Handbook of High Temperature Superconductor Electronics Part 6 ppsx

Conductance noise of a superconductor material, either in the conductor–super-conductor transition temperature region or in the superconductor–super-conductor state, is a non-trivial iss

Conductance Noise in High-Temperature

Superconductors

László Béla Kish

Texas A&M University, College Station, Texas, U.S.A.

5.1 INTRODUCTION

High-T csuperconductor (HTS) materials have the potential to revolutionize low-noise electronics, because superconductor electronics can be realized at relatively high temperatures These temperatures might soon be achievable by solid-state cooling elements at the commercial level if both the HTS and the cooling element development continues at the current level However, it should be kept in mind that this task might not compromise the requirements of low-noise The real con-dition of success of HTS materials will always be the potentially low level of noise, because their high-speed semiconductor and nanoelectronics are strong competitors at convenient working temperatures Due to the different physics of superconductors, their potentially achievable noise properties seem to be unbeat-able However, as far as HTS devices are concerned, although their noise proper-ties are good, the same properproper-ties are achievable by semiconductor circuits also, but the circuits need a carefully designed circuitry Therefore, understanding the source of noise and reducing the noise level in the HTS materials has a high pri-ority in research

Although the details of the mechanism of noise generation in HTS materi-als is not fully understood, it has been proven that percolation effects are the key

to understanding the characteristic behavior of the noise in the conductor–super-conductor temperature regime The percolation models are superior also in the sense that they can predict the behavior of normalized noise in a very wide range, with variations up to nine orders of magnitude Such a success of a model to pre-dict behavior can rarely be found in condensed matters physics

In the present survey, we give a basic overview (based on Refs 1–3) of low-frequency (ƒ 105

Hz) conductance noise in the conductor–superconductor tran-sition region of HTS materials We are concerned only about the “essence,” namely understanding the origin and mechanism of dominant and generally oc-curring noise effects For readers who are interested in learning more about the base of other, more sample- and material-specific noise effects and their theories,

we recommend Refs 1–3 and the references therein The original concept has

been enriched by the inclusion of the biased percolation effect; see Section 5.4.4,

which explains the lack of (universal) scaling when the control parameter is not the temperature but the current or magnetic field Those who are interested in learning some fundamental limits of the present approaches and some relevant un-solved problems, we recommend Ref 4 Since the publication of the percolation picture (1–4), the field has become very active; however, up to now, no compara-ble breakthrough has appeared Some interesting experimental and theoretical ad-ditions can be found in Refs 5–18 It can also be of interest to review Refs 19–27 Noise results relevant for applications, such as bolometers, can be found in Refs 28–33

Finally, some concern to those readers who go beyond reading the present chapter As the aim of this chapter is to show a coherent, and reliable frame of thinking which can be the base of further studies, I will not deal with and will not take any responsibility for all the materials described in Refs 5–33 More-over, in certain cases, I have some strong reservations about the reliability of some of the published data and theories, see the above relevant comments about the number and nature of mistakes in this field However, as the evolution of sci-ence has been manifested by disputes, when some inspiring thoughts were pre-sented in an article, I decided to include it even if I could not always fully trust its content

5.2 BASIC TERMS

Conductance noise in a normal conductor material (34) means a fluctuation of the resistance which can be described as a stationary, random, stochastic process Conductance noise of a superconductor material, either in the conductor–super-conductor transition temperature region or in the superconductor–super-conductor state, is a non-trivial issue, as the material is non-Ohmic and the energy dissipation can contain components due to vortex motion Therefore, in order to avoid any

misunder-standing, in this section, we clarify what we call conductance noise and relate the defined quantities to well-known quantities of classical noise research

For the measurement of the conductance noise of the superconducting ma-terial, it is assumed that a four-terminal measurement method is used (see Fig 5.1)

or an equivalent arrangement to avoid contact noise Otherwise, contact noise could dominate the noise due to the low resistivity of HTS samples when being close to the superconducting state For simplicity and due to the low resistivity, here we neglect thermal noise, however, the thermal noise voltage of the voltage contacts can be a problem The measured resistance fluctuation is defined as

whereU(t) is the measured voltage noise and I is the dc current through the

cur-rent contacts Therefore, the power density spectra of the resistance noise is re-lated to the power density spectrum of U(t) by

SR(ƒ)  SU

I2

(ƒ)

Both the measured noise and the resistance R(T ) of the sample strongly depend on

the temperature and the dependence varies between samples made by different technologies or made of different materials The normalized resistance noise

spec-F IGURE 5.1 Four-terminal sample arrangement The current is fed by a low-noise current generator via contacts 1–3 (current contacts), and the voltage and voltage noise are measured between 2–4 (voltage contacts), presumably

by a device which does not have dc input current Thus the resistance fluctu-ations of contacts 2 and 4 cannot be seen because of the lack of dc current, and the resistance fluctuations of contacts 1 and 3 cannot be seen because of the current generator driving.

trum is defined as

History has shown that to find a coherent, conclusive, and reproducible behavior

of the noise of various HTS samples and materials, the temperature has to be used

as a control parameter (hidden variable) and the C(T ) versus R(T ) curve should be

analyzed In this way, the various temperature dependencies are put out of the

pic-ture and the C[R(T )] or C(R) curve with a hidden temperapic-ture variable supplies

in-formation about the spatial distribution of the microscopic current density distri-bution in the sample This sort of plot made it possible to identify percolation in HTS films already at an early stage of HTS technology in 1989 (35)

5.3 TEMPERATURE DEPENDENCE OF THE MEASURED

NOISE 5.3.1 General Temperature Dependence of the Normalized

Noise

We study the most characteristic temperature-dependent behavior of the noise at

a fixed frequency According to thorough investigations, in a significant fraction

of samples two fundamental temperature regimes exist; see Figure 5.2

SR (ƒ, T )

R2(T )

F IGURE 5.2 Qualitative temperature dependence of the normalized noise and the resistance In the bulk regime, the normalized noise is not increasing, and sometimes it is even decreasing with decreasing temperature In the perco-lation regime, the normalized noise is radically increased while the tempera-ture is decreased.

5.3.1.1 “Bulk” Temperature Regime

At the high-temperature part of the conductorn–superconductor transition region, when the resistance has significantly decreased (the onset of superconductivity

has started), C(R) can be constant, or sometimes even decreasing, with decreasing

R (i.e., with decreasing T ) In this regime, the microscopic current density is

spa-tially homogeneous in the sample (as in a bulk conductor) or, at least, the

distri-bution is independent of the temperature Note: Although a significant part of HTS

samples show this behavior especially materials of lower quality, the bulk regime

is often missing from the C(R) curve of the films (2).

5.3.1.2 “Percolation” Temperature Regime

C(R) increases many orders of magnitude with decreasing R (i.e., with decreasing

T ) This regime is reported in almost all articles in the literature In this regime,

the microscopic current density is spatially random in the sample and the distri-bution randomly changes when the temperature is varied The distridistri-bution of cur-rent density has the properties (percolation) of conductor–superconductor random composites

5.3.2 Scaling of C with R

This is a very frequently occurring behavior (see Figs 5.3–5.5) which can often

be quantitatively explained (1–4) When, at fixed dc measuring current, the

nor-F IGURE 5.3 Scaling with universal exponents through nine orders of magni-tudes of the normalized noise The sample (4) is an Y-based HTS film, repre-senting low-noise and good technology (laser ablation technique with in situ annealing).

malized noise and the resistance R is controlled by varying the temperature, C can

be approximated as a power function of R(T ):

where the exponent x can have various values in different temperature ranges and

in different samples The occurring x values are usually close to the following

val-ues:
2.74,
1.54,
1, 0, and  2 These values are predicted by a simple

the-ory (1) Note that the existence of relation (4) remains hidden if only the R(T ) and

SR (T ) curves are plotted.

F IGURE 5.4 Universal scaling in Y-based superconductor films (1) represent-ing an older technology (evaporation with ex situ annealrepresent-ing).

F IGURE 5.5 Universal scaling in a Y-based superconductor film (1,2) repre-senting a middle-class technology (evaporation with in situ annealing).

5.4 SIMPLE THEORETICAL PICTURE

5.4.1 Framework of the Model: Two-Stage Transition

Picture

For simplicity, we call normal conducting charge carriers “electrons” and super-conducting ones “Cooper pairs.” The model is based on a physical picture of the conductor–superconductor transition (1), which is a slight modification of the two-stage transition picture of HTS materials In sufficiently homogeneous HTS mate-rials, at the high-temperature part of the transition there are no superconducting grains present yet The conductance increases with decreasing temperature due to the increasing number and lifetime of Cooper pairs The current density distribu-tion is homogeneous, so the name “bulk regime” is used here In the low-temper-ature part of the transition region, there are superconductor grains of random sizes

at random locations, because the Cooper pairs are very “fragile” due to their ex-tremely short coherence length, which means that even an atomic scale disorder can prohibit superconductivity in small subvolumes That implies a random distri-bution of current density and naturally leads to percolation effects in this regime The neighboring grains can form superconducting islands via Josephson coupling, and the lower the temperature, the larger the mean linear size (percolation length)

of these islands When the percolation length reaches the thickness of the film, a three-dimensional (3D)/pro-dimensional (2D) crossover occurs At the effective

Tc, where the macroscopic superconductivity sets in (then the system is at the “per-colation threshold”), there is at least one large island between the electrodes

Note: If the Tcin the microscopic subvolumes of the material is strongly inhomogeneous (T c tr, where Ttris the width of the transition region in the subvolumes and T c is the root mean square spatial fluctuation of T c), the bulk region does not exist Percolation occurs in the whole transition region This be-havior can be observed at high-tech HTS materials with a very narrow transition region

5.4.2 Effects at the High-Temperature End of the Transition

Region (Bulk Region) Number of Fluctuations of Electrons and Cooper Pairs Mobility Fluctuations of Electrons

In the present model, it is assumed that, in the bulk region, the noise basically orig-inates from the electrons, as in normal conductors The knowledge of low-fre-quency noise in conductors (36) implies that the microscopic origin of the noise is rather independent of the temperature in the few Kelvin range around 100 K, which is the typical width and location of the conductor–superconductor transition region This condition makes the explanation of the generality of the scaling be-havior easier

5.4.2.1 Number of Fluctuations of Electrons and Cooper

Pairs

It is assumed that the density n of free electrons fluctuates due to trapping When

the proportion of the Cooper pairs is small, at a given temperature the density of Cooper pairs is proportional to the density of electrons In that way, the fluctua-tion of the electron density causes a correlated fluctuafluctua-tion of the Cooper-pair den-sity and the normalized fluctuations n/n of electrons and Cooper pairs will be

equal It can be easily shown (1) that this effect leads to a normalized conductance noise which is independent of the temperature, no matter which class of carriers dominates the dc current Thus,

An example can be seen in Figure 5.4

5.4.2.2 Mobility Fluctuations of Electrons

It is assumed that the mobility  of free electrons is the only fluctuating quantity

As the mobility of electrons (unlike their density) is not coupled to the Cooper pairs, the resulting system can be modeled as two parallel conductors: one of them (the electronic) is noisy and its conductance is independent of the temperature, whereas the other one (the Cooper pairs) is noise-free and strongly temperature dependent It can be easily shown (1) that this effect leads to a normalized con-ductance noise which satisfies the following relation:

C(R) & R2

(6)

An example can be seen in Figure 5.5

5.4.3 Effects in the Percolation Regime Classical

Percolation Noise and p-Noise

Percolation effects in random resistor networks have been intensively studied dur-ing the last two decades (see Ref 3 and references therein) The study of HTS

noise enriched the field of percolation by the appearance of p-noise, which is a

new type of percolation noise (see Sec 5.4.1.2)

5.4.3.1 Classical Percolation Noise

The relevant classical model is a random resistor network, where some resistors

at randomly located places are short-circuited The resistors represent the normal conducting materials, whereas the short circuits represent the supercon-ducting grains, so the lower the temperature, the larger the number of short

cir-cuits The noise originates from the normal conducting material The resistors

are noisy and their resistance fluctuations are uncorrelated and independent of

the temperature In such a system the normalized noise satisfies the following relation:

C(R) & R x

(7)

with x close to
1 and its value depending on the geometrical dimension of the sample Examples are presented in Figure 5.4, at the low-resistance ends of the curves

5.4.3.2 Novel Percolation Noise Effect: p-Noise (1,2)

Assume that we have the same random resistor network as above, except that the resistors are noise-free and some of the short circuits are “noisy”; that is, they ran-domly switch “on” and “off” in time, controlled by independent random processes

w(t) These switching elements represent unstable superconductivity in grains or

intergrin junctions Such switching may occur because of defect motion, electron trapping, or flux motion at the unstable elements The large number of switching

elements causes a fluctuation of the volume fraction of superconducting material, which is called p in earlier works on percolation (0 p 1) We call this fluctu-ation “p-noise.” In order to calculate the behavior of the resultant normalized noise C, it is assumed that in a given temperature range, the number and

dynam-ics of switching elements do not change For that temperature range, simple cal-culations yield (1,2)

C(R) & R x

(7)

with x values given as
2.74 in three dimensions,
1.54 in two dimensions, and

0 in one dimension Figures 5.3 and 5.5 are examples for the 2D and 3D behavior and the 3D/2D dimensional crossover, respectively, described earlier The

differ-ence between classical percolation noise exponents and p-noise exponents is

re-markable

5.4.3.3 Summary of the General Model

Under the applied assumptions, the normalized noise can be approximated as C(R)

& R x

, where the x exponent can vary as the conditions determine (Fig 5.6) Apart

from p-noise, the microscopic origin of the noise is the noise of the normal con-ducting charge carriers If the normal conductor phase is noise-free, then only

p-noise would exist in terms of this model

Figures 5.7 and 5.8 show a rough prediction based on this simple theoreti-cal picture A number fluctuation n of charge carriers and a p-noise p are

as-sumed For simplicity, these microscopic noise sources (n and p) are assumed

to have a constant strength in the whole temperature range An exponentially

de-caying resistance R(T ) is assumed The 3D/2D crossover of percolation is

as-sumed to be abrupt, which makes the relevant peak on the noise curve sharper than real peaks of that kind

5.4.4 When the Temperature Is Kept Constant and the

Current or Magnetic Field Is Varied: Biased Percolation

Until now, we assumed that the temperature was the control parameter which

causes the change of R and C Another interesting questions is what happens when

F IGURE 5.6 Summary of the power exponents by which the C(R) function can

most frequently be approximated.

F IGURE 5.7 Illustration of the scaling of normalized noise versus resistance and the behavior of measured noise voltage in terms of the simple model The dashed hairline shows the change due to temperature fluctuations (not inherent in the general model).