Z. Phys. B - Condensed Matter 81, 299-303 (1990) Condensed ffir Physik B Matter 9 Springer-Verlag1990 Conductance noise spectrum of mesoscopic systems Lfiszl6 B. Kiss 1, J~nos Kert6sz 2, and Jfinos Hajdu Institut fiir Theoretische Physik, Universitfit zu K61n, Ziilpicherstrasse 77, D-5000 K61n 41, Federal Republic of Germany Received July 4, 1990 We investigate the shape as well as the size- and temperature-dependence of the conductance noise spectrum of a small system containing electrons and both fixed and mobile scatterers. If the number of mobile scatterers within a phase-coherent region is sufficiently large, the temporal variation of the conductance can be viewed as a random walk process limited by the universal conductance fluctuations, resulting in a practically Lorentzian power spectrum. We discuss the conditions under which the noise spectrum of a system consisting of many phase-coherent regions is either Lorentzian or 1/f-like. The temperature-dependence of the power spectrum is determined by the hopping mechanism and the variation of the phase breaking length. As a function of temperature the spectrum satisfies power law scaling relations with exponents depending on the dimension and the temperature range; the spectral intensity can both increase and decrease with decreasing temperature. 1. Introduction The conductance noise of small conducting samples containing a sufficiently high number of scatterers has recently been investigated both experimentally [-1-6] and theoretically [7-12]. These systems show universal conductance fluctuations (UCF) which is basically a quantum interference phenomenon [13, 14]. Therefore mobile scatterers - changing the interference patterns - give rise to a special time dependence of the conductance [7, 9, 15]. It has been pointed out that the time dependent conductance behaves like a random telegraph signal if the system contains a single mobile scatterer bound to a double well potential [7, 12]. This leads to a Lorentzian power spectrum of the conductance fluctuations. In the 1 On leave from Institute of Experimental Physics, University of Szeged, D6m t6r 9., H-6720 Szeged, Hungary 2 On leave from Institute for Technical Physics, HAS, H-1325 Budapest, Hungary presence of many mobile scatters the resulting power spectrum has been assumed [7, 12] to be given by the D u t t a - H o r n formula 1-16] which is the sum of individual Lorentzians weighted by the distribution of hopping rates. As is well known, reference to this formula is the standard way of obtaining the 1/f noise spectrum. A crucial requirement for this procedure to be applied is that the resulting signal is a sum of independent parallel contributions. However, in systems showing U C F this requirement is generally not satisfied. In fact, the DuttaH o r n formula applies only in the special case when there is at most one mobile scatterer within a phase-coherent region and the whole system consists of a large number of such regions. One of the aims of the present work is to investigate the effect of many scatterers within a phase-coherent region on the noise spectrum. Whereas in a macroscopic system the individual hopping events switch the conductivity independent of the state of the other scatterers, a mesoscopic system behaves in an essentially different way. The change of the conductivity due to an individual hop depends on the other scatterers which may move and therefore alter the interference conditions for the electron waves. These considerations led us to describe the time-dependence of the conductance as a limited random walk process, the limits being given by the UCF. In opposition to the DuttaHorn mechanism the resulting power spectrum is not l/f-like. Both the number of mobile scatterers within a phasecoherent region and their dynamics depend on temperature. Consequently, the power spectrum has been studied in different temperature regimes. The result is that, over many frequency decades, the total power spectrum S(f) obeys general scaling behaviour with respect to the temperature T s(y, TO:S(d, T~)(~) * (1.1) where the exponent q depends on the dimension of the system and the hopping mechanism. In most cases q > 0, 300 i.e. the amplitude of the spectrum increases with decreasing temperature as already mentioned in [10, 11] and found experimentally in I-3, 5]. The paper is organized as follows: In the following Sect. 2 we describe the physical model and relate it to the power spectrum. In Sect. 3 and 4 we investigate the power spectrum as a function of the number of mobile scatterers and the temperatures, respectively. The final Sect. 5 presents a discussion of the results. 2. Noise model and power spectrum We consider a d-dimensional mesoscopic system consisting of electrons and fixed as well as mobile scatterers. The physical situation in the system depends on the relation of the following characteristic length scales: the linear size of the system L, the elastic mean free path I, the inelastic mean free path lin, the phase breaking length L o = / ~ , , and the thermal phase breaking length lo =hvr/k B T where vr is the Fermi velocity. The system is divided into phase-coherent regions of linear size Lbox min (L~, I~) (for Lbox~ L). Let us denote the number of fixed and mobile scatterers within a phase-coherent region by N and M, respectively. At high temperature the mobility is due to thermal activation whereas at low temperatures tunneling in double well potentials dominates. Whenever hopping takes place the conductance of a phase-coherent region is changed by a certain amount 6G1 [7, 9, 12], = 6G 2 ~ L22xda G 2 (2.1) nl 2 where n=N/L~o, is the density of the scatterers and 6G2=e2/h is the UCF 1-13, 14]. Obviously, ]JGII<=]c~GI puts a limitation on (2.1). The time-dependent fluctuation of the conductivity AG(t), is also limited by the UCF; therefore the noise produced by the mobile scatterers is a stationary stochastic process which is characterized by the correlation function T/2 ~ dt'AG(t')AG(t'+t)=(AG(O)AG(t)) q~(t)= T--~ lim T1 -- T/2 (2.2) where AG(t)= G(t)- ( G). The power spectrum S(f) is related to the correlation function by O9 S ( / ) = 4 Re S dteiZ~:t~(t). (2.3) 0 Since the fluctuation of the conductance is due to instantaneous hopping processes at certain switching times t~, the time-dependent conductance can be written as G(t)=G(ti) for ti<=t<ti+~i, G(h + zi)- G(t~)= A x~ (2.4) (2.5) being the change of the conductance caused by one hop of a single scatterer. Both % and Ax~ are stochastic variables. Supposing that the hoppings of different scatterers are uncorrelated, the probability density for the time between two adjacent switchings is given by a Poisson process P('ci) ~ e -~il~ (2.6) : T where 1/~ is the total rate of the switching process oo 1 S dvvp(v). "C 0 (2.7) Here p(v) is the density of scatterers on the v-scale having the rate v. Equation (2.7) expresses the fact that the total rate of the conductance switching process is the sum of the rates of the individual hopping events of the scatterers. Let us note in passing that the probability density (2.6) with (2.7) corresponds to a distribution of waiting times which is significantly different from the one suggested in [8]. The distribution proposed in [8] contradicts the additivity of the rates of independent stochastic processes. Moreover, it violates the normalization condition. Determining the distribution of the other stochastic variable Ax~ requires further assumptions. This point will be discussed next. 3. Power spectrum for a phase-coherent system Let us start by investigating the case where the length Lbo, is larger than or equal to the system size L. First we assume that there is only one mobile scatterer in the system. The simplest example for the motion of the scatterer is repeated hopping in a symmetric double well potential. Obviously, [Ax~I=IAXI~aG1 (3.1) and G(t) is a random telegraph signal [7, 12], AG(t) = +_]AX[/2. (3.2) The spectrum of this signal is known to be Lorentzian: "G S (f) = 4q~(0) 1 + (2 zcfT)2" (3.3) If the moving scatterer is not bounded to a double well potential then its motion is still a nonergodic process with respect to the defining statistical ensemble of the UCF. This ensemble consists of an infinite number of identical systems with different configurations of the scatterers. Starting from one configuration, one hop of a single scatterer gives rise to a new configuration which is also an element of the ensemble. However, the set of elements scanned by the motion of a single scatterer has zero measure. Since the hopping sites are statistically distributed over the system and the hopping distances are at least of the order of the Fermi wavelength, the effect of subsequent hopping events can be considered to be uncorrelated. Consequently, the time-dependent conductance behaves like a generalized telegraph signal 301 with random amplitude of the order of ~ G1. The resulting power spectrum is again Lorentzian (3.3) with 1/z replaced by 2/z. We are turning now to the case when the fraction of mobile scatterers is of the order of unity, M = (9(N). Two extreme possibilities can be distinguished. If in (2.1) the prefactor is larger than or equal to unity, i.e. 6G~ = 6 G2 the situation is similar to the one discussed above. The conductance switches essentially between two fixed values (+ c5G1), however, according to (2.7), with a rate equal to the sum of the individual rates. The alternative limit IAxil~lbGl=(e2/h) is more elaborate and requires a careful analysis based on the U C F theory. In this paper we restrict ourselves to approximating the problem by a limited random walk in which G(ti)=x i plays the role of the coordinate. This can be justified as follows: Let us assume that, at a certain initial time t = 0 , the conductance of the system takes a typical value G(0). At the first switching instant the conductance will be changed by an amount of +I6GlJ. Due to the large number of moving scatterers, subsequent switches are caused by different scatterers. Moreover, during the time between two subsequent switches at t~ and t 2 caused by the same scatterer, the distribution of the other ones is considerably rearranged so that the increments of the conductance at t~ and t2 can be viewed to be statistically independent. Thus, for short times, the temporal change of the conductance is approximately a random walk process with average step length 16Gll. The time sequence of this process follows the Poisson law (2.6) with rate 1/~. However, as in the course of time (AG(t)) 2 increases, the global limit ~ G 2 imposed by the U C F becomes effective. Whenever G(t) approaches (G)++_If G[ the process looses its random walk character: the mechanism responsible for U C F prevents the conductance to exceed much beyond the above limitations. This leads us to approximating the problem by a limited random walk process with reflecting boundaries at + [6GI. To calculate the power spectrum, we go to the continuous limit of the limited random walk problem, ri~0, Axi-+O, ((Ax~)2)/z=const-O~. (3.4) In this limit 1 q~(t)=2-f~ [ dx ~ dx , K(x, t; x', O) x .x' (3.5) where the integrals are extended over the interval -16G] < x < ]OGI, and K(x, t; x' 0) is the Green's function of the one-dimensional diffusion equation with reflecting boundary conditions, OK/Ox=O for x = +_136[. From this: S ( f ) = ~ t~G2 ~, n = 1 (2 F/-- ZRW 1)6 + (2 n-- 1)2 (2 ~zf'CRW)2 (3.6) where rRW = (2/7z)4 6 G2/Dc,. (3.7) Thus, S ( f ) ~ f ~ for f,~LZbo~a/nl2z and S ( f ) ~ f -z for f >>LZo~a/nlz z. In general, (3.6) can be well approximated by a Lorentzian. If M<N12/L?, the random walk cannot fill out the fluctuation band _+]6G]. Yet, the arguments leading to the random walk character of the conductance are still valid, but the limits are now given by __+M~-M-~. The characteristic time required to reach the limit of the random walk is now ZRW~ Mz. Again, the power spectrum is essentially Lorentzian. 4. Power spectrum for a system of many phase-coherent parts In general, a macroscopic sample consists of more than one phase coherent regions, the number of which is (LLboOa. Then the total conductance ( G ) of the sample is given by the conductance (Gbox) of a phase-coherent region multiplied by a geometrical factor, ( G ) ~ (Gbox) (L/Lbox)d-2. (4.1) Since the different phase-coherent regions contribute independently to the total relative fluctuations, (AG 2) _ (AG2ox) 1 (Gbox) 2 (L/Lbox)d (G)2 (4.2) where (AG~,o~) is the quadratic fluctuation of the conductance of a phase-coherent region. Using this and (4.1) we obtain ( AG2) = ( AG2ox) (~x-) 4-a. (4.3) As discussed above, the value of (A G~ox) is less or equal to the UCF, 6G z, depending on the number of mobile scatterers and on the value of 6G~. If the system consists of many phase-coherent regions but there is at most only one mobile scatterer in each of them, then the Dutta-Horn formula can be applied, i.e. the power spectrum is given by oo 7; S ( f ) ~ ~ dzg(z) 1 +(2-Tcfz) 2" (4.4) 0 Since there is at most one mobile scatterer per phasecoherent region, the distribution of rates 1/z occuring in the Lorentzian contributions of individual regions is the same as the distribution of rates for single defect hopping. Assuming as usual g(z),-~ l/z, and taking into account that for M < 1 the right hand side of (4.2) is multiplied by M, equation (4.4) yields 2 IL b o x \4-a _1_ :. (4.5) In contrast to this, no 1/f power spectrum results when there are many mobile scatterers per phase-coherent region. This is because the total rate is the sum over the rates of the individual hopping events (2.7) and, therefore, the distribution of the characteristic frequencies occuring in the individual Lorentzian contributions is rather narrow. Accordingly, (4.4) leads now to a Lo- 302 rentzian. The integral of the spectrum can then be approximated by f * S ( f ) with arbitrary f < f * where f , ( T ) = 1 6G2 ~. (4.6) Turning to the case of many mobile scatterers per phasecoherent region we assume that the fluctuation (A G2ox) takes its limiting UCF value (saturation). This always occurs if M is sufficiently large. Then the spectrum is approximately Lorentzian and Using (4.3) we finally get S(f rl)=S(f S ( f ) - ( A G ~ o , , ) (~-~2) 4-d f-u l (4.7) 5. Temperature dependence The temperature enters the problem via the length scales l~. and le as well as the density of moving scatterers m and the rate v, of activated processes. For the temperature-dependence of the inelastic mean free path we use the approximation l l n ~ T -p with p ~ l . This leads to L4,~ T -p/2. On the other hand I ~ 1/T. Therefore Lbox crosses over from Le to lo at some temperature Tx ~ 1 K. Above a certain temperature T* the motion of the scatterers is dominantly activated. Below this temperature tunneling processes determine the dynamics. The density m of tunneling defects at temperature T is given as kB T m ( T ) = m o AE" (5.1) Here AE is the average barrier height in the double well potentials. For the sake of simplicity we assume that the barriers are uniformly distributed between E~i. and Emax. The individual tunneling rates are independent of temperature. For the activated processes the individual rate v, is given by the Arrhenius law Va = V0 e- ~/k,r (5.2) Again, we assume that the activation energy E is uniformly distributed between Emi, and Ema~. The quantities T* and Eml, are related by ks T* ~ Eml,. The total number of mobile scatterers within a phasecoherent region depends both via re(T) and Lbo x o n the temperature. Therefore, since in general Tx < T*, T -dp/2 M~ T 1-d for T< Tx for T > Tx" (5.3) We shall see that in all cases considered the scaling relation S(f, T~)=S(f, T:)(T2-~q \ T1] (5.4) holds. For M=<I tunneling scatterers (4.5) and (5.1) lead to ,~, r~ [Lbox(r2)~ -4 S(f ro=s(f 12)T22~) " (5.5) Thus, q ={2p-- 1 3 for T< Tx for Tx < T < T*" (5.6) _ , f * (T2) {Lbo~(T2)] d-4 12) f ~ i ) \Lbo~(T1)] (5.7) with f * being the frequency at which the spectrum changes its behaviour from f o to f - 2 . If one hop is sufficient to reach the UCF limit, then f * = 1/z (case i.), whereas f * ~ ( 1 / z ) L 2 o f if many random walk steps are necessary (case ii.). Inserting the temperature dependence o f f * and Lboxinto equation (5.7) the scaling relation (5.4) results. The exponent q depends on the dimension of the system, the range of temperature and the fluctuation mechanism. This will be investigated in the following. At low temperatures tunneling processes dominate. The total rate (2.7) of the switching process is given by 1 - = m ( T ) L~ox ~ z (5.8) where ~ is the average individual tunneling rate, which is essentially independent of the temperature, and for the density of tunneling scatterers (4.1) is used. At very low temperatures, T< Tx(< T*), the phase-coherence length Lboxis given by L~ ,,~ l I T p/2. Finally we get ~2p + 1 - dp q = ~(p + 1 - dp/2 for case i. for case ii. (5.9) In the low temperature range Tx< T < T*, when Lbox = ~ l/T, (5-2d q = ~3 - d for case i. for case ii. (5.10) At high temperatures the dominant process is thermal activation. Assuming a flat distribution of the activation energies between Emin and E . . . . the temperature dependence of the total rate is 'k T _ - Em~x/"kBT -z1= v o m L ~ o x -kAE~ -s T - ke - EmJ~,~ e ] (5.11) where A Ea = E m , x - E ~ , . Since Em,x>>ks T in all practical cases, for high temperatures, Em~. ~ kB T, the temperature dependence of the total rate will be the same as for tunneling, 1/z~ TL~ox. Since at high temperatures Lbo x = 14, equation (5.10) remains valid. 6. Discussion In this work we have studied the influence of the concentration of mobile scatterers, the system size and the temperature on the noise spectrum of systems showing UCF and have arrived at the following results. It the system consists of a large number of phase coherent regions with at most one mobile scatterer in 303 each, the D u t t a - H o r n analysis applies, leading to a 1If power spectrum [7] for a certain intervall fmin < f < fmax" However, in the case of many mobile scatterers the quantum phase coherence prohibits treating the contributions of such individual scatterers independently. Whenever the fluctuation of the conductance of a phase-coherent region caused by a single mobile scatterer 6G 1 is approximately equal to the U C F 6G, the total conductance behaves like a generalized telegraph signal. The switching rate is the sum of the individual hopping rates, c.f. (2.7). As is well known, the corresponding power spectrum is Lorentzian. This is also the case when the system consists of many phase-coherent regions. The DuttaH o r n mechanism also fails to apply for 6 G I < 6 G , the time-dependent conductance of a phase-coherent region being then like a random walk process limited by the UCF. The spectrum is again essentially a Lorentzian. The frequency f * given by (4.6) separates the regimes in which the spectrum behaves like f o and f - 2 , respectively. Again, this is the case when the system consists of many phase-coherent regions, the distribution of the spectral contributions of different regions being narrow. In the literature [7, 8, 11, 122 these possibilities have not been considered appropriately. The temperature-dependence of the power spectrum is a rather complicated matter. Therefore, different temperature regions should be distinguished. A scaling relation (1.1) is shown to be valid for f < f * if the spectrum is Lorentzian and for f m ~ n < f < f m , x if it is 1/f-like. The exponent q depends both on the temperature region and the dimension of the system. In most cases q <0, i.e. the amplitude of the spectrum increases with decreasing temperature. This effect [3, 5], which is mainly due to the decrease of the number of phase-coherent regions with decreasing temperature, has already been pointed out in [10, 112 . However, to obtain the correct temperature-dependence of the spectrum one has to take into account also the variation of f * , the density of mobile scatterers and the activation rates with temperature. In particular, for three dimensions and sufficiently high temperatures ( T > Tx, i.e. Lbox = 14~) the behaviour is different from the previous one: If the U C F limit is reached by single hops, the amplitude decreases with decreasing temperature whereas the limited random walk yields an amplitude which is independent of temperature (c.f. (5.10)). Unfortunately almost nothing can be said about the different very interesting intermediate cases. A detailed microscopic analysis is required to describe the crossover between the D u t t a - H o r n and the limited random walk behaviours. It seems plausible that in three dimensions a 1/f-type spectrum may occur even if the phase-coherent regions contain more than one (but not too many) mobile scatterers. Non-trivial frequency dependence of the power spectrum can also be expected in the crossover regime of tunneling and activated hopping (T ~ T*). 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