Conductance noise spectrum of mesoscopic systems

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, ]JGIILZo~a/nlz z. In general, (3.6) can be well approximated by a Lorentzian.

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=