Z. Phys. B 94, 487492 (1994) ZEITSCHRIFT FOR PHYSIK B �9 Springer-Verlag 1994

On the additivity of Flicker contributions to the excess noise in quantum conductors J. Hajdu, G.B. Lesovik*

Institut fiir Theoretische Physik, Universit/it zu K61n, Zfilpicher Strasse 77, D-50933 K61n, Germany

Received: 23 December 1993

Abstract. We investigate the quantum interference in- duced non-additive contribution to the excess noise due to several mobile scatters in the diffusion/cooperon ap- proximation. For weak impurity coupling and standard assumption on impurity hopping the relative correction to the noise spectrum is proportional to l og f Although the constant of proportionality is small the logarithmic deviation from the main additive part showing 1If be- havior seems rather remarkable.

PACS: 73.50.Td; 74.40 + K

1. Introduction

The discovery of universal conductance fluctuations (UCF) initiated a number of investigations devoted to the noise in quantum conductors brought about by mo- bile impurities. It has been demonstrated [,1] that at low temperatures the standard flicker noise theory [2] yields a 1If power spectrum if the disordered conductor is build up of a large number of phase coherent boxes, each of which containing at most one mobile scatterer. The aim of the present work is to determine the non- additive contribution of several mobile impurities to the noise spectrum within the framework of mesoscopic fluc- tuation theory [-1, 3, 4]. In the weak coupling limit the relative correction to the noise spectrum due to non- additivity turns out to be c~ ln(c0/Tmin)/ln(Tm.x/Tmin ) where c~ is a certain parameter and Ym~n, Ym,x are the extreme values of the switching rate between two possible posi- tions of the mobile impurities. If both the fixed and the mobile impurities are point-like with the same coupling constant, the parameter ~ is just the inverse conductance

Work performed within the research program of the Sonder- forschungsbereich 341, K61n-Aachen-Jfilich

* Permanent address: Institute of Solid State Physics, Russian Academy of Sciences, Moscow Distr., Chernogolovka 142432, Russia

in units of e2/h. Since this is normaly very small the result given above is still valid even if the logarithmic factor is rather large.

It should be kept in mind that the non-additivity due to quantum interference is not the only one occuring in mesoscopic conductors. We demonstrate this in Sect. 2 by means of a simple model. Our calculation of the quan- tum interference contribution to the noise spectrum (Sects. 3-5) is based on (a slightly modified version of) the equation of motion of the two-particle Green's func- tion in the diffusion/cooperon approximation [-4]. The derivation of this equation is outlined in the Appendix. In Sect. 6 we summarize and comment on the main re- sults.

2. Single-mode channel

Let us consider first a ballistic wire with only one trans- port channel occupied, i.e. the Fermi energy E F is situ- ated between the lowest and the first excited quantized transverse energy level. If there is a static scattering po- tential inside the wire and an external potential V is applied, then the current through the wire is given by the Landauer formula [-5]

2e a I(0 = v W - T (1)

where T is the transmission coefficient associated with the scattering potential.

Suppose now that the scattering potential is due to a nmnber of impurities, two of which are moving whereas the others are fixed. What is the appropriate current expression for such a time-dependent scattering poten- tial? For slowly moving impurities we can use an adia- batic approximation, i.e. calculate T for the instantaneus impurity configuration at time t and insert T(t) into (1). The corresponding current-current correlator reads

( I ( t ) I ( t + r ) ) = V 2 {T(t) r ( t + z ) ) (2)

488

where <. . . ) denotes averaging with respect to the sto- chastic motion of the mobile impurities�9

Obviously, when calculating the current in the adia- batic approximation we omit the current fluctuations due to the discreteness of the electron charge and the random character of quantum transmission which lead to shot noise�9 This part of the excess noise, however, remains finite at small frequencies and its spectral density is always less than the classical value e I [61�9 Therefore, in what follows we will not pay any attention to this contribution to the current fluctuation�9

If the impurities happen to change their positions fast, say infinitely fast, then, within some interval of time fol- lowing each individual jump of an impurity, the current cannot be written in the simple form given by (1) (since electron exitation etc. can occur)�9

In the wave packet picture the time during which (1) is invalid can be estimated as the time zf the wave packets needs to traverse the scattering region�9 This has to be compared with the typical time z~ between subse- quent jumps�9 If

%>>r: (3)

the current expression (1) is practically exact and the omitted contribution to the current-current correlator, averaged over the time t, gives rise to a relative correc- tion of the order z : / % . In the following we shall always assume that inequality (3) is satisfied.

Suppose now that each of the two moving impurities can occupy two positions, so that we are dealing with four different potential shapes and four corresponding transmission coefficients T+ +, T+_, T_ +, T__. Here the first index indicates the position of the first impuritiy and the second that of the second one. For the sake of simplicity we shall assume that the switching of the positions is a Poisson process with some rate 7j for impu- r i tyj = 1, 2. The time average in (2) can then be calculated in the following way: at time t we find the system in one of the four states, e.g. (+ +) with probability 1/4. Then, at time t + z the system is with some probability in state ( i , j ) with transmission T~,j, which can easily be calculated. Indeed, the probability e.g. for T~,j= T_ + is equal to the probability for the first impurity having changed its position an odd and the second one an even number of times. Since, for the Poisson process, the prob- ability of having exactly n events during the time t is

P, = (71~ exp(-- zT), (4)

the total probability for the transition (+ + ) ~ ( - + ) in time "c is

o(I) p(2)_,~x~r_ z 71) 1 [exp(.c 71)_ exp(_ rT1) ] a 2 n + l X 2 m - - ~ " b'~ i/,/tl = 0

�9 exp ( - r 72) �89 [exp (z ]/2) + exp ( --'c 72)1

(5)

Considering in the same way all possible transitions we finally obtain for the current-current correlator up

to the additive term <I(t)> 2

<I(t) l ( t + z)> e 4 V 2

- 4h 2 { e x p ( - - Y ~ z ) ( T + + + T + - - - T - + - - T - - ) 2

+ exp(--72 z ) (T+ + + T_ + - T+ _ - T_ _)2

+ e x p ( - [71 +72] z ) ( T + + + T_ _ - T+ _ - T_ +)2} (6)

To get an idea about the relative value of the additive and non-additive contributions let us calculate explicite- ly the coefficients T~,~ in the presence of only two moving point impurities using perturbation theory. The first order correction 6T to the unit transmission is just the square matrix element of the impurities potential for the incident and reflected plane waves,

6 T = - ( U ( + UZ + 2 c o s [ 2 p v ( x l - x2)] U 1 U z ) / K (7)

where xi are the positions of the impurities, U~ the poten- k~ h 4

tial coupling constants and K = ~ . Substituting (7) into (6) we find

16e 4 V 2 U 2 U 2 <cSI(t) 6 I ( t + z ) > = h 2 K 2 �9 {exp(--T1 z)

sin 2 [PF (x [ - - x ~)]

c o s 2 [ p F ( x ~ - x;)]

�9 sin 2 p F [ ( x [ + x [ - x ~ - x j ) ]

+ exp ( - 72 z)

sin 2 [ p v ( x ] - x2)]

c o s 2 [p~(x~ - x i - ) ]

�9 sin2 [pF(x~ +x[- --x~- - x 2 ) ]

+ e x p ( - [71 +72] 7)

sin 2 [ p f ( x + -- X])]

sin 2 [pF(X + -- X[)]

�9 cos 2 [p~ (x + + x ; - x + - x ~ ) ] } (8)

Note that the non-additive term decaying with rate 7 =7~ +72 can be of the same order of magnitude as each of the additive terms. It can even happen that the non- additive term is the only non-vanishing one. This occurs if

p~(x?~2~- x;~a~) = ~/2 + ~ n, (9)

where n is an integer. Let us now suppose that the distance between the

impurities is sufficiently large so that we can treat the system as two quantum wires separated by a reservoir�9 According to the appropriate Landauer formula the po- tential drop across the scattering regions is given by I h ( 1 - T~)/2e 2 T~, i= 1, 2. Adding the voltage drop at the entrances of the wire we get

v = 1 2 ~ [ i + I - T~ + I - T2 ~ ] (lo)

Obviously, in the adiabatic approximation, the impuri- ties contribute independently to the voltage-voltage cor- relator (V(t) V( t + ~)) and thus no non-additivity occurs. In contrast to this the current-current correlator exhibits non-additivity simply because the conductance

r2 + r~ - r2 r~

is non-additive with respect to the transmissions. This holds true even if the intermediate reservoir is much wider than the two single-channel quantum wires so that

G = 2eZ/h(T1 - 1 _}_ T2- 1) (12)

This trivial non-additivity which occurs in all kinds of conductors does not concern us here. Instead we will focus only on the non-trivial non-additivity which is en- tirely due to quantum interference effects.

3. Dirty conductors

Our next task is to express the current-current correlator for a quasi-ld disordered conductor in terms of the aver- age two-particle Green's function.

The general expression for the local current-current correlator reads

e 2 h 2 ( I ( x , t l ) I ( y , t2)> = - lira

x - x , 4 m 2 Y~Yl

�9 ( G G ) ( v , - v , )

(gJ* (x, q) g~(xl, q) ~*(y, t2) ~(Yl, t2)> (13)

Here 7J*(x, t) and ~g(x, t) are fermion creation and anni- hilation operators, respectively, in Heisenberg picture with respect to the total Hamiltonian, including both fixed and moving scatterers, and ( . . . ) denotes averaging with respect to the electron degrees of freedom. Accord- ing to Week's theorem

<~*(x, t~) ~(xl , h) ~t(y, t2) ~(yl , t2)> = < ~'* (x, t,) ~(x~, tl)> < ~* (y, t~) ~(yl , t~)>

- - (~ t ( X, tl) ~(Yl, re)> (I/fi'(Xl,/1) ~(Yl, re)> (14)

The second term on the rhs of (14) can be shown to yield a shot-noise like contribution to the current-current correlator. Since it does not lead to a 1I f behaviour in the power spectrum we will omit this term. An appro- priate approximation for the impurity averaged correla- tion function (g**(x, t)~F(y, t)> can be obtained [7] by using the Keldysh diagram technique [8]. This involves the retarded, advanced and Keldysh Green's functions defined by

GR(A)(1, 2)= - ( + ) i O ( + ( - ) t l - ( + ) t2) (g-'(1) 7'*(2) + ~P*(2) ~(1)5, (15)

and

GK(1,2)= - - i (~(1) ~g*(2)--t/'t(2) hu(1)>, (16)

489

respectively, and connected in equilibrium by the rela- tion

G~= [G# - G~] [i - 2n(E)] (17)

where n(E) is the Fermi distribution function. Within the wire the impurity averaged Keldysh

Green's function satisfies the Laplace equation [7],

v ~ d~(r, 70 = 0 (18)

At the ends of the wire

g~(r, r)1(~_ 0,L)= 2 ~z iv {1 - 2 n (E +_ eV /2 ) } (19)

Here v is the density of states at the Fermi level. With the boundary condition (19) the solution of (18)

for rl = r reads

af(7, 71 = of(0, 0)+ -~[#f(c, L ) - of(0, 0)] (20)

Using (15) and (16) we can express the correlation func- tion ( f*(1) ~f(2)} in terms of Green's functions,

i (7J*(1) 7'(2)) = --~[GK(2, i)--GR(2, 1)+ Ga(2, 1)] (21)

It is easily seen that only GK(2, 1) contributes to the total current across the wire,

eh [ = - i 4ram ~ d E ~ d S ( V 1 - V2) G ~ ( r l , r2)1~ =~2 (22)

which still depends on the impurity configuration. As- suming that all impurities are fixed and averaging with respect to the impurity distribution we get

= e D S V ~ G~(r, r) dE = ~ - ~ d E v D A n(E) (23) I

Here is D = v~ ze~/3 the diffusion coefficient ('ce~ being the elastic mean-free time), S the cross section of the wire, and A n (E) = n (E - e V/2) - n (E + e V/2).

The average current-current correlator consists of two parts which can be expressed by the diffusion-type and cooperon-type two particle Green's functions

pal(c)(71, ~ r2) = a e + o , (rl, r2) G~ (r; (1), rl (2)), (24)

respectively. The bar denotes averaging with respect to the configuration of fixed impurities.

If there are both fixed and slowly moving impurities the average current-current correlator will depend on time via the paths of the mobile impurities which are not averaged over. For a single mobile impurity this problem was investigated by Fal'ko and Khmel'nitskii [4] who derived an operative equation for P~. Essentially the same equation holds also for several mobile impuri- ties provided their number Nm is small in comparison to the total number of impurities N (cf. Appendix).

If both the inelastic length Lin = (D zin) 1/2 and the ther- mal length L r = (D/k~ T) 1/2 are larger then the transverse size of the system L• the problem becomes essentially

490

one-dimensional and the equation for P,o (multiplied by a numerical factor) takes the form

I--leo DA +~, Cj c~(x-Rj)fj(t, z)-- 1/zin ] Po~(x, x') J

= - a ( x - x ' ) (25)

In (25) the coefficients C~ depend on the strength of the potential and the hopping distance of the mobile impuri- ty with (average) coordinate Rj. If all mobile impurities are of the same kind as the fixed ones and the hopping distance is much larger then the Fermi wavelength, typi- cally C ~- L/N'r d. Furthermore, f~(z) is a telegraph signal switching between zero and one with some rate 7j and initial value f~(0)=0. The derivation of (25) is outlined in the Appendix.

Expressed in terms of Po, the average current-current correlator

K (t, ~) = ( I (t) I (t + v)) (26)

reads an(E) an(E +co)

K(t ' z )=(ev)a~dEd~ ~E ~3E

"\ hL 2 ] ~ dx~dx2{lP~

+ 1/2ne(Po,(t, z, xl, x2)) 2} (27)

s(~o) = S(~ + s ( % o ) + s(~)(o)) + . . .

In the limit k B T/h, zi, ~ 4~ D/L a

215 dx, dx2 P(~ x2) e(~)(x,, x2) 4 L s sin 2 (R i n n/L)

- D3 n 6 .~ Ci fi n 6 t ,n

(33)

(34)

Restricting ourselves to the mode n = 1 which gives the main contribution we obtain

S(1)((D) = V2 24e 4 L 27i sine(Ri re~L) ( hg3)2 0 ~i Ci 4Y{ +~~ (35)

If in this expression the sum is approximated by an inte- gral and the rates 7~ are assumed to be distributed ac- cording to the law

P(7)=Po/7 (36)

with some constant Po, a 1/co-type power spectrum re- sults. However, in general this is not the case [93. In second order

e4 L2 ~C, CJi(t, z)f~(t, z) F(Ri, R~) (37) K(2)(t, z) = V 2 h2 D2 ' . 1,3

where

The limit K(t, z = Go) describes corrections to the station- ary current and must be substracted when calculating the noise power spectrum.

F (x, y) = sin z (x n/L) sin E (y n/L)

+ n a sin (x n~ L) sin (y n/L) - - x(L--y)

L 2 , y > x (38)

4. Weak coupling mobile impurities

For a vanishing number of mobile impurities the solu- tion of (25) with boundary conditions

P~,(x,y)=O (x,y=O,L) (28)

reads

2 sin(x I nn/L) sin(x 2 nn/L) (29) P(~ x2) v

For sufficiently weak mobile impurity potentials (25) can be solved in a perturbative way. In first and second order in Cj

pO)= _ ~ Cjf~ P~ R j)P~ x2) (30) J

p(2) = ~ Ci C , fH / p(0)(xl, Rj) P(~ Ri) P(~ x2) (31) j , i

This perturbative calculation of P~, generates a power series expansion of K and the noise power spectrum

�9 1 T / 2 oo

S(co)=hrn ~- I dt2Re 5 d~exp(iooz)K(t,'c) (32) --7"]2 0

in terms of Cj,

Let us now investigate the additivity property of S(2)(co). Consider a definite term of the sum. The product fi(t, z)fj(t, r) is non-zero only if both impurities, i and j, jumped an odd number of times during the interval z. For two independent Poisson processes the probability of such an event is

(~]i "C) 2n+ 1 exp(_Ti z) (Tj .[)2m + 1 e xp ( -T j z)

( 2 n + l ) ! (2m+1)! (39)

This yields for the average offi(t, z)f~(t, z) over the time t

fi (z)fj (z) = 1/4 [ 1 - exp ( - 22i z) -

exp(-- 27j z) + e x p ( - 2 (7i + 7j)z)] (40)

Consequently a sum of three Lorentzians occurs in S(2) ((D),

7z VJ t- 7~ + 7j (41) 4,~2q-o,) 2 4~2 +co 2 4('Yi + 7j)2 + O) 2

Whereas the first two are additive the third obviously incorporates non-additivity. Replacing again the sum by an integral and using the distribution (36) we end up with

S(2)(o)) Po 2 rc ln(~/7mln) (42) 4co

Here 7m~. is a cut-off frequency introduced to avoid infra- red divergence. Equation (39) is valid for co > 7mi.. The constant Po can be expressed by the number of mobile impurities Nm and the ultraviolett cut-off 7 . . . . Po = Nmov / ln (Tmax / 'Ymin ) . The correction S (2) (co) deviates loga- rithmically from the 1/(o behaviour and its relative order of magnitude

S <2)(co) _ Nm CLln [co/~)min] S~ D in [Tmax/"/min "]

(43)

is small as long as co ~ 7max and

c~ = CL/~ < 1 (44)

The case ~ ~ 1 corresponds to the situation when the number of mobile impurities is large enough and their coupling is sufficiently strong to change the value of the conductance by an amount of the order e2/h. Assuming that fixed and mobile impurities couple approximately by the same potential, c~ can be estimated to be

1 L 2 ~ N 1 ~- (45)

For strong impurity coupling 1 ~ k~ SL/N and

e 2 c~ ~ - - (46)

hG

5. Strong coupling mobile impurities

The strong mobile impurity coupling is characterised by CL ~ - > 1. If the temperature is high enough so that ks T/h,

T~1>>C/L perturbation theory can be applied just as in the weak coupling case. In fact it is intuitively obvious that at sufficiently high temperatures the impurities act practically independently.

The case of low temperatures is much more difficult. To get an idea what is going on we consider a cluster of Nm impurities situated in a distance R from one of the edges of the wire. The solution of (25) can be estimat-

r 9

cd to be P - ~ - i f all thc impurities are "switched off"

R 2 and P - D - if at least one impurity is "switched on".

The probability for the first case is the product of the probabilities for all impurities be at the initial position,

�9 (r) = I~ 1/2 [1 + exp (-- 2 ~i z)] (47)

The probability of the second case is obviously 1 - r Therefore, the time average of K(t,~) is

K('c)=(e~-)2~--~[O(l+exp(--2~iz))--I ] (48)

If all rates are equal, 7i = 7,

(~)2 N k K(T)= k_~l 2 @ exp(-2k';~)

491

(49)

N~ where C~ is the binomail coefficient C~-(N_k) !k ! .

The spectral density at vanishing frequency is

S(O) ~ N7

Equations (49-50) reveal non-additivity. Indeed, additi- vity implies

[e2\ 2 K ( z ) ~ ) N exp(-2Uc), (51)

and for the spectral density

s(o) ~ 7 (52)

As we see, the spectral density at small frequencies is much less in the non-additive limit (50).

6. Summary

We investigated the non-additive flicker contribution to the excess noise in disordered mesoscopic systems brought about by phase coherent scattering of electrons by mobile impurities. The relevant two-particle Green's function was calculated in the cooperon/diffusion ap- proximation for the fixed impurities while the mobile impurities were taken into account by lowest order per- turbation theroy. The dynamics of the mobile impurities was approximated by a two-site hopping process (e.g. tunneling) generating a telegraph switching signal. To get an estimate for the relative amount of the non-addi- tive contribution to the power spectrum (as compared to the leading additive part) we assumed the transition rates 7 to be distributed according to a 1/7 law as usual in the theory of 1If noise [2]. For weak coupling impurities we obtained the relative non-additive correc- tion

S (2)(co) _ Xm CLln [co/3/min ] S(1)(co) D ln[Tmax/Tmin ]

which may be relevant due to the logarithmic frequency dependence even if the prefactor is small. This is in accor- dance with a previous phenomenological analysis [9]. Therefore we may conclude that mesoscopic effects gen- erate both a l / f noise and a non- i / f correction. One has to keep in mind, however, that phase coherent scattering by several mobile impurities is not the only source of non-additive contributions to the excess noise (cf. Sect. 2). Other sources may be even more effective. On the other hand the importance of mesoscopic interfer- ence effects is clearly indicated by experiments showing a 50 per cent reduction to the noise when applying a

492

weak magnetic field [10]. This observation is most likely explained by the destruction of the cooperon mode.

As mentioned before the prediction of 1If noise in mesoscopic conductors [ l J is based on the U C F theory and the Du t t a -Horn approach which assumes a large number of mobile scatterers bound to double-well poten- tials and a homogeneous distribution of the barrier heights leading to a 1/7 distribution of the hopping rates [2]. Several experiments on noise in mesoscopic conduc- tors [11] show at low temperatures a transition from the 1If behavior to a frequency dependence which can well be approximated by the sum of a few Lorentzians. This indicates that relatively high temperatures are re- quired to have a sufficient number of relaxation pro- cesses. The aim of the present work can be viewed as investigating the range of validity of the mesoscopic 1If noise assertion by estimating the corrections which ap- pear when the average distance between the mobile scat- terers is less than the phase breaking length - even if the Du t t a -Horn approach is valid.

We are indebted to Dr. V. Fal'ko for valuable discussions and to the Deutsche Forschungsgemeinschaft for supporting this work within the Sonderforschungsbereich 341.

7. Appendix

Since mobile impurities do not give rise to singular inter- ference modes they can be treated as a perturbation. In lowest order the diagrams

have to be added to the rhs of (A1) where the small circles correspond to mobile impurity potentials [1, 4]. Since

G ~ exp (-- r/l),

assuming l to be small as compared to the typical range of variation of P, (A4) contributes for a single mobile impurity at position R the term

C(z) 6 ( r - R )

to the equation for Pod(r, r'). The coefficient C can be obtained by integrating the equation with respect to r. If the impurity fails to move C vanishes, of course. In the case of several mobile impurities at positions R i a sum over contributions Cj(r)6(r--Ri) occurs. Writing Cj(z) = cffj(z) we end up (in one dimension) with (25).

In this appendix we outline the derivation of (25). Being interested in the non-additivity brought about by quan- tum interference we restrict ourselves to the most diver- gent diffusion and cooperon modes. In terms of diagrams these correspond to ladders generated by the equation

i

- ' ' , -t-

and a similar equation with arrows pointing in the same direction. The impurity line x x contains as a factor the impurity correlator V(r0 V(r'0. We assume

V(rl) V(r'~)= 5(r 1 -r'l)cim U z (A2)

where Cim is the impurity concentration and C~m U 2 =h/rcvz. As is well known 1-12] the approximat ion (A1) leads to

0 1 (i09 +DArl ) P~ (rl, r2)= z2 rCV (5(rl --r2) (A3)

References

1. Feng, S., Lee, P.A., Stone, A.D.: Phys. Rev. Lett. 56, 1960 (1986); ibid. 56, 2772E (1986); ibid. 59, 1062 (reply to Comment) (1987)

2. Dutta, P., Horn, P.M.: Rev. Mod. Phys. 53, 497 (1981); We- issman, M.B.: Rev. Mod. Phys. 60, 537 (1988)

3. Al'tshuler, B.L.: JETP Lett. 41, 648 (1985); Lee, P.A., Stone, A.D.: Phys. Rev. Lett. 55, 1622 (1985)

4. Fal'ko, V.I., Khmel'nitskii, D.E.: JETP Lett. 51, 189 (1990) 5. Landauer, R.: In: Kramer, B., Bergmann, G., Bruynserade, Y.

(Eds.): Localization, interaction and transport phenomena. So- lid State Sciences, vol. 61, p. 38. Berlin, Heidelberg, New York: Springer 1985

6. Lesovik, G.B.: JETP Lett. 49, 593 (1989); Bfittiker M.: Phys. Rev. Lett. 65, 2901 (1990); Martin Th., Landauer, R.: Phys. Rev. B45, 1742 (1992)

7. Larkin, A.I., Khmel'nitskii, D.E.: Soy. Phys. JETP 64, 1075 (1986)

8. Keldish, L.V.: Sov. Phys. JETP 20, 1018 (1965) 9. Kiss, L.B., Kert~sz, J., Hajdu, J.: Z. Phys. B81, 299 (1990)

10. Birge, N.O., Golding, B., Haemmerle, W.H.: Phys. Rev. Lett. 62, 195 (1989); Mailly, D., Sanquer, M.: Surf. Sci. 229, 260 (1990)

11. See Ralls, K.S., Ralph, D.C., Buhrman, R.A.: Phys. Rev. B40, 11561 (1989)

12. Lee, P.A., Ramakrishnan, T.V.: Rev. Mod. Phys. 57, 287 (1985)