 1, is introduced, by a factor -exp( - Lv/21,) (v being the harmonic index). This, similarly to temperature and disorder, I, may serve as a cutoff on the number of relevant harmonics. The noise spectrum is averaged over an ensemble of scattering potentials. It is related to the conductance of the system, G(o), through the fluctuation-dissipation theorem

{K(o)} = 1$w

[G(o) + G( -w)l.

W.1)

NOISE

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83

When neglecting localization corrections, the conductivity of the system, a(o) = G(o)L, for finite values of rc/A,z = L/X, is thus (cf. Eqs (111.7) and (111.9))

I&.&z!

1

sinh( L/21) m L 1+ (02)~ cosh(L/U) T cos 2714’

W.2)

where the T stands for even (odd) number of electrons. When L/21 + cc (the thermodynamic limit) this reduces to the Drude expression for the one-dimensional ac conductivity. (However, in this limit, localization corrections are crucial and Eq. (V.2) is not the leading condtribution then !!). Similarly, the expansion of the ac conductivity in harmonics of 4 is obtained from (111.13) in conjunction with (V.l). In the weak localization correction regime (l/L + 0, or < 1 ), we obtain from Eq. (IV.23) that the conductivity vanishes as o 2. This is characteristic of insulators [31-331. Few technical comments are due, concerning the difference between K(w) and the ensemble-averaged quantity, {K(o)}. One might expect that the former consists of d-function spikes at those frequencies which correspond to spacings between the energy levels of the particular realization. If we bias this member of the ensemble by an external ac field at such a frequency, resonant absorption is made possible, which gives rise to a b-function spike in G(o). By the fluctuation-dissipation theorem, this will show up as a spike in the noise spectrum. We note that the ensemble averaging procedure smoothens this spike spectrum [33-341. It is interesting to compare our analysis with previous works. The first studies along this direction were by Landauer and Biittiker [14] and by Biittiker [ 15, 28, 291, who have analyzed a two-level model for a ring. The noise spectrum for a flux-periodic two-level system (assuming a single relaxation time) is derived in Appendix A and is found the be compatible, within the fluctuation-dissipation theorem, with their results for the conductance. Imry and Shiren [16], Imry [l], and Triveldi and Browne [9] have also considered dissipation in Aharonov-Bohm geometries. Some of analytical results presented here are compatible with their tindings. One notable exception concerns the question whether the conductance, as a function of the flux, has a minimum or a maximum at C$= 0. (The fact that the single-electron energies have extrema at 4 = 0 implies that the conductance has an extremum as well). Our results (e.g., Eqs. (111.7~(111.9) and Eqs. (111.13)-(111.16)) imply that the noise spectrum, hence the real part of the conductance, may be either a maximum or a minimum, depending [35] on the parity of N (the electron number). (Within the framework of the grand canonical ensemble, we define N as the number of electrons at T= 0, 4 = 0.) We believe that the significance of this work lies in adding to our understanding of submicron systems. By putting our results for the noise spectrum within the context of transport properties (including localization effects) on the one hand, and thermodynamic properties (e.g., fluctuations, persistent currents) on the other hand, one may hope that we are closer to a unified picture of electronic properties of mesoscopic samples. One may also view our analysis as a starting point for a

84

GEFENAND ENTIN-WOHLMAN

systematic study of higher-dimensional Aharonov-Bohm systems. Beyond the theoretical significance of our study, we note that the predictions we make are amenable to experimental investigation. At present, semiconducting systems consisting of very few transverse modes (conducting channels) may be fabricated [24]. Studying the flux and frequency dependence of the noise spectrum may be of considerable interest. From Eq. (V.l) it is straightforward to derive the expression for the conductance, and, in turn, its various limiting behaviours as discussed in this paper. APPENDIX NOISE IN A FLUX-PERIODIC

A: TWO-LEVEL

SYSTEM

Here we calculate the noise spectrum of a two-level system. Such a system, whose spectrum is a periodic function of the applied flux, has been considered by Landauer and Biittiker [14] and by Biittiker [ 15,28,29]. In these works the energy dissipation was calculated for a system biased by a time-dependent flux. Here we evaluate the fluctuations in that system; comparison with the previously calculated dissipation shows that the results are in accordance with the fluctuationdissipation theorem [ 11. We follow the model put forward by Landauer and Biittiker. The two levels are denoted by subscripts + and -. These levels carry the currents I, and I-, respectively [36]. Associated with this system is a 2 x 2 density matrix, p, whose diagonal elements are denoted by p+ and p _, respectively. The off-diagonal elements are assumed to be zero. In the presence of a time-dependent flux, the latter assumption amounts to neglecting Zener transitions between these levels. We also use p”, and p” to denote the instantaneous equilibrium occupation probabilities. Evidently, p+(t)+p-(r)=pO,

+p”

= 1.

(A.1)

We assume that the dynamical equations are governed by a single equilibrium time [13-151, req. The resulting equation for, say, p- is dpdt=

P--PO+P+-PO+ Teq Teq

(A.21

By Eq. (A.l) we obtain f$+

-2(p-

-p”)/z,,.

The respective solutions of Eq. (A.3) with the initial (B):p-(t=O)=O, are ~“(t)=exp(-2t/z,,)+p0(1--xp(-2t/r,,)), p”_(t)=pT(l

-exp(-We,)).

(A-3) conditions

(A): p-(t = 0) = 1;

(A.4)

NOISE

SPECTRUM

IN MESOSCOPIC

Our goal is to calculate the currentcurrent

correlation,

K(t) = ((Z(O) - (OMQ

85

RINGS

K(t),

- (0)).

(A.51

In calculating K(t) we have to account for conditional probabilities. For example, if at t = 0 the system is in the state - (the probability of this to happen is p”_ ) then, at a later time, t, its probability to be in the state -is p”p”_(t). This is the weight factor that appears with the contribution (I- - (I))* to K(t). Similarly, the system can be found in the state + at time t with probability p” (1 - p!(t)), which is the weight factor of the (I_ - (Z))(Z+ - (I)) contribution. Overall we obtain

K(t)=pOC(z~)*pA(f)+z-z+(l

-PA(t))1

+(I -P”)Cz+z-PB(f)+(z+)2(1

-PBw)14(0)*,

(A.61

which yields K(t)=exp(-2t/z,,)p0(1

-pO)K

t>o.

-Z+)*,

64.7)

Let us now define the noise spectrum, K(o), as = 2 Re O” K(t) eiwf dt. s-cc

(A.8)

Teq ,pO(l -p”)(Z- -I+)*. 1+ Weq/2)

(~4.9)

K(w)

Using Eq. (A.7), we then find

K(0) =

Assuming now that p” and p” obey the Maxwell-Boltzmann

statistics, (A.lO)

we obtain e-SAE K(0) = where AE= E, - Ep . The fluctuation-dissipation

% 1 + weqm2

(I- -Z+)’

(1 + e--BAE)2’

theorem, in the classical limit, K(w) = 4k, T/R(w),

(A.ll)

reads [ 111 (A.12)

where R(o) is the frequency-dependent resistance. Combined with Eq. (A.ll), it yields an expression for the “conductance” as obtained by Btittiker (see Eq. (11) of [28] and Eq. (4.6) of [29]).

86

GEFEN

APPENDIX

B:

AND

ENTIN-WOHLMAN

EXAMINATION

OF OTHER

DIAGRAMS

Here we consider the contributions of other diagrams to the noise spectrum. We present a dimensional analysis, which aims at evaluating the possible divergence of these diagrams as a function of the system dimensionality. No attempt is made at computing their contributions in detail. The types of diagrams considered here are depicted in Fig. 5. They consist of maximally-crossed ladder boxes (cf. Figs 5a and 5~) and a ladder of maximally-crossed ladder boxes (cf. Figs. 5b and 5d). Such graphs have been previously discussed in various contexts [26, 37,381. The boxes alluded to above are given by r(k,

w) =

u

(B.1)

1 - UX(k, E, E + co)’

a

P’+p.Pz

P’+P-P,

b

FIG.

5. (a)

Maximally-crossed

types of boxes are defined

ladder boxes and (b) a ladder in (c) and (d), respectively.

of maximally-crossed

boxes.

The two

87

NOISE SPECTRUM IN MESOSCOPIC RINGS

where U and X are defined in (IV.5) and (IV.6), respectively. Consider, for example, the diagram depicted in Fig. 5a. It gives rise to a contribution ‘zF c (GR(p, E+ 0) GA(p’, 8))’ P.P'

P.2)

xGA(p+p'-p2,&)GA(p+p'-p,,&).

To study the possible divergence, we focus on small momentum case (B.2) is approximated by

transfers, for which

2

~WR(~>~+4GA(~,4)2

1P

In the thermodynamic

1

c

n411 r(4d rcq1+

q2

+ P’ - PI.

(B.3)

41.w

limit (see discussion in Section IV) r attains a diffusion pole r(q,o)=i~

1 z 0 + it(u&)2’

(B.4)

The square brackets in (B.3) give rise to powers of z (in the small wr limit). The summations over q, and q2 may be transformed into integrals. These integrals might depend on the upper cutoff of the integrations (for the specific diagram discussed here this is the case for d > 3, d being the dimensionality). The upper cutoff is the inverse of some microscopic length scale in the problem. More importantly, we consider the small q contribution to the integrand. Standard dimensional analysis shows that this diagram contains a divergence of the form ~~~~~~~~~~By comparison, a diagram of the type exhibited in Fig. 5a but with two boxes crossing [38] contains a divergence -&‘-‘)‘*, and generally, if we have n boxes crossing, the divergence is -o (n-3/2)d-n . It turns out that for d > 1 the n = 1 diagram (which is the standard ladder diagram portrayed in Fig. 4) is the most divergent of the family of diagrams just considered. Analyzing the family of diagrams depicted in Fig. 5b in the same way, we conclude that for d > 1 the most divergent member is the n = 1 one, namely, the standard maximally-crossed diagrams. When finite-size systems are considered, one has to perform the summations in (B.3) over discrete values of q1 and q2. r then does not contain a pole (see discussion in Section IV and [ 81). Higher orders in n will contain higher powers of As the denominator of r contains hyperbolic functions of L/l (see UT2 -I/L. Eqs. (IV.7) and (IV.8)), these may not be accounted for. ACKNOWLEDGMENTS We gratefully acknowledge discussions with Y. Imry on the relation of this study to previous works. Discussions with L. Levy on related experiments, and correspondence with N. Triveldi, D. Browne and H. Bouchiat are appreciated. In particular, we would like to thank R. Landauer for his involvement in

88

GEFEN

AND

ENTIN-WOHLMAN

various stages of the work, his critical reading of the manuscript, and his comments that were determinant for the final form of Appendix A. The research was supported by the U.S.-Israel Binational Science Foundation, the Minerva Foundation, Munich, Germany, the German-Israeli Foundation for Scientific Research and Development, and by the Fund for Basic Research administered by the Israel Academy of Sciences and Humanities.

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