Coupled nanostructure lattices: electron dynarnics, experiments and future prospects

Andrew John Bennett

A thesis submitted in conformity with the requirements for the degree of hlaster of Applied Science

Graduate Department of Electrical and Cornputer Engineering

University of Toronto

@Andrew John Bennett 1999

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Coupled nanostructure lat tices: elec tron dynamics , experiments

and future prospects

Master of Applied Science, 1999

Andrew John Bennett

Department of Electrical and Computer Engineering

University of Toronto

Abstract

Modern microelectronic devices have changed little since t heir invention. Insteaci, exponential

improvement in performance has arisen prirnarily from miniaturization. However, the cost of the

lithographie fabrication required has also grown exponentially, and problerns associated with heat

dissipation and interconnection increase with packing density.

Non-lithographie template synthesis is a prornising method of hbricating orciered, regular struc-

tures on the nanoscale (< 100 nm). Such nanostructiires could enable new device and system

designs, but basic physical issues need to be resolvecl first.

To this end, we studied dissipative charge dynamics in a model system of a coupled nanostruc-

ture lattice. Additionally, ive investigated electron tunneling in coupled lattices using Monte Car10

simulations, and using o new continuum model.

The long-term objective of this study is to use nanostructure lattices in computational applica-

tions. We therefore constructed analogies between coupled lattices and the automata network, an

abstract and highly generalized model of corn pu tation.

Acknowledgement s

Thanks to my supervisor, Professor Jimmy Xu, for providing me with the opportunity to do

research in the Emerging Technologies Laboratory, lor introducing me to the study of nanoelec-

tronics, and for well-timed encouraging words.

Thanks to the members of rny examining cornmittee (Profs. J. Long, E. H. Sargent, and S.

Zukotynski) for the valuable time taken out of their full schedules to read this text, to attend my

thesis defense, and to provide helpful comments and constructive criticism.

Thanks to the members of the Nanostructures/Nanophysics group in our laboratory for inter-

esting discussions, criticisrn and collaboration, and to Mark Roseman and Peter Griitter a t McGill

University for help with scanning probe microscopy.

Thanks to my family (Jessica, John, Kulla) for general support, encouragement, and good ad-

vice.

Thanks to Tara for being an ideal cornpanion throughout the research and writing of this text,

and for her perspective, optimism, and example.

Funding for research and travel was provided by the Naturai Sciences and Engineering Research

Council (NSERC), Mitel Serniconductor through the Canadian Advanced Technology Association

Microeiectronics Scholarship, Norte1 Networks, the European Cornrnunity, NATO and the Japan

Society for Applied Physics.

Contents

1 Introduction 1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The end of Moore's Law? 1 . . . . . . . . . . . . . . . . . . . . 1.2 Present and future challenges in microelectronics 3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . 2 . Interconnection 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Heat dissipation 4

. . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Mesoscopie and quantum effects 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Fault tolerance and reliability 5

. . . . . . . . . . . . . . . . . . . . 1.3 Collective dynamics in physics and computation 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Organization of the text 7

2 Non-lit hograp hic nanost ruct ure arrays: fabrication and material properties 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Fabrication 9

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Experiment 12

3 Dissipation and the environment in electron tunneling 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction 14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Tunneling and dissipation 14

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Tunnel rates 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Properties of J ( t ) and P ( E ) 20

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Dissipative dynamics 20 . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Zero-temperature formulation 21 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Finitetemperature formulation 25

3.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4 Single electron tunneling junctions and circuits 30

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 4.2 Nanostructures and tunneling junctions . . . . . . . . . . . . . . . . . . . . . . . . . 31 4.3 Electron tunneling in Monte-Carlo simulation . . . . . . . . . . . . . . . . . . . . . . 36

4.3.1 Change in the electron energy due to tunneling . . . . . . . . . . . . . . . . . 39 4.4 Continuum approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

. . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Derivation of continuum mode1 43 4.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

5 Computation schemes employing ordered nano-arrays 48

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.2 Poten tial advantages of "p hysically" parallel corn pu tation . . . . . . . . . . . . . . . 49 5.3 Computation as evolution of phase space trajectory . . . . . . . . . . . . . . . . . . . 50 5.4 First order dynamics of computational machines . . . . . . . . . . . . . . . . . . . . 51 5.5 Analogies between neural networks and ordered nano-lattices . . . . . . . . . . . . . 52 5.6 Analogies between cellular au tomata and ordered nano-1st tices . . . . . . . . . . . . 53

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Future opportunities 54 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Notes 55

6 Conclusion 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Contribution of the present work 56 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Future work 56

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.21 Initial efforts 57 6.2.2 Phase transitions and other cooperative behavior . . . . . . . . . . . . . . . . 57

. . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 The vertical degree of freedom 57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Experimen tal investigations 58

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Conclusion 58

List of Figures

Moore's law for transistor coun ts. (after [BH93]) . . . . . . . . . . . . . . . . . . . . SEM image of Cu VLSI interconnects fabricated using IBM CMOS 7s process, with

minimum 0.12 Fm feature size. (after [Gep98]) . . . . . . . . . . . . . . . . . . . . .

Schernatic diagrarn of ordered nanowire array. (after [RTH+96]) . . . . . . . . . . . . Top view AFM image of highly ordered, close-packed array of nanopores. . . . . . . Perspective view AFM image of highly ordered, close-packed nanopore array. . . . . Magnetic force microscopy image of cobalt nanowire array. (M. Roseman, McGill

Univ.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . + . .

Schematic of single tunneling junction of effective tiinneling resistance R and capac-

itance C, with source (k) and drain (q) electrodes. . . . . . . . . . . . . . . . . . , P(E) at zero temperature for an ohmic resistor R in serieï with a single tunnel

junction of capacitance C. (parameters: R = 0.05 R,, 0.5 Rp, 5.1 Rq, C = 5 x 10-17) . P ( E ) a t zero ternperature for an ohmic resistor R and an ideal inductor L in series

with asingle tunnel junction of capacitance C. (parameters: R = C = LO- '~R,

L = (l /C)(e/~tcC)~, such that hw, = E, = e2/2C)) . . . . . . . . . . . . + . . . . . . Convergence of iterative solution procedure for P ( E ) . In a successful solution, the

integrated probability ends up very close to one, while the relative error IPi+r (E) - ( E ) 1 / 1 &+l (E) 1 converges to zero. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

P ( E ) at finite temperature for an large O hmic resistor R in series wit h a single tunnel

junction of capacitance C. (parameters: R = 1.25 x 10'0, C = 5 x 10-I'F) . . . . . P ( E ) (log-scale) at finite temperature for an ohmic resistor R and an ideal inductor

L in series with a single tunnel junction of capacitance C. (parameters: R = 103R,

C = 10-16F, L = (~/C)(e/2hC)~, such that tw. = Ec = e / 2 C ) . . . . . . . . . . . .

Double junction system showing a single isiand coupled to source and drain electrodes

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . by tunneling junctions. 33 Equivalent circuit diagram of dou bIe ju nct ion. . . . . . . . . . . . . . . . . . . . . . . 33 Equivalent circuit diagram of paired nanostructure (parallel, coupled double junctions). 35

1-V characteristics of coupled nanostructure pair for elastic and inelastic tunneling. . 35 Energy cost associated with electron tunneling into different sites on an array of N =

LOO coupled nanostructures, for Cg = 0, 10-la, IO-", 1 0 - l ~ ~ . Smoother changes

in space are associated with stronger coupling between lattice sites. (parameters:

Ri = 1o8R, Rz = 10% and Cl = C2 = 1 0 - l ~ ~ ) . . . . . . . . . . . . . . . . . . . . . 37 Monte-Carlo simulation of I-V characteristics of coiipled nanostructure array for

coupling capacitances Cg = 0, 10-18, IO-'?, 10-16F. The Coulomb charging steps are

most pronounced for zero coupling, and disappear in the case of strongest coupling,

where Cg > CL, C2. (parameters: RI = 1o8R, Rz = 10" and Cl = C2 = l0-l7F) . . 38 Schernatic of mode1 system of coupled nanowire array . . . . . . . . . . . . . . . . . 39 Self-consistent numerical calculations and exponential fit of poteritial shift due to an

excess charge at lattice index O. High coupling capacitances lead to a low, broad

potential shift, and a good exponential approximation. Low coupling capacitances

lead to a sharply peaked potential shift and a poor exponential fit. . . . . . . . . . . 42 Analytical solution for k-dependent decay of normalized charge density perturbation

n(k) in an array of nanostructures. The curves are spaced in time by 10-L Is beginning

with log(n(k)) = O at t = O. (parameters: Cg = 10-15F, Ri = Rz = 1o6R, Ci =

Cz = 10-"F, a0 = 6 . 0 2 ~ (a is the lattice constant) and = -4.93 x 10-") . . . . . 46

Highly ordered hexagonal array of carbon nanotubes grown in durnina rnatrix (af-

ter [BLL+99]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

vii

List of Tables

4.1 Energy changes for forward electron tunneling events occuring in a nanowire pair. . 34 4.2 Energy changes for al1 possible electron tunneling events occuring at lattice site i. . . 41 4.3 Energy changes for a11 possible electron tunneling events occuring at coordinate r. . 44

viii

Chapter 1

Introduction

The great success of microelectronics has not occurred due to technological revoltitions during

its development, but due to the remarkable performance improvements enabled by incremental

rniniaturization. In fact, few fundamental changes have been niade to the planar fabrication pro-

cess devised by Noyce in 1958 or to the MOS transistor scaling principles developed in the early

1970s [HM72, DGY+74]. Instead, t hese initial breakthroughs have been refiried to an unprecedented

tevel. In the midst of its undeniable success, the microelectronics industry is coming to a cross-

roads. Previously routine serniconductor process improvernents now dernand consitlersble research,

development and investment, and fabrication plants now cost billions of dollars to build [Sem97]. In

parallel with Moore's law of exponential growth in semiconductor system performance (Figure 1.1)

operates anot her, less desirable rule, which is sornetimes called Moore's second law. Corn bined,

these "laws" state that both system performance, aiid the capital investment reqiiired to fuel it,

double every eighteen mont h s [Mal%].

1.1 The end of Moore's Law?

This second law suggests that even if physical and technological problems are overcome, the semi-

conductor industry must address the rising costs of research and equipment and must balance

them against the diminishing benefits of the one-dimensional track of miniaturization. As costs

increase exponentially while return-on-investment in hardware technology decreases, al1 but the

largest players will be squeezed out, and capital will 0ow to the areas in which radical innovation

and significant performance improvement are still possible: for example, novel silicon architectures,

system desips, and software. This will consequently slow down innovation in hardware technologies

Figure 1.1: Moore's law for transistor counts. (after [BH93])

and lead to the plateauing of Moore's law.

The saturation of Moore's law may yet be a blessing in disguise. Since no one can predict

the final outcome as hard econornic and technical iimits are reached over the next two decades,

the future of cornputing need not be determined exclusively by the major players in the industry,

or by others with billions of dollars to invest. Instead, one can anticipate a burst of creativity,

such as new devices, architectures, and system designs, which will be required to address the

burgeoning problerns arising frorn miniaturization, interconnection and heat dissipation. The search

for alternative paths to nanoelectronics opens up the future to msny more potential players, and

likely to new ones as well.

Alternatives to the obvious path of rniniaturization do exist. Noniithographic nanofabrication is

one example, and a particularly attractive one given that lithography is the most costly part of the

current technology. It is becoming increasingly apparent that "natural" approaches such as self-

organization and self-assem bly can produce well-ordered, high quality, densely packed (101° cmw2)

structures on the nanoscale [LMPX]. In such fabrication processes, physical and chernical laws,

acting and/or competing on small characteristic length scales, conspire to build or grow structures

on the nanoscale. In cornparison to precision photo or electron beam lithography, such approaches

typically require relatively little direct engineering effort to generate complex structures.

Indeed, i t is possible that while circumventing some of the di-fficulties faced by lithography,

non-lithographie nanofabrication techniques can also address some of the rnost significant problems

facing current and near-future microelectronic systerns. Interconnection and heat dissipation are

two of the most significant and pressing challenges; these are intrinsic to the binary and serial

paradigm of computation on which the design of current-day systems is based. Future challenges

on the horizon include mesoscopic and quantum physical effects, and the statistical problem of

finite device yields.

Computational schemes based on coiiective interactions and dynarnics in macroscopic arrays

of nanoscale structures could circumvent the presen t challenges of heat dissipation and in tercon-

nection by storing, manipulating and transmitting information in an entirely different way than

in current microelectronic systems. Additionally, such systems could be engineered to account for

and even take advantage of the inevitable mesoscopic physical effects encountered in nanoscale

structures [BLL+99].

1.2 Present and future challenges in microelectroiiics

1.2.1 Interconnection

In order to keep the resistance of circuit interconnects low, it is desirable to make them as broad as

possible. However, this requirernent contradicts the philosop hy of miniaturization, which improves

the functionality and power of semicond uctor circuits and systems by increasing packing density.

In fact, some highly interconnected circuits have over ninety percent of their "real estate" taken

up by interconnections, leaving only ten percent for transistors and other devices [HI

problem for two reasons. First, the prospect of interconnecting LO'O devices of < ,100 nm dimension in the space of 1 cm2 would overwhelm current, and likely future, interconnection technologies. This

would demand a completely different approach to corn pu tation whic h would take locality and the

cost of signal transmission into account. Second, one could utilize the "wireless" coupling between

lattice sites-e.9. Coulomb, dipole-dipole, and possibly even magnetic interactions-to transmit

and process information.

1.2.2 Heat dissipation

With today's fastest processors dissi pating approximately 30 W/cm2-a greater power density

than a typical cooking elernent-heat dissipation will become an even more significant problem as

packing density of devices continues to rise [Keygï].

With respect to ultimate linlits, there is certainly plenty cf room for irnprovcmcnt upon current

systems. Modern-day microprocessors dissipate approximately 10-~.1 per bit switched, while one's

brain is estirnateci to dissipate only 10-l6J per bit for a total of 4 W in the average human [tlay94].

This ten-order of magnitude disparity arises from the fiindamental difference i n operating very fast,

binary switching elements versus a greater nurnber of relstively slow, densely packed and highly

interconnected analog (integrateand-fire) elements. For cornparison, we note that the minimum

energy to switch a bit irreversibly was computed to be k T ln(2) or approxirnately 3 x L O - ' ~ ~ [LanSG].

From a head dissipation point of view, it is growing increasingly costly to comrniiriicate between

binary switching logic blocks using narrow, and hence resistive RC interconnects. Coupled nanos-

tructure lattices may have an advan tage in solving t his problem, since little current, and thus almost

no resistive heating, is associated with signal transmission through dipole or Coulomb interactions.

Similar capabilities might be intrinsic to cornputers based on quantum dot cellular autornata (QCA)

which also operate through electronic polarization of nanostructures [LTg?, SOA+98]. The dif-

ference between these two alternatives is that while coupleci nanostructure lattices will not use

a seriallbinary approach to computation, QCA still uses this approach. Consequently, QCA has

significant obstacles to overcome before it can be used at room temperature.

1.2.3 Mesoscopic and quantum effects

A problem which will affect future nanoscale silicon devices is the failure of the fundamental as-

su m ptions undeilying the semi-ciassical Boltzmann transport equation . This is a direct consequence of the overwhelming success of singletrack miniaturization [Dat95, FG97j. As device dimensions

drop below the dephasing length andior the mean free path of the electron(- 100 nm or greater

in high-quality semiconductors), we enter the realm of 'bmesoscopic'~ transport. In t his regime,

many unusual effects are manifested: for example, ballistic transport, waveguiding effects, and

phase-dependent phenornena such as coherent interference and reflection [Dat95]. At still srnaller

size scales (- 1-2 nrn), quantum tunneling through thin oxide layers, like transistor gates, becornes

signifiant. A11 of these miniaturization effects disrupt many of the most fundamental design de-

mands of digital systems such as signal cascadability, fan-out, binary iogic level restoration, and

input-output isolation [I

correct solution to a problem may be obtained in the presence of both noise and defects.

1.3 Coiiective dynamics in physics and computation

We envision a goal of computing using non-lithographic, self-organized nanoscale lattices, although

not by simply con tinuing to exploit the heretofore successful paradigm of miniaturization. For an

alternative technology to compete with the incumbent or dominant technology, it needs to address

different applications, or else to address the same needs from radically different angles. Many new

technologies such as Josephson junction cornputers, resonant tunneling logic, biastable optical logic,

and so forth, were heralded in the last few decades as the "inovitable" suceessor to silicon VLSI,

usually because of extrernely high switching speecl. Despite the early promise of many of these

technologies, none fulfilled t heir proponents' expectations [l

Figure L.2: SEM image of Cu VLSl interconnects fabricated using IBM CMOS 7s process, with

minimum O. 12 pm feature size. (after [GepSB])

Despite the fascinating technological possibilities raised by such new computation concepts and

architectures, there exist nurnerous challenges to the development of devices and systems based

on non-lithographie nanostrcict ures. At the presen t time, a VLSI microprocessor is a physical

system with a phenomenal degree of engineered complexity and information conterit, as shown in

Figure 1.2. In contrast, present day nanostructured materials tend to be either randomly patterned,

ordered in too small a region, or else to have a very simple lattice structure repeated rnany millions

of tirnes. Unfortunately, neither a randorn (amorphous) lattice, nor the ordered lattices which we

con fabricate, effectively re plicates the engineered com plexity of a micro processor* T herefore, while

the opportunities and potential rewards are significant, challenging work lies allead. An accessible

review of some of these issues is given in [CII

has been performed over the past few years.

In Chapter 3 we examine the probiem of electron tunneling in a model system based on the

non-lit hographic nanostructure lattices described in Chapter 2. Since we wish to use discrete,

wirelessly coupled nanostructure lattices in computation, it is necessary to isolate the lattice sites

from one another. Thus, the issue of electron tunneling is raised as a way to change the charging

characteristics, and thuq the relative interaction strength, of the individual lsttice sites. A formalism

for electron tunneling in the presence of dissipation is developed and applied to several model

eiectromagnetic environ men ts.

In Chapter 4, we study the problem of electron tunneling in coupled arrays of nanostructures,

using a discrete (Monte-Carlo) simulation technique, as well as a new 'continuum' model.

Chapter 5 leads into the idea of applications: using a coupled nanostructure lattice as a computer

based on a cellular automaton or neural architecture. Analogies between both neural networks and

cellular automata, and cou pled nanostructure lattices, are described. The ideas presented are highly

exploratory and speculative, and siiggest n umerous opport u ni ties for work in the near fu turc.

Finally, in Chapter 6 we present sorne possibilities for extensions of and fiiturc work on the

problems considered in this text, as well as our conclusions.

Chapter 2

Non-lit hographic nanost ruct ure

arrays: fabrication and material

propert ies

2.1 Fabrication

Porous aluminium oxide films formed by electrochemical anodization processes in various elec-

trolytes have been studied for more than Forty years [I

poreus alumite loyer

mr tol aluminum

f layrr

Figure 2.L: Schematic diagram of ordered nanowire array. (after [RTH+96])

force microscope image of a typical example of the self-organized anodic alumina template with

highly ordered pores, as grown by the author with significant assistance from J. Li.

After fabricating the AAO template, nanostriictures may be grown in the pores by a number

of different met hods:

a electrochemical deposition of metals, alloys and cornpounds into the pores of the template

electrophoretic filling of the pores with colloids

e filling with sol-gels via capillary action or with metals at high pressure

a filling through CVD or polymerization.

Deposition of materials in the AAO ternplate has been used to produce nanostructure arrays

with curious properties often differing from those of the bulk rnaterials. For example: A I I B ~ CO,

pounds (e.9. CdS and CdS,Sel-,) have been deposited into the AAO template [RBMX96]; gold

clusters and colloids (quantum dots) have been introduced into the AAO template via electrophore-

sis [HKP97]; bismuth quantum wire arrays have been deposited by a vacuum rnelting and pressure

injection proces [ZYD98]. Using a new and very promising method, Li et al. first produced and

subsequently filled arrays of carbon nanotubes in the AAO template. The desired interior metal,

such as nickel or cobalt, was deposited within the nanotubes by "electroless" deposition [LMH98].

Figure 2.2: Top view AFM image of highly ordered, close-packed array of nanopores.

Figure 2.3: Perspective v i e ~ AFM image of highly ordered, close-packed nanopore array.

The AAO template nanofabrication method passesses rnany advantages. It provides a simple

and inexpensive way to fabricate large area, highly ordered, high density (- 10'' cm-*) arrays

of closely-packed nanopores that can be filled with wires, dots, and even tubes in a large range

of materials. Magnetic and non-magnetic metals, superconductors, semiconductors, other optical

materials, and recently carbon nanot ubes, have al1 been successfully deposi ted.

The potential applications for such a technology are numerous, and include ultra-high density

rnagnetic storage, field-emission displays, and optical/infra-red detectors. These possibilities have

been described in the literature [TRa96, RTHf 961. The recent progress in forrning highly ordered

and uniform array structures rnakes these possibilities al1 the more promising.

2.2 Experiment

A number of optical, magnetic, and electronic experimen ts have been performed on nanowire arrays

fabricated using the procedure described above. Conductance values and 1-V chsracteristics were

obtained for metal nanowires at different temperatures, and cornparisons were made with the theory

of Nazarov describing coupling tunneling electrons to the electromagnetic environment [D HR+98].

Coulomb blockade effects were observed in nanowires capped with oxide layers [AMEM+]. Other

measurements include X-ray diffraction, resonant Raman spect roscopy and p hotoliiminescence to

study crystal structure and quantum size effects, and magnetic coercivity measurements to assess

potential for high-density magnetic storage applications [RBMX96].

Recent experimental efforts have been directed toward the development and use of nanoscale

probing and measurernen t capabilities. Scanning probe rnicroscopy is an important tool in t his con-

text, in order to perform topographical, magnetic and electrical measurements on the sub-100 nm

scale of the lattice sites themselves. In particular, magnetic force microscopy was used to determine

the magnetic dornain size in cobalt nanowires, as shown in Figure 2.4. Such measurernents may

have importance in determining the coupling of electronic and magnetic degrees of freedom, with

possible influence on the information handling capabilities of coupled lattices.

Figure 2.4: Magnetic force microscopy image of cobalt nanowire array. (M. Roseman, bfcGill Univ.)

Chapter 3

Dissipation and the environment in

electron t unneling

3.1 Introduction

In Chapter 1 we raised the possibility of using ordered nanostructure lattices to perform computa-

tion. For example, it might be possible to exploit self-organization of charges in long nanowires (via

interaction energy rninimization) to compute the solution to optimization-type problems. Such a

computation scheme would require some means of loading the array with charges, such as through

a scanning tu nneling micrcscope (STM) tip. In addition to such highly speculative technological

schemes, much could be learned about the individual nanowires through broad-area contact trans-

port measurements, or through local probe STM current spectroscopy measurements. For these

reasons, it is important to consider the issue of electron tunneling in the presence of a dissipative

environment, such as an array of nanostructures.

In this chapter, we consider single electron tunneling in the presence of an arbitrary electro-

magnetic environment, represented by an irnpedance Zt (E). In the following chapter, we will build

upon this to study electron tunneling in a mode1 system of a coupled array of nanostructures.

3.2 Tunneling and dissipation

It is a well-known fact in physics that electrons may temporarily disobey the law of conservation

of energy and ''tunnel" through a potential barrier despite the existence of classical turning points.

Such effécts have been observed in vacuum, in junctions fabricated from thin (1-2 nm) insulating

oxide layers, and in twedirnensional electron gases shaped by lithographically-patterned surface

contacts, among other systems. Indeed, this is one of the most unusual and most celebrated results

of quantum mechanics. In the text to follow, we will be primarily concerned with electron tunneling

in solid-state systems (Le. simiiar to the nanostructures described in Chapter 2). In such systems,

the tunneling behavior of the electmns is influenced significantly by the cornplex structure of the

surrounding environment.

In general, electron tunneling processes occur stochastically, obeying a quantum rate equation.

A metal-insulator-rnetal (MIM) tunnel junction behaves like a "lesky" capacitor. In fact, according

to the lowest (trivial) order of perturbation theory, such a junction is a pure capacitor since the

tunneling cornponent of the Hamiltonian is neglected entirely. Successive perturbation terms lead

to ordinary tunneling processes ( i e . the Fermi golden rule), the second order terms representing

two-particle processes ( "cotunneling" [ANSO]), and so forth [IN92].

In addition to the multiplicity of tunneling processes, there also exists the possibility of the

tunneling electron coupling to modes of the electromagnetic environ men t-that is, inelastic tun-

neling. We will see that inelastic tunneling rnay be described by a probability distribution function

associated with the energy loss of the tunneling electron, and we will make extensive use of this

function, denoted by P(E) . For an electron tunneling in a cornplex environment or surroundings,

inelastic tunneling is the rule rather than the exception.

A full description of the dynamics of electron tunneling in coupled arrays of nanostructures

must clearly take the possibility of inelastic tunneling into account. In such an array, there exist

electrornagnetic modes to which an electron may couple and dissipate some of its energy; this in

turn affects the tunneling probabilities and hence t lie average tunneling rates.

In order to characterize the inelastic character of a particular electromagnetic environment, we

will fint write down the Harniltonian of the system ( i e . contacts + tunneling + environment). The environment felt by the tunneling particle enters the formulation as an ensemble or 'breservoir"

of harrnonic oscillators which can receive energy from the tunneling electron, and, for T > O, give energy to the electron. After some formal analysis i t is possible to relate the inelastic behavior of a

particular environment, represented by P ( E ) , to the impedance &(E) of the environment as seen

from the junction.

junctio n 9

drain

Figure 3.1: Schematic of single tunneling junction of effective tunneling resistance R and capaci-

tance C, with source (k) and drain (q) electrodes.

3.2.1 Tunnel rates

The follwing formulation of single electron tunneling is due to Ingold and Nazarov [IN92]. The

quasiparticles in the electrodes are describcd by the following Hamiltonian:

where k refers to the wavevector of quasiparticles in the top electrode, q to the wavevector of

those in the bottom electrode. cLo (ck,,) is the creation (annihilation) operator for quasiparticles of

wavevector k, spin o, and energy c k . The system of the tunneling junction and electrodes is shown

in Figure 3.1.

The tunneling Harniltonian is given by

where the action of the operators removes one quasiparticle from a given electrode and creates one

on the other electrode, as expected in a tunneling process.

The total Hamiltonian is the sum of the two cornponents (3.1) and (3.2), plus the Harniltonian

of the environment.

The phase 9 is denoted by

where U = Q/C.

The variables in (3.3) are bit) = d ( t ) - ( e l i i ) Vt, the fluctuation about the average phase, and Q = Q - CV, the fluctuation about the average capacitor charge. Note that these variables are selected to be canonically conjugate, that is, [4, QI = i e .

In this situation described by fI,,, the electron loses energy to the environment bu t the en-

ergy remains present in the form of photons. The introduction of a dissipative (resistive) term

corresponding to in homogeneity in the crystal will couple electromagnetic oscillations to the oscil-

lations of the crystal lattice. This is similar to classical darnping or friction, in which the forces are

proportional to velocity, in that the motion of electrons leads to dissipation.

In the subsequent formulation, the assumption that the effective tunneling resistance RT is

large compared to the quantum resistance Rq = h/e2 z 25.8 kR allows u s to treat the tunneling

component of the Hamiltonian as a perturbation to the remaining parts. This effective resistance is

derived from fundamental constants, and can be viewed as a characteristic measiire of the mixing of

the quasiparticle states on the two electrodes. In the case of high resistance, the states are distinct

from each other.

Since, by assurnption, the tunnelling Hamiltonian is srnall, it can be treated as a perturbation

to the other components of the total Hamiltonian. Thus, it is possible to use the Fermi golden rule

(first-order perturbation) to determine the tuniieling rates [Sakg41

The equilibrium (unperturbed) states may be written as the product of a quasiparticle state

and a charge state, the latter being associated with the coupling to the environment or reservoir.

Letting Ji) = IE)I R) and 1 f) = 1 E') IR) where IE) and 1 E3 are quasiparticle states of given energy and IR) and IR') are reservoir (charge) states of energy ER and EtR1 we find that

where H+P is the factor of the tunnel Hamiltonian which operates on the quasiparticle component

of the state.

In order to compute the total tunneling rate, a surn over al1 possible k and q states, weighted

by PB, their probability of occupation a t inverse temperature /3 = l /kT , must be performed.

An arbitrary term in H+p is T ~ , . & c ~ ~ ; this term couples filled states IL, o) on the top electrode

to empty states 19, o) on the bottom electrode. Al1 other state occupation numbers are irrelevant,

and so this allows factorization of PB(E) in terms of Fermi distribution functions.

This simplification yieids

where PLI(E), the probability to find the initial state, has been replaced with the product of Fermi

functions,

Assuming that eV is much snialler than the Fermi energy, it can be assurneci that al1 tunneling

events occur near the Fermi energy. If Tk, is approximately independent of s,: and E,, then the sum

over rnatrix elements can be replaced with an average matrix element lT12 which incorporates the

density of states a t the Fermi energy pIBJJS88].

Merging al1 of these factors into an effective tunneling resistance RT possessing the desired units

of Ohms, we obtain

Turning now to the sum over reservoir states, we write the probability of finding initial state

1 R) in terrns of the quilibrium density rnatrix as:

w here

pp = z;L eV(-PH,,)

and Zg = Tr(exp (-P Hm)) is the environmental partition function [Pat72].

18

Rewriting the delta function in terms of its Fourier transform, we obtain

and this gives

for the tunneling rate. Substituting the definition of the equilibriurn correlation hnction

into the rate expression we obtain the foilowing

The environmental Hamiltonian is hsrmonic can be seen from Equation (3.3); by Wick's the-

orem, it is therefore possible to express the correlation function (3.14) i n terms of phase correlation

functions of second order or less ptYS9.51. This permits the simplification

to be made. Performing the substitutions

P ( E ) = Jm dt erp ( ~ ( t ) + F) 2nb ,, we can simplify considerably the tunneling rate expression

By the properties of the Fermi function and some basic algebra, one can obtain the final result

for tunneling through a single junction in the presence of dissipation. This result was derived by

Ingold and Nazarov [IN92].

This function P(E) is very important, and is a very useful way of incorporating the influence of

the surrounding environment into the tunneling rate. In future sections, we will see that P ( E ) rnay

be obtained from the impedance Z(w) of the macroscopic circuit in which the tunneling junction

resides. In fact, there exists a highly intuitive and easily understood interpretation of P ( E ) : it rnay

be viewed as the probability for the tunneling electron to give up energy E to its surroundings.

Computed P ( E ) functions for various mode1 circuits validate this interpretation, and we will see, for

example, that a circuit with resonont frequency wo will typically possess a P ( E ) with pronounced

peaks at E = h, 2&, . . . and so forth.

3.2.2 Properties of J ( t ) and P ( E )

There are a nurnber of usefd and general properties of J ( t ) and P ( E ) whicli are maintained

regardless of the environmen ta1 impedsnce. ln particulsr, we note the following

P(E)dE = exp[J(O)] = 1

from the definition of the delta function and (3.18).

Another useful integral property is obtained from the tirne derivative of Equation (3.18).

A final property is the "detailed balance syrnmetry" .

In words, the probability of the electron absorbing energy from the environmest is equal to the

probability of the environment absorbing energy from the elect ron, multiplied by a Boltzmann

factor.

These properties were detived in detail by lngold and Nazarov [IN92].

3.3 Dissipative dynamics

A number of analytical approaches exist for determining P ( E ) from Z t ( E ) , the effective impedance

of the equivalent circuit of t h e environment, as seen From the tunneling junction. While many

of these approaehes are effective in certain important regimes such as high bias or Iow bias, it is

desirable for u s to obtain a quantitative description of this function P(E) over a range of energies.

Since our ultimate goal is to describe the dynarnics of electrons in a complex environment-coupled

nanostructure lattices-we want a large measure of flexibility in our formulation. Fortunately, it

is possible to derive general equations describing P ( E ) over a wide range of energies. The cost is

complexity and cornputational time-these equations are generally not analytically tractable and

must be solved through numerical approaches.

3.3.1 Zero-temperature formukt ion

Integral equation

We seek

over the

solutions of the integral equation

domain E 2 O. This equation is obtained from the definition of J ( t ) at zero temperature,

following a transformation from the t-domain to the (E = hw)-domain [Min76]. Since it is not

possible for the tunneling electron to absorb cnergy from the environment at zero temperature, it

is sufficient to consider only positive electron energy losses. We sought to develop a method for

the solution of this equation for arbitrary Z t ( E ) . Çince an analytical method was not a realistic

objective for cornplicatecl environmental impedances such as those encountered in a coupled array,

especially over a wide range of voltage biases, we chose to investigate numerical solutions of the

equation.

Numerical solution

Converting the integral equation to a discrete sum equation, we obtain the following result

where Pi = P(E;), w; is a weighting function arising from the (as yet unspeciled) quadrature rule, and E; = iAE from the fixed lower integral boundary of zero and the assumed equol spacing of

quadrature points. Our objective is to determine the value of Pi for al1 integers i E [O, N - LI. We define the kernel of the discrete integral equation to be

The argument of this function, Ei - Ej, is simply (i - j )AE, and depends only on the difference of the indices of the two points. We therefore write the kernel as Ki-j.

In order to finish the discrete version of the integral equation, a quadrature rule for the integral

must be selected. We chose a trapezoidal integration approach since it is relatively simple, and

thus amenable to algebraic manipulation, without sacrificing numerical accuracy [Coh92, PFTV921.

The trapezoidal rule is characterized by a lincar interpolating function between quadrature points.

In the case of N equally spaced quadrature points, we have the following formula for the integral

from points a = xo to b = Z N - 1 .

Note that the endpoints oiily contribute to the surn once, wliile al[ other points contribute twice.

First applying the trapezoidal ride, we tlien write the integral equation in s somewhat more

compact form

After some algebraic manipulation, it is possible to bring the equation into the following form

We have defined Po = 1 knowing that P ( E ) is a probability distribution, and can be normalized later on. The scaling factor is not necessary for the initial solution of the equation.

This necessitates making an approximation to the infinite upper bound of the integral; the lower

bound is always zero. In general, i t is possible to determine an appropriate balance between the

competing goals of (i) numerical accuracy and stability, which demands relatively narrow spacing

of quadrature points, and (ii) an effective approximation to the upper bound, which demands either

wide spacing or else a very large number of points. Since the number of parameters to be varied

is smaii, it is usually easy to find an ( N , A E ) pair which gives good results with relatively little

computation time.

Our final equation can be expressed in rnatrix form.

/ -Ko O O O ... O Kt 2 - K o O O ... O

2& 2Kt 4 - & O W . . O

2K3 2K2 2K1 6-1

0.2 Energy (eV)

Figure 3.2: P ( E ) at zero temperature for an ohmic resistor R in series with a singlc tunnel junction

of capacitance C. (parameters: R = 0.05 R,, 0.5 R,, 5.1 R,, C = 5 x 10-17)

Figure 3.3: P(E) at zero temperature for an ohrnic resistor R and an ideal inductor L in series with a

single tunnel junction of capacitance C. (parameters: R = 10%, C = 10-l6R, L = ( / C) (e/2ftC)* , such that b, = Ec = e2/2C))

Another way of describing this system considers the junction as a capacitor. When an electron

tunnels through the junction, the electrostatic energy cost due to the change of charge on the

capacitor is e2/2C. For very low impedances, the voltage source supplies charge to fix the resulting

im balance on a very short timescale-effectively instantaneously-meaning t hat a tu n neling process

taking place for V < ea/2C does not cause violation of energy conservation. In contrast, the high impedance environment lengthens the timescale over which the voltage source rnay recharge the

junction, meaning that energy conservation is violated; consequently, no tunneling is permitted

until V > e2/2C [GGJ+90]. In Figure 3.3, we see that the electron lias a high probability of coupling into the elastic peak

at E = 0, but also has significant probabilities of coupling into the circuit resonances at E =

Iwo, Zhu0, . . . where wa = (LC) I l 2 . Increasing the resistive component of the impedance dirninishes the effect of the resonances and spreads out the energy lost over many more environmental modes.

3.3.2 Finite-temperature formulation

Integral equation

In the finite temperature case, the equation to be solved to obtain P ( E ) is also an integral equation,

although a more cornplex one than in the zero temperature case. The inhomogeneous Minrihagen

integrai equation is written as

where I(E) is the inhornogeneity defined by

with D = rRe(Zt(0))/(phRq) [IG91].

K ( E , E') is the integral kernel defined by

where v, = 2lrn/h/3 [GSI88].

The matrix inversion approach used to solve the zero-temperature case is not suitable for the

finite temperature case. In particular, the simple lower triangular form which arises from the

Numerical solution

We used an iterative approach

PO(^ =

P m =

P2(E) =

.

integral boundaries is not present here, since it is in principle possible for the electron to gain or

lose arbitrary amounts of energy to the environment. Additionally, it is necessary to approximate

both the upper and lower (infinite) integral boundaries by finite numbers, in contrast to the zero

temperature case where the lower boundary was fixed. By the detailed balance symmetry (3.23),

it can be seen tha t the lower boundary should be much smaller than the upper boundary at low

temperatures; as the temperature is increased, this condition no longer holds.

to solve this integral equation, with I ( E ) as the first test function.

Irn d E 1 K ( E , E') Irn dE"K(Et, E")I(Et - Eu)

This can be written more cornpactly as a surn over repeated in tegrals,

in which the map F is defined as

F is a rnap acting on the set of probability density functions-functions which are (i) non-negative

and (ii) have unit area. Thus, a further way to view the problem is as a search for fixed points of

the equation

Iterating this map should lead to a correct solution for P ( E ) , and this was the approach we

In the finite temperature case, the procedure of discretization was very similar to the one used

in the zero temperature case. The key differences are the inhomogeneity, the more complicated

Figure 3.4: Convergence of iterative solution procedure for P ( E ) . In a successful solution, the

integrated probability ends up very close to one, while the relative error 1 Pi+l (E) - e(E) III Pi+, (E) 1 converges to zero.

kernel, and the variability of upper and lower boundaries. These are technical corn plications, but

are not conceptually any more demanding t han those encou n tered in the zero temperature case.

In general, we found that wllen the lower and upper energy boiindaries were suitably selected,

and the number of quadrature points was chosen in the range N = L000-4000, convergence was

achieved in no more than LOO iterations. It was observed tliat higher teniperatures caiised greater

probiems with convergence, especially in the RLC circuit environment.

By monitoring I I PIIi, the value of the integral over P ( E ) , we were able to assess the validity of the integral boundaries. With a good selection of the integral boundaries, iterating the function

P ( E ) led to an unit integral value, as expected For a probability lunction. In contrast, when the

boundary approximations were too small, the integral value waa mucli less than one. In order to

assess the convergence, we also monitored I I Pl lz-the integral over 1 P ( E ) 12-as well as the relative error, erer, defined as the integral over 1 Pi+r (E) - Pi(E)I/I fi+I (QI. Unsuccessful convergence was marked by exponentially increasing oscillations in the value of the I I PI 11. On the other hand, large initial fluctuations in the value of I I PI12 and er,i did not necessarily signify poor convergence, and indeed in some cases seemed to be characteristic of good convergence. Such an example is shown

in Figure 3.4; 11 Pllz is not shown since its magnitude is much greater than the other two quantities. In general, when the iterative procedure convergeci properly, I I Pllit I I Pl l 2 and erec approached the values of one, some positive constant, and zem, respectively.

Energy (eV)

Figure 3.5: P ( E ) at finite temperature for an large ohmic resistor R in series with a single tunnel

junction of capacitance C. (parameters: R = 1.25 x L O ~ R , C = 5 x W L 7 F )

P(E) at finite temperature

In Figure 3.5, we show P(E) for a large Ohmic resistance at different ternperatures. While at very

low temperatures, the Coulorn b blockade peak is q u i te strong, when the tem perature is increased,

the tunneling electron can couple to many more environmental modes-tliat is, botli thermal fiuctu-

ations and quantum fluctuations can suppress the Coulomb blockade. This plienomenon represents

a challenge to experirnentalists, who must maintain tunrieling junctions at extremely low temper-

atures in order to observe these efrects. For example, for a jiinction with capacitance C = IO-I'F,

the temperature must be kept well beiow e2/2kC = 0.9K. In many practical cases, rnillikelvin

temperatures are often reqiiired.

In Figure 3.6, the Iow temperature behavior is very similar to that at zero temperature. As the

temperature is increased, the peaks are broadened and diminished; more interestingly, we can see

(on the logarithmic scale) the emergence of peaks at E = -fiwo, -2hwo, . . . in which the electron absorbs quanta of energy frorn the circuit environment. This is a direct illustration of the detailed

balance symmetry relation (3.23).

3.4 Notes

The issue of dissipation in quantum tunneling was studied early on by Caldeira and Leggatt [CL81,

CL831.

Important theoretical studies of the Coulomb blockade were performed by Likharev, Averin and

Figure 3.6: P ( E ) (log-scale) at finite temperature for an ohrnic resistor R and an ideal inductor

L in series with a single tunnel junction of capacitance C. (parameters: R = 103R, C = 10-'~l?,

L = ( l / ~ ) ( e / 2 1 i C ) ~ , such that hw, = Ec = e/2C)

6.0 I I

others beginning around 1984. Their predictions gave experimentalists a specific observable effect

for which to look [AL86, BJG85, Lik881.

With increasing experimental interest in single electron tunnelirig in rnesoscopic systems [FD87],

there was a significant response by theorists to account for the dissipative effects arising in tunneling

junctions as weli as other, more general systerns [GS188, Naz89b, Naz89a, GCJ+90, DEG+90]. In

the context of single electron tunneling, these new results were given a more organized bais in

s u bsequen t articles by Grabert, I ngold and Nazarov [G IDf 9 1, IN921.

Recent work in this field has attempted to determine the influence of multiple conduction

channels ("electron-wave modes") on tunneling [J EDM]; to apply sop histicated non-equilibrium

Green's function techniques to singleelectron t unneling [H097]; to develop new and im proved

models of environmental effects on tunneling, especially in the low-impedance regime [WDCH98,

J E97].

4.0

- 0.1 K .-..........

7 1 K s 1

---- 10 K 2 P Y

A - 2.0 - k!!. 0 -- --_

8 @ -------*-iI

u3 O 1 - 8

#

0.0 - ,' #

#

-2.0 1 I -0.005 0.000 0.005 0.010

Energy (eV)

Chapter 4

Single electron tunneling junctions

and circuits

4.1 Introduction

In recent sections, we have focused our attention on single tunneling junctions embedded in an

electromagnetic environment, the latter being represented by macroscopic circuit parameten R, L

and C. Depending on the bias applied by the external circuit, these junctions may behave very

differently. For example, a voltage-biased junction will have an ordinary O hmic (linear) current-

voltage (1-Y) characteristic, while a curren t-biased junction will exhibit a srnall offset of e / 2 C now

known as the Coulomb blockade. This scenarios are equivalent to a voltage-biased junction with

low and high series impedance, respectively, as we showed in Chapter 3. Considerable work was

undertaken in the mid-1980s to describe these phenomena, as described in Section 3.4.

However, the system of greater interest to us, as described in Chapters 2 and 1, is a coupled

system of isolated nanostructures ordered in a regular lattice; such systems could exhibit self-

organized charge dynamics which would be of use in new computational systems. A starting point

in the study of these complex systerns is a pair of tunneling junctions in series, representing the

partial isolation of a nanastructure from rnacroscopic contacts by thin oxide barriers.

4.2 Nanostruct mes and t unneling junctions

The double junction

A system of two junctions in series can exhibit Coulomb blockade efFects in addition to other

interesting phenornena. In fact, one junction can behave as a high impedance environment for

its neighbor, permitting the Coulomb blockade effect to be observed rnuch more easily than in

the single junction case. Additionally, for very low capacitance tunnei junctions, the elect rostatic

energy cost associated with a single electron charging the capacitance of the "island" between

the junctions may exceed the energy available to the electron through work done by the external

source. In such a case, only a very small, discrete number of electrons rnay reside on the island. In

conjunction with an asymmetry in the effective junction resistances which tends to retain electrons

on the junction, this effect leads to plateaus in the 1-V characteristic-the "Coulomb staircasel'.

Each plateau corresponds to a particular (discrete) equilibriurn numbcr of electrons residing on the

island, where the specific number of electrons permitted is governed by the external bias and by the

junction parameters. In such a situation, transport proceeds by alternation of two favored tunneling

events-e.g. on via junction 1, off via junction 2, and so forth, with occasional unexpected events

due to the stochastic dynarnics of the systern.

In usual operation, an electron tunnels through the first barrier, resides on the island for a time,

and then tunnels through the second barrier, in the direction corresponding to the applied external

bias. Since the process is stochastic, it is possible that electrons rnay tunnel in the reverse direction,

although the probability is reduced since the exterrial bias opposes this type of motion. In some

situations, however, the reverse process is favored. For example, when the electron population

on the island is high, the tunneling rate for an electron moving onto the island may actually be

considerably less than the tunneling rate for an electron moving ofl the island ont0 the source

eiectrode, and both rates shoiild be less than the tunneling rate off the island ont0 the drain

elect rode.

Another less obvious, but very important, difference between the double and single junctions

arises in the analysis and simulation of these two different classes of systems. The current Aow

through a single junction rnay be computed by taking the difference between the forward and

backward tunneling rates4.e. by substituting the desired voltage bias V into (3.20). The double

junction is not so straightforward. Since for sufficiently srnall total island capacitance the number

of eiectrons on the island has a strong influence on the tunneling rates through the junctions, the

approach used for the single junction does not work. That is, by permitting an electron to tunnel

onto or off the island, a tunneling junction changes its own voltage bias, as well as that of its

neighbor. Thus, a double junction can be described as a stochastic, discrete-event Markov system.

A Monte-Carlo approach is an effective way to deal with this type of system, and we will describe

out efforts in this direction in Section 4.3.

Why is the double tunneling junction of such interest in the current work? The main reason is

that such nanostructured systems have been fabricated using the AAO template synthesis procedure

outlined in Chapter 2, and it is important to be able to explain the dynarnics of electrons in

these structures [AMEM+]. Additionally, wbile the techniques ou tlined in Section 3.2 have been

frequently applied to single junctions, t here has been relatively little attention paid to double

junctions. Finally, the case where the junctions are widely separated so that sorne component

of the electromagnetic environment resides between the two junctions-as in the case of a long

nanowire-has been little studied.

A number of related cases will be considered in the following sections-a single nanostructure,

a coupled pair of nanostructures, and a linear array of coupled nanostructures.

Single Nanostructure

The simplest case is that of a single, isolated nanostructure or particle located between two macro-

scopic electrodes. The transport of electrons on to and off' the nanostructtire csn be characterized

by the macroscopic resistance and capacitance of the two tunnel junctions: & and Ci ( i = 1,2).

It is elementary to solve for the potential difference over the two different tunneling junctions, or

equivalently, for the electrostatic potential of the island, in terms of the applied bias, the capaci-

tances, and the number of electrons on the island. A schematic of the singie island (doiible junction)

arrangement is shown in Figure 4.1.

For a single nanostructure, the governing equations 17

are PIBJ JS88]

where Ci is the capacitance of junction i, V is the externally applied bias, n is the number of

positive elementary charges residing on the island, and e is the elementary charge. The circuit

diagrarn is shown in Figure 4.2.

junction

junction

R I , Cl

island

R2, C2

source

Figure 4.1: Double junction system showing a single island coupled to source and drain electrodev

by tunneling junctions.

P

Figure 4.2: Equivalent circuit diagrarn of double junction.

drain

/ Junction 1 Energy Change 1

Table 4.1: Energy changes for forward electron tunneling events occuring in a nanowire pair.

In this mode1 system, with applied bias electrons can "fiIl up" the nanoparticle until the forward

bias across the source junction is sufficiently reduced, at whicii point a dynamic equilibrium is

reached. This assumes that the tunneling resistance across the second junction is greater than that

of the first ( R z > Ri), so that charge does not leak out before the island is fully charged.

Paired nanostructures

The problern of electron dynarnics and transport in a coiipled pair ol nanostructures (Figure 4.3)

is conceptually and analytically more corn plex than t hat of a single nûnost ructu re. The advantage

of considering this problern is thst it is the simplest case in which charged islands intersct. The

dependence of the island potentials, and thus the tunneling rates, on the junction parameters can

be analytically solved to provide physical insight for the niore general problem of arbitrarily-sized

arrays. In Table 4.1 we show the energy changes associated with forward tiinneling in a nanowire

pair; reverse tunneling energies are obtained by substituting (ni, n2, V) -t (-ni, -n2, -V).

The quantities are defined as follows: the cou pling capacitance is Co, the pairwise product

of capacitances is Z = CoCL + CICz + C&, Cj = Cji + Cjz, the net source capacitance is Cs, = CI* + CZLt and the net drain capacitance is Cdr = Ct2 + CZ2.

In the case of two coupled nanostructures, transport through each island is influenced not only

by the electron population of the island, but also by that of its neighbor. For example, depending

on the junction parameters, the presence of several excess electrons on one island may hinder

transport through the other island. This case was investigated in detail by Tager et al. to study

the possibility of spontaneous dipole formation in parallel nanowires [TX97]. This work revealed

that unequal drain junction resistances led to a minimum energy "lateral dipole state" in the

presence of external bias, in wirhich electrons would preferentially leave the island with the Iower

drain resistance.

Figure 4.3: Equivalen t circuit diagram of paired nanost ruct ure ( parallel, cou pled dou ble junctions) .

y elastic tunneling - - - -

28-07 inelastic tunneling

le-07 - _-.--- - * d m

* d a -

* *

Oe+oO 0.0 0.5 1 .O 1.5

Voltage (V)

Figure 4.4: 1-V characteristics of coupled nanostriicture pair for elastic and inelastic tunneling.

In some of our most recent work, we have attempted to apply the results described in Chapter 3

to a nanostructure pair such as that investigated in [TX97]. In particular, we have sought to extend

that work in order to account for the influence of the electrornagnetic environment in a nanowire

pair. Indeed, in a coupled array of nanostructures, the influence of inelastic coupling of the electron

energy into the environmental modes should be significant, and might even dominate the electron

dynamics in some cases.

In Figure 4.4 we see the influence of the electrornagnetic environment, represented by a trans-

mission Iine model. While the elastic tunneling exhibits a Coulomb staircase as would be expected

from a pair of iow capadtance islands, the inclusion of inelastic effects changes the 1-V characteristic

significantly. Work is currently in progress to determine wby this occurs.

Latt ice of Nanostruct ures

The general situation for coupled nanostructures in one dimension is a linear lattice of N islands

coupled to their nearest neighbors on either side, and to the top and bottom (source and drain)

electrodes (Figure 4.7). For metal islands it is adequate to consider only nearest neighbors, since

the large number of conduction electrons perrnits very effective screening, but the validity of this

approximation may break down for semiconductor islands [WW096].

It is possible to solve analytically for the potentials of the islands, given an arbitrary distribution

of charges in the array, but the result is very complicated For large N. Tlius, numerical approaches

are more appropriate for investigating these systems.

4.3 Electron t unneling in Monte-Carlo simulation

A stochastic Monte-Carlo approach is perhaps the most direct and straightforward rnethod of

sirnulating electron transport in large scale coupled nanostructure lattices? and is also very useful

for the simpler cases described above. We will first describe the aigorithm used in our sirnulations,

and explain some end results of our numerical experimcnts on large arrays ( N - 100). Following this, we will go into more detail in describing the precise formulation of the energy change seen by

the tunneling electron, which are the key calculations underlying the success of the Monte-Carlo

scheme,

The Monte-Carlo algorit hm used for simulating single electron tunneling in nanost ructures has the

following steps.

a Compute the total electrostatic energy associatecl with the current state (initial), and the

state after each possible tunneling event (final).

a Convert these energy changes to transition rates via (3.20), the Fermi golden rule formulation

of tunneling.

0 Evaluate the probability of each possible event i occuring in a time step At, based on the

rate r i : P;: = PiAt. ( O < ft K 1)

a Test each event to find out whether it occurs in a particular tirne interval.

4.10 I t 1 1 1 1 0.0 20.0 40.0 60.0 80.0 100.0

Position index

Figure 4.5: Energy cost associated with electron tunneling into different sites on an array of N = 100

coupled nanostructures, for Cg = O, L O - ~ ~ , 10-17, ~ 0 - l ~ ~ . Smoother changes in space are associated

with stronger coupling between Iattice sites. (parameters: Ri = ~ 0 ~ 0 , Rz = 106R and Cl = C2 =

Update the

There are a n

current state and repeat.

umber of assurnptions associat ed with this recipe. It is assumed th

sufficiently smail to respond instantly to the presence of a new charge distribution:

at the array is

in psrticular,

the new electrostatic equilibrium must be able to be propagate to the most distant parts of the

array before the next time step occurs. On the other hand, for reasonable results we must also

assume, and indeed ensure, that the tirne step is small enough to encompass the tirne scales of

the fastest tu nneling processes being considered. Additionùlly, we will return to the assumption of

elastic tunneling, such that al1 energy associated with the potential energy difference between the

Fermi surfaces of the electrode and the nanoparticle is available to the electron in the tunneling

process.

Figure 4.5 depicts the energy cost associated with tunneling onto the different lattice sites of a

coupled array. The curves describing the energy cost are random since they were generated as a

snapshot after 104 Monte-Carlo time steps. From the figure it is clear that a lattice with stronger

coupling between sites leads to a smoother effective energy cost with lower variance. Such a lattice

averages the contributions of many different nearby charge sites when determining whether or not

an electron may tunnel ont0 a particular location. In contrast, the sites of a lattice with weaker

coupling possess more autonomy, since there is l e s influence From charge on nearby sites on the

Figure 4.6

4.00 - - O aF -*---,- 1 aF ---- 10 aF

3.00 - --- 100 aF Y

C C 0) t , 2.00 -

1.00 -

0.00 0.000 0.010 0.020 0.030 0.040

Voltage Bias (V)

nte-Carlo simulation of 1-V characteristics of coupled nanostructure a r r y for coi

pling capacitances Cg = 0, 10-18, IO-", IO-16F. The Coiilomb charging steps are most pronounced

for zero coupling, and disappear in the case ofstrongest coupling, where Cg > CI, fi. (parameters: RI = 108R, R2 = 10% and Cl = C2 = 10-"F)

electron tunneling ont0 a given lattice point. Thus, one sees greater relative swings in the energy

cost of tunneling over a few lattice sites in the case of weakly coupled lattices.

In Figure 4.6 we see an externally observable result ol the spatial averaging seen in Figure 4.5.

For an uncoupled or weakly coupled lattice, the single electron effect known as the Coulomb staircase

is evident since each lattice site operates independently, blocking transport of electrons until the

maximum permitted number of electrons is one or greater (Le. until the energy cost of placing the

fint electron on the island is favorable). This assumes that R2 ) Ri, so that each island is usually

fully charged and transport is blocked until an electron tunnels forward off the island through the

second junction.

For an increasingly coupled lattice, the staircase is gradually washed out: an electron is exposed

to oot only the (low capacitance) single lattice site onta whicli it is tunneling, but also to its

neighbors, reducing the electrostatic energy penalty. This may also be viewed as an increase in the

effective capacitance of the aggregate structure into which the electron is tunneling; this increased

effective capacitance dirninishes the signature of single-electron effects. In principle such effects

could be observed experimeotally if 1-V curves could be obtained for several different arrays of

nanostructures with different lattice constants, and hence different coupling capacitances.

Figure 4.7: Schematic of mode1 system of coupled nanowire array

4.3.1 Change in the electron energy due to tunneling

We wish to descri be the tirne dependence of the charge population ni in an array of nanostructures

or islands. The islands are coupled to top and bottom electrodes via tunneling junctions, and to

each other via effective capacitances, as shown in Figure 4.7. In the usual perturbation-theoretic

formulation of tunneling through thin oxide junctions, the elastic tunnel rate is given by rij = E i j / e R , in which Eij is the energy change of the tunneiing electron, e is tlie electron charge

and Rij is the effective tunneling resistance. Rij shocild be sufficiently Iiigh so that first-order

perturbation theory is adequate to describe the tunneling pmcesses, and cotunneling does not play

a major part.

In order to describe the evolution of ni, we first recognize that the tiinneling rate r i j describes the rate of change of ni in tirne. To obtain a differentid equation describing the beïiavior of ni, we

therefore need to formulate the right-hand side of the equation in terrns of ni, if possible.

Derivation of the elect ron energy change expression

In the (physically realizable) case of discrete nanostructures, the expression for the energy change

seen by a tunneling electron is

1 Eij = - - C ((Q~ + AQki) (vk + AVk) - QkVk) 2 (4.3)

k l

In this expression, the first term represents the work done by the voltage source, and would be

equal to the energy change of the electron in the absence of any Coulomb charging effects. The sum

over k represents the correction to the tunneling electron's energy due to the electrostatic charging

of the nanostructure array. This accounts for the changes in potential both at the site ont0 which

the electron tunnels, and elsewhere. In this expression, i indexes the nanostructure and j = 1,2

indexes the junction. Vk and Qk are the pret unneling elect rostatic potential and charge associated

with island k, and A h and Agki are the changes in potential and charge of island k resulting from

the tunnaling of an electron onto island i. This expression can be simplified using a combination

of approac hes.

Writing out the sum in full, we obtain

First, we note that

meaning that hQki

a change in charge occuw only at island i, ont0 which the electron tunnels,

= -ebi, and the second term can be removed from the sum. This yields

when we substitute Qk = -enk. By obtaining an expression for the spatial dependence of the

potential shift due to a charge added to one island, it is possible to derive an expression for vi

in terms of the bias voltage and the junction parameters-al1 constants-with the only variables

being the charge populations nk. This requires one to specify AV& for each k. Then we need only

know the charge popdations nk (our independent variables) in order to describe cornpletely the

energy change of the tunneling electron, and thus the tunneling rate. Althougti this formulation

is useful in the case of the discrete array, it will be essential to the development of the continuum

approximation.

The expression for the potential of an island i is denoted by

in which V o is the island potential in the absence of any excess charges. The equation describes

a superposition of the potential changes at a selected site i , due to the charges on each site k

(including k = i). The value of AVki does not depend on the absolute values of i and k unless they

are very close to boundaries, and so we can make the simplification A k = AKmk. This quantity

is also symrnetric-i.e. AK-k = Avk-i-sin~e the Coulomb interaction is symrnetric in space.

We can now write the expression for the energy change of the tunneling electron

[on , 2 1 -eV; + ;(v: - 40)

Event

on, 1

off,2

off, L

Table 4.2: Energy changes for al1 possible electron tunneling events occuring at lattice site i.

Energy Change

eV: - e zk nkA6-i + :(v: - h) e v ~ + e ~ k n k A V k - i - q ( V f + ~ o )

- e v + e x k n k A b - i - %(vp+ 40)

In this equation,

to simplify further.

still depends on the island position i, so we can use the result of Eqimtion (4.6)

This yields

To solve for E,, we need to obtain a general form for &K-k-all other quantities in the

equation are either specified constants, or else are the tirne-dependent Functions nk wliose behavior

we ultimately wisti to describe. We c m write the energy changes Eij for al1 possible tunneling

events (e.g. forward through source junction ( L ) , etc.) at a particular lattice site as shown in

Table 4.2.

Through self-consistent numerical simulation, it can be shown that the potential shift on an

island k due to a charge located at island i may be written as

in which & = lAVol = lAqil is the magnitude of the potential change a t site i due to one excess

charge at the same site, and aa is a characteristic decay length which depends on the strength of

capacitive coupling between adjacent nanostructures. This expression is most valid in the regime of

strong coupling between adjacent structures. In the opposi te c.ue of zero coupling, the exponential

fit to the full numerical solution is poor since the islands do not interact and the numerical solution

is a discrete delta function centered on the a t which tunneling takes place. In this case it is possible

to derive a much simpler analytical solution for AEij. An example of the self-consistent numerical

solution for the potential change in a coupled array, together with the exponential function fit, is

shown in Figure 4.8.

In Monte-Carlo simulation of a finite array, it is suitable to solve self-consistently once for al1

possible cases of AC

Structure Index

Figure 4.8: Self-consisten t numerical calculations and exponen tial fit of poten tial shift due to an

excess charge at lattice index O. High coupling capacitances lead to a low, broad potential shift, and

a good exponential approximation. Low coiipling capacitances Iead to a sharply peaked potential

shift and a poor exponential fit.

a means of cornputing the tunneling rates for al1 possible events, randomly selecting tunneling

events, and then updating the array, al1 within the usual M~nte-Carlo simulation framework. We

used this approach in our 1-V curve simulations for different coupling strengths (C' values), as

shown in Figure 4.6.

4.4 Continuum approximation

Why continuum?

It is cornmon for the physics of single-electron tunneling systems to be simulated using Monte-Carlo

techniques, as we have done in Section 4.3. However, in the case of a very large array, especially in

two or three dimensions, the computational time required for this approach is prohibitive due to

the great number of tunneling rates needing to be calculated at each iteration. Taking a different

approach, we can mode1 a large array using a macroscopic approximation, recognizing that we lose

t h e details of individual electron behavior on the microscopie level [BX98].

4.4.1 Derivation o f continuum model

Electroa energy change in the continuum model

We use the expression for the energy change seen by a tunneling electron in the elastic tunneling

case, using the exponential expression obtained for AK-k

If we view the array on a relatively large scale (e.g. millirneters) the summation will be taken

over - 105 nanostructures. In this case, it is reasonable to replace the scim by an integral. This gives us the following expression

In this equation, a0 now represents a physical distarice or "influence Iength" and n(r) repre-

sents a linear particle density. Since the exponential cutoR is quite rapid For physically reasonable

parameter sets (usually a* - O(lo0 - 10') islands), it is appropriate to extend the boundaries of the integral to f oo.

We expand n(ro + 6) in a Taylor series about the point PO in order to perform the integral over

Now it is possible to evaluate the integral term by term. Since dmn/drm(ro) does not depend

on 6 we bring it outside the integral, leading us to a series of integrals of the form

{ ( 2 a m even m odd

It is straightforward to show that the &en solutions are valid for positive integers m. Therefore,

the full solution of the equation For Ej(ro) is

1 Event 1 Energy Change 1

Table 4.3: Energy changes for al1 possible electros tunneling events occuring at coordinste r.

in which we sum all non-zero (even index) terms of the term-by-term integral over the Taylor

series. For physically reasonable, smooth functions, this siimmation should be convergent: the

specific requiremen t is t hat even-order derivatives of the elect ron densi ty shouid converge to zero

more rôpidly than powers of a;*. For physically possible n(r) , this requirement is acheivable, silice

this approacli cannot predict the physics of n(r) over distance scsles shorter than ao. In other

words, we assume smoothness of n(r) over this length scale, and must question any results which

change rapidly over distances of a0 or smaller.

The series expansion of the function (1 - 2) - * is given by

We use this expression to rewrite the eqiiation for E j , where the variable x is now replaced with

the differential operator D aid2/dr2

Depending on the junction participating in the tunneling (top, bottom), as well as the direction

of electron motion (forward, backward), we will have different expressions for the energy change,

as shown in the discrete case in Table 4.2. We show the corresponding energy changes for the

continuous case in Table 4.3.

Differential equat ion

We now seek to use this expression to formulate a differential equation that will describe the

behavior of charge density in the array. In general, the time dependence of the charge on an island

can be described as

assuming low temperatures, so that there is little or no backflow (reverse tunneling) oielectrons.

There are some potential problems associated with using this formulation. I n effect, we are

treating the charge as a continuous fluid, which is not really correct in view of the fact that we

have tunnel junctions isolating the islands from the contacts, giving rise to discrete charge states

in the islands. However, since we seek to study the largescale behavior of charge in the array, we

can view our approach as an ensemble average over a suitable number of individiial islands: while

each island individually contains discrete charges, the change of a single charge i n the ensemble

will not drastically perturb the average charge density in the region. Thus, we treat the charge in

the ensemble as efFectively continuous. Since our analytical approach is not intendcd to describe

the behavior of individual islands, invoking the idea of an ensemble dom not sacrifice any of the

model's most important features.

At zero temperature and in the presence of a voltage bias, electrons will move in only one direc-

tion, and we need only account for the forward tunneling rates of each junction. This was assumed

in (4.17), where only one tunneling rate was written for each junction. Using the expressions for

the rates ri and î2 we can write the equation describing the time dependence of n(r, t )

where ci = (e R , ) - ~ (V: - &~/2) - (e R$l (Vf + 4012) and c2 = 2#0ao/e( RF' + R;').

Solutions of the equation

Using periodic boundary conditions n(r) = n(r + L) for large L, we can describe the solutions of the equation. The presence of ci indicates the existence of a non-zero steady-state solution,

the behavior of which is not very interesting. Without loss of generality we therefore neglect the

existence of cl in our analysis.

Figure 4.9: Analytical solution for k-dependent decay of normalized charge density perturbation

n(k ) in an array of nanostructures. The curves are spaced in time by lO-"s beginning with

log(n(k)) = O a t t = O. (parameters: Cg = 10-16F, Ri = R2 = 1o6R, Cl = C2 = ~ o - ~ ~ F ,

a0 = 6 . 0 2 ~ (a is the lattice constant) and do = -4.93 x L O - ~ )

We substitute the solution n(r, t) = ~ k ( t ) & , for some wavevector k and time-dependent

wave amplitude Ak (t). This solution gives rise to a wavevector-dependent decay constant for

perturbations to the equilibrium charge density

This effect causes short-wavelength (high-k) perturbations to the charge density to decay somewhat

more slowly t hat those of long-wavelengt h (low-k) . Physically, this effect is consistent with expectations: long-wavelength perturbations are a p

proximately '?xnipolar", so that a given island is surrounded by charges of the sarne polarity-a

relatively unstable situation which wouid tend to increase the tunneling rates of charges off the

island; in contrast, short-wavelength perturbations are approximately "dipolai' so t hat a given is-

land is close to several oppositely charged islands, so that the tunneling rate off the island increases

relatively little even with the extra charge present. Such a prediction could be cornparecl relatively

easily to the predictions of the Monte Cario approach described earlier in this chapter.

4.5 Notes

In the literature, the type of singleelectron tu nneling circuits which are most commonly considered

tend to be single or multiple junctions driven by a circuit with some environmental impedance, as

well as linear and two dimensional arrays of junctions. Systems in which some component of the

environment is contained between two junctions, as in Figure 4.4, tend not to be studied. The

reason for this may be that it is relatively difficult to fabricate good resistors on GaAs wûfers,

the system in which most SET experiments are performed. Consequently, it is rather complicated

to fabricate in GaAs the system which arises naturally using the template synthesis procedures

outlined in Chapter 2.

Additionally, while it is possible to fabricate several tens or possibly even hundreds of coupled

SET islands in an array using elect ron beam lithograp hy, the n u rn ber of closely- pac ked islands w hich

may be produced using template synthesis exceeds this number by several orders of magnitude.

Thus, the continuum mode1 developed here would not be useful in the description of SET structures

fabricated by electron beam techniques.

In general, our modeling and simulation approach in this chapter has been giiided by