Contents - Professional · PDF fileContents List of Authors ... Kabir Mirpuri, Jerzy Szpunar, ... Zhiyong Fan University of California at Berkeley, Berkeley, CA

Contents

List of Authors . . . . . . . . . . . . . . . . . . . . . . . . xiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Part I Nanowires

1 Electrical Properties of Metallic Nanowiresfor Nanoelectronic Applications . . . . . . . . . . . . . 3Carmen M. Lilley and Qiaojian Huang1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 31.2 Electrical Resistivity of Metallic Nanowires . . . . 71.3 Failure Properties of Metallic Nanowires . . . . . 161.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Texture and Microstructure Dependence ofElectromigration Defect Nucleation in Damascene CuInterconnect Lines Studied In Situ by EBSD . . . . . 29Kabir Mirpuri, Jerzy Szpunar, and Horst Wendrock2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 292.2 Electromigration . . . . . . . . . . . . . . . . . . . . 312.3 Texture in Metals . . . . . . . . . . . . . . . . . . . . 332.4 Experimental Setup . . . . . . . . . . . . . . . . . . . 422.5 Case 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.6 Case 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.7 Case 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.8 Failure Mechanism . . . . . . . . . . . . . . . . . . . 53

3 Carbon Nanotube Interconnects in CMOSIntegrated Circuits . . . . . . . . . . . . . . . . . . . . . . 61Gael Close3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 613.2 Trends in Interconnect Scaling . . . . . . . . . . . . 633.3 Carbon Nanotube Interconnects . . . . . . . . . . . 693.4 CMOS Platform for Benchmarking Carbon

Nanotube Interconnects . . . . . . . . . . . . . . . . 793.5 On-Chip Performance Analysis of Multiwall

Carbon Nanotube Interconnects . . . . . . . . . . . 843.6 Conclusion and Outlook . . . . . . . . . . . . . . . 87

v

Copyri

ghted

Mate

rial

vi C o n t e n t s

4 Progresses and Challenges of NanowireIntegrated Circuitry . . . . . . . . . . . . . . . . . . . . . 93Zhiyong Fan, Johnny C. Ho, and Ali Javey4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 934.2 Synthesis of Single-Crystalline Nanowires . . . . 944.3 Characterization of Nanowires . . . . . . . . . . . 974.4 Nanowire Assembly . . . . . . . . . . . . . . . . . . 1054.5 Printable Nanowire Arrays for Electronics,

Optoelectronics, and Sensors . . . . . . . . . . . . . 1154.6 Conclusion and Outlook . . . . . . . . . . . . . . . 123

Part II Molecular Electronics

5 Printed Organic Electronics: From Materialsto Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . 131Manuela La Rosa, Nunzia Malagnino, Alessandro Marcellino,Donata Nicolosi, Luigi Occhipinti, Fabrizio Porro, GiovanniSicurella, Raffaele Vecchione, Luigi Fortuna, Mattia Frasca,and Elena Umana5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1315.2 Materials for Organic Electronics . . . . . . . . . . 1335.3 Stamp-Based Fabrication Processes . . . . . . . . . 1365.4 Organic Thin-Film Devices . . . . . . . . . . . . . . 1435.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 155

6 One-Dimensional Nanostructure-EnabledChemical Sensing . . . . . . . . . . . . . . . . . . . . . . 161Aihua Liu6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 1616.2 Semiconducting Metal Oxide

Nanowire-Based Sensing . . . . . . . . . . . . . . . 1626.3 Metal Oxide Nanotube-Based Sensing . . . . . . . 1766.4 Polymer-Based Nanowires or Nanotubes

for Sensing . . . . . . . . . . . . . . . . . . . . . . . . 1856.5 Metal Nanowire-Based Biosensing . . . . . . . . . 1906.6 Concluding Remarks . . . . . . . . . . . . . . . . . . 1916.7 Future Perspectives . . . . . . . . . . . . . . . . . . . 192

7 Cross-Section Fabrication and Analysis of NanoscaleDevice Structures and Complex Organic Electronics 203David W. Steuerman and Erich C. Walter7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2037.2 Device Cross-Section Fabrication and

Imaging Considerations . . . . . . . . . . . . . . . . 205

Copyri

ghted

Mate

rial

viiC o n t e n t s

7.3 Case Studies . . . . . . . . . . . . . . . . . . . . . . . 2097.4 Future Opportunities and Conclusions . . . . . . 222

8 Microfabrication and Applicationsof Nanoparticle-Doped ConductivePolymers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227Bonnie L. Gray and Ajit Khosla8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2278.2 Fill Factor and Percolation Threshold . . . . . . . 2298.3 Nanoparticle Shapes and Materials . . . . . . . . . 2318.4 Conductive Nanocomposite Polymers

for Microsystems: Preparation andMicropatterning . . . . . . . . . . . . . . . . . . . . . 235

8.5 Applications of Conductive NanocompositePolymers in Microsystems . . . . . . . . . . . . . . 248

8.6 Summary and Future Directions . . . . . . . . . . . 256

9 Single-Electron Conductivity in OrganicNanostructures for Transistors andMemories . . . . . . . . . . . . . . . . . . . . . . . . . . . 263Sandro Carrara9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 2639.2 Transistors Working at 4 K . . . . . . . . . . . . . . 2659.3 Room-Temperature Inorganic Transistors . . . . . 2689.4 Room-Temperature Organic Transistors . . . . . . 2729.5 Room-Temperature Memories with

Organic SETs . . . . . . . . . . . . . . . . . . . . . . . 2779.6 The Patents . . . . . . . . . . . . . . . . . . . . . . . . 2809.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 281

10 Recent Developments toward the Synthesisof Supramolecular BioelectronicNanostructures . . . . . . . . . . . . . . . . . . . . . . . . 289John D. Tovar, Stephen R. Diegelmann, and Brian D. Wall10.1 “Supramolecular Synthons” Used to Construct

Self-Assembled Materials . . . . . . . . . . . . . . 29010.2 One-Dimensional Supramolecular Assemblies

of Pi-Electron Materials . . . . . . . . . . . . . . . 29210.3 Electrically Conductive Polymers as

Biomaterials . . . . . . . . . . . . . . . . . . . . . . . 30110.4 Peptide-Oligothiophene Conjugates for

Bionanostructures . . . . . . . . . . . . . . . . . . . 30810.5 Concluding Remarks . . . . . . . . . . . . . . . . . 314

Copyri

ghted

Mate

rial

viii C o n t e n t s

Part III Nanodevices

11 New Developments in Nanostructured ElectrodeMaterials for Advanced Li-Ion Batteries . . . . . . . . 321Ying Wang, Chuan Cai, and Dongsheng Guan11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . 32111.2 Nanostructured Cathode Materials . . . . . . . . 32311.3 Nanostructured Anode Materials . . . . . . . . . 33811.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 348

12 Quantum-Dot Devices Based onCarbon Nanotubes . . . . . . . . . . . . . . . . . . . . . 361Ali Kashefian Naieni and Alireza Nojeh12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 36112.2 Theory of Single-Electron Devices . . . . . . . . . 36212.3 Fabrication of Quantum-Dot Devices

Based on Carbon Nanotubes . . . . . . . . . . . . 37212.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 338

13 Individual Carbon Nanotubes as ElectromechanicalActuators: Simulations and Initial Experiments . . . 395Tissaphern Mirfakhrai and John D. W. Madden13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 39513.2 Theoretical and Simulation Work . . . . . . . . . 40113.3 Experimental Studies on Actuation

in Individual Carbon Nanotubes . . . . . . . . . . 41613.4 Conclusions and Future Directions . . . . . . . . 422

14 Low-Level Electrical Measurements atthe Nanoscale . . . . . . . . . . . . . . . . . . . . . . . . . 427Jonathan Tucker14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 42714.2 Nanotechnology Testing Overview . . . . . . . . 42814.3 Low-Level Measurement Techniques

for Nanoscale Measurements . . . . . . . . . . . . 43814.4 Electronic Transport Characteristics

of Gallium Nitride Nanowire-BasedNanocircuits . . . . . . . . . . . . . . . . . . . . . . 473

14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 478

15 Nano ESD: Electrostatic Discharge in theNanoelectronic Era . . . . . . . . . . . . . . . . . . . . . 481Steven H. Voldman15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 48115.2 Photomasks . . . . . . . . . . . . . . . . . . . . . . . 482

Copyri

ghted

Mate

rial

ixC o n t e n t s

15.3 Magnetic Recording . . . . . . . . . . . . . . . . . . 48615.4 MEMs . . . . . . . . . . . . . . . . . . . . . . . . . . 48915.5 Transistors . . . . . . . . . . . . . . . . . . . . . . . . 49715.6 Silicon Nanowires . . . . . . . . . . . . . . . . . . . 50115.7 Carbon Nanotubes . . . . . . . . . . . . . . . . . . 50115.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . 502

16 Nanopackaging . . . . . . . . . . . . . . . . . . . . . . . . 509James E. Morris16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 50916.2 Nanoparticles . . . . . . . . . . . . . . . . . . . . . . 50916.3 Carbon Nanotubes . . . . . . . . . . . . . . . . . . 51316.4 Health and Environment . . . . . . . . . . . . . . . 51816.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 519

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525

Copyri

ghted

Mate

rial

List of Authors

Chuan Cai Department of Mechanical Engineering, Louisiana StateUniversity, Baton Rouge, LA (chap. 11)

Sandro Carrara EPFL—Swiss Federal Institute of Technology,Lausanne, Switzerland (chap. 9)

Gael F. Close IBM Research Laboratory, Zurich, Switzerland (chap. 3)

Stephen R. Diegelmann Johns Hopkins University, Baltimore, MD(chap. 10)

Zhiyong Fan University of California at Berkeley, Berkeley, CA(chap. 4)

Luigi Fortuna Dipartimento di Ingegneria Elettrica Elettronica e deiSistemi, Universita degli Studi di Catania, Italy (chap. 5)

Mattia Frasca Dipartimento di Ingegneria Elettrica Elettronica e deiSistemi, Universita degli Studi di Catania, Italy (chap. 5)

Bonnie L. Gray Engineering Science, Simon Fraser University,Burnaby, BC, Canada (chap. 8)

Dongsheng Guan Department of Mechanical Engineering, LouisianaState University, Baton Rouge, LA (chap. 11)

Johnny Chung Yin Ho University of California at Berkeley, Berkeley,CA (chap. 4)

Qiaojian Huang University of Illinois at Chicago, Chicago, IL(chap. 1)

Ali Javey University of California at Berkeley, Berkeley, CA (chap. 4)

Ajit Khosla Engineering Science, Simon Fraser University, Burnaby,BC, Canada (chap. 8)

Manuela La Rosa Technology R&D, ST Microelectronics, Italy(chap. 5)

Carmen M. Lilley Department of Mechanical and IndustrialEngineering, University of Illinois at Chicago, Chicago, IL (chap. 1)

xi

Copyri

ghted

Mate

rial

xii L i s t o f A u t h o r s

Aihua Liu Qingdao Institute of Bioenergy and Bioprocess Technology,Chinese Academy of Sciences, China (chap. 6)

John D. W. Madden Department of Electrical and ComputerEngineering, University of British Columbia, Vancouver, BC, Canada(chap. 13)

Nunzia Malagnino Technology R&D, ST Microelectronics, Italy(chap. 5)

Alessandro Marcellino Technology R&D, ST Microelectronics, Italy(chap. 5)

Tissaphern Mirfakhrai Department of Electrical and ComputerEngineering, University of British Columbia, Vancouver, BC, Canada(chap. 13)

Kabir Mirpuri Intel Corp., Chandler, AZ (chap. 2)

James E. Morris Department of Electrical & Computer Engineering,Portland State University, Portland, OR (chap. 16)

Ali Kashefian Naieni Department of Electrical and ComputerEngineering, University of British Columbia, Vancouver, BC, Canada(chap. 12)

Donata Nicolosi Technology R&D, ST Microelectronics, Italy(chap. 5)

Alireza Nojeh Department of Electrical and Computer Engineering,University of British Columbia, Vancouver, BC, Canada (chap. 12)

Luigi Occhipinti Technology R&D, ST Microelectronics, Italy(chap. 5)

Fabrizio Porro Technology R&D, ST Microelectronics, Italy (chap. 5)

Giovanni Sicurella Technology R&D, ST Microelectronics, Italy(chap. 5)

David W. Steuerman University of Victoria, Victoria, BC, Canada(chap. 7)

Jerzy Szpunar Department of Mining and Materials Engineering,Montreal, QC, Canada (chap. 2)

John D. Tovar Johns Hopkins University, Baltimore, MD (chap. 10)

Jonathan Tucker Keithley Instruments, Inc., Cleveland, OH (chap. 14)

Elena Umana Dipartimento di Ingegneria Elettrica Elettronica e deiSistemi, Universita degli Studi di Catania, Italy (chap. 5)

Raffaele Vecchione Technology R&D, ST Microelectronics, Italy(chap. 5)

Steven H. Voldman IEEE Fellow, Vermont (chap. 15)

Copyri

ghted

Mate

rial

xiiiL i s t o f A u t h o r s

Brian D. Wall Johns Hopkins University, Baltimore, MD (chap. 10)

Erich C. Walter 3M Display and Graphics Business Laboratory, 3MCenter, St. Paul, MN (chap. 7)

Ying Wang Department of Mechanical Engineering, Louisiana StateUniversity, Baton Rouge, LA (chap. 11)

Horst Wendrock IFW-Dresden, Dresden, Germany (chap. 2)

Copyri

ghted

Mate

rial

C H A P T E R 1Electrical Properties ofMetallic Nanowires for

Nanoelectronic Applications

Carmen M. Lilley and Qiaojian Huang

Many researchers have focused on metallic nanowire (NW) materials,such as Cu, Ag, and Au nanowires, as basic material building blocksfor nanoelectronics. This chapter was written with the intention ofproviding an overview of electrical properties of metallic nanowirematerials useful to researchers and engineers. Researchers may findthe information useful in designing nanoscale electrical systems be-fore they fabricate devices. The chapter is divided so that we beginwith an overview of basic electrical properties, such as electrical re-sistivity and thermal coefficients of resistivity. In addition, we discussparameters that influence the electrical properties at the nanoscale,such as size and surface effects. Finally, this chapter concludes witha discussion on the failure properties of metallic nanowires, such asfailure from Joule heating and electromigration, so that the reliabil-ity of material system components can be considered in the design ofnanoscale electrical systems.

1.1 IntroductionNanowires have a wide range of applications in electronic systems,for example as interconnect wires in field-effect transistors,1 reso-nators,2−5 nanomagnets,6−8 and spintronic systems.9,10 Nanowiresare also critical components used in the design of nanoelectrome-chanical systems (NEMSs), where they have applications as intercon-nects in circuits and sensors to detect chemical or biological agents.

3

Copyri

ghted

Mate

rial

4 N a n o w i r e s

A considerable challenge is the successful integration of these types ofnanotechnology into larger scale systems with multiple platforms ofintegration, so that the nanosystems can interact with the macroscaleworld. Microscale electrical systems are a logical choice as the firstplatform for integration of nanosystems, because they are the closestto the nanosystem length scale among current manufactured elec-tronic products. This makes microsystems the best candidate as aplatform for integration and a link for controlled interaction with themacroscale world. The successful integration of nanosystems into mi-croscale electronics depends on stable material properties that are reli-able for at least a 10-year life cycle (with greater than a trillion cycles ofoperation).11 However, most nanoscale systems fabricated to date areprone to material instabilities (for example, oxidation of surfaces oragglomeration of quantum dots or carbon nanofibers) that negativelyaffect their usefulness.

There has been much research in the area of modeling and char-acterization of surface properties and crystalline structure of metallicnanowires and how these properties influence their electrical proper-ties. Within the nanoscale domain, for instance, there has been con-siderable research on the size dependence of electrical resistivity. Forexample, as a nanowire’s critical dimension decreases from hundredsto tens of nanometers, their electrical resistivity will increase becauseof size and surface effects. As the critical dimensions decrease further,to dimensions smaller than 1 nm, their electron transport propertieswill exhibit a unique shift to quantized behavior that can only bemodeled by quantum mechanics. To date, however, there is limitedresearch on the long-term stability of these properties. It should benoted that the surface and structural properties of nanomaterials maylead to unexpected material failures, which are a significant obstacleto the reliability of these material systems.

Some of the most common metallic nanowire materials used byresearchers are face-centered cubic (FCC) metals such as copper (Cu),silver (Ag), and gold (Au) synthesized by bottom-up and top-downapproaches. For a monolithic nanowire, the electrical and failureproperties of the nanowire will vary with size and surface proper-ties. Therefore, this chapter gives an overview of electrical proper-ties of Cu, Ag, and Au nanowire materials with critical dimensionsof a 10- to 100-nm size scale. The influence of size and surface onthe material properties is also discussed. The aim of this chapter isto provide the reader with practical information on material prop-erty considerations of metallic nanowires when planning to designa nanoelectronic system. In addition, by providing information onfailure properties, scientists and engineers will be able to incorpo-rate reliability of material components into their design processes. Itis expected that scientists and engineers will find this information

Copyri

ghted

Mate

rial

5E l e c t r i c a l P r o p e r t i e s o f M e t a l l i c N a n o w i r e s

useful for the practical design and fabrication of nanoelectronicsystems.

1.1.1 Size and Surface Effects on Electrical Propertiesof Nanowires

Figure 1.1 illustrates the percentage of surface atoms with respectto bulk atoms for an FCC close-packed nanoparticle of Cu with di-ameters ranging from 0.25 to 50.9 nm.12 As can be seen, the ratio ofsurface atoms to bulk atoms becomes increasingly large as the parti-cles become smaller. There has been much focus on how surface in-fluences the material properties of nanoscale structures, where someof these surface effects may lead to changes in the electrical13,14 ormechanical15−18 properties of NWs. For example, apparent variationsin measured elastic modulus for NWs have been attributed to a com-bination of effects from surface stress and surface elasticity; see forexample Refs. 15, 16, and 18. In addition, size has been found to in-fluence the electrical resistivity of nanowires. For example, it is nowwell known that nanowires with dimensions below 10 nm can ex-hibit quantized resistivity behavior; and this behavior has been stud-ied extensively using molecular dynamic simulation methods. Fornanowire sizes larger than 10 nm, electron surface scattering and elec-tron grain boundary scattering have been shown to cause a nonlinearchange in the electrical resistivity. For these larger nanowire diame-ters, the well-established kinetic theory (commonly referred to as theFuchs-Sondheimer theory) has been used to model the effects fromelectron surface scattering, and Mayadas-Schatzkes theory has beenused to model the effects from electron grain boundary scattering.

00 5040302010

10

20

30

40

50

60

70

80

90

100

110

% o

f Sur

face

Ato

ms

Diameter (nm) of FCC Cu Nanoparticle

FIGURE 1.1 Graph of the percentage of surface atoms with respect to bulkatoms for FCC close-packed Cu nanoparticles.

Copyri

ghted

Mate

rial

6 N a n o w i r e s

In Section 1.2, we present a review of these theoretical models for sizeand surface effects on the electrical resistivity of a nanowire.

1.1.2 Stability of Nanomaterial Properties—SurfaceMatters

Adsorption of surface contaminants is another type of surface effectthat may cause changes in material properties at the nanoscale. Forexample, variations in tensile strength of Au NWs were attributedto the presence of carbon (C), oxygen (O), and nitrogen (N).17 Sim-ilarly, exposure to air or other sources of contaminants may affectthe measured resistance and long-term reliability of metal NWs.14,19

For example, researchers have found that the aging mechanisms forpermalloys, such as those of nickel (Ni) and iron (Fe), were a result ofoxidation and diffusion of this oxygen into the bulk NW material. Thisoxidation and the subsequent diffusion of oxygen cause the electricalresistivity of the nanowire to increase. Consequently, it was foundthat capping the nanowire surface, such as with a thin surface film ofgold (15 nm), prevented surface oxidation and thus also the increasein electrical resistance.19 However, Au has very high mobility and canreadily contaminate silicon-on-insulator (SOI) devices at the waferlevel.20 Therefore, it is unlikely that Au would be used to fabricatenanoscale electronics that would be integrated into microelectronicsin a commercial fabrication setting. As previously mentioned, thereis limited published research on the stability of nanoscale materialproperties. However, researchers should take note: surface contami-nants may affect their design of nanoscale electronic systems, and theymay find it useful to identify the surface composition of the nanoma-terials as a way to identify whether surface contaminants have anysignificant impact on their designs.

As discussed previously, the relative ratio of surface atoms to bulk(or volume) atoms in nanoscale systems increases dramatically asthe nanostructures decrease in size. This may also influence mate-rial diffusion at the surface, because there are more atoms positionedat the surface, and these surface atoms may be detached more eas-ily from their equilibrium positions. This would result in materialshaving surface diffusion at lower temperatures as compared to bulktemperatures.21 This lower diffusion temperature (or enhanced dif-fusivity) has an important impact on the stability of a nanowire. Forexample, Karabacak et al. recently published a premelting tempera-ture of 673 to 773 K for copper nanorod arrays with a diameter ofapproximately 100 nm.21 This premelting temperature range is lowerthan the bulk melting temperature and is believed to be a result of thenanowire size. Other researchers have also found that although thecurrent-to-failure density increases as wires become smaller, the meantemperature to failure may actually decrease as compared to the bulk

Copyri

ghted

Mate

rial


melting temperatures.21,22 A decrease in the temperature for materialdiffusivity therefore may affect the material lifetime where they mayfail from such effects as electromigration.

1.2 Electrical Resistivity of Metallic Nanowires

1.2.1 Electron Surface ScatteringAs previously mentioned, size effects on electrical resistivity for metal-lic nanowires have been modeled by various researchers using twowell-established theories, Fuchs-Sondheimer for electron surface scat-tering, and Mayadas-Schatzkes for grain boundary scattering. Thissection provides a brief overview of these theories, and their appli-cation to experimental results is discussed in the following section.Fuchs-Sondheimer theory is also called kinetic theory and is a well-established theory to model nonlinear changes in electrical resistivityfor thin films.23,24 Chambers applied this theory to model the elec-trical resistance of wires.25 Herein, we will present the analysis fora trapezoidal nanowire system using Chambers’s approach, becausethis more general geometry can be applied to model square, rectangu-lar, trapezoidal, and triangular nanowires (all of which can result fromvarious nanofabrication methods). The variable p is used to charac-terize the probability of elastic electron reflection at the surface wherep = 1 for a total elastic reflection and p = 0 for a purely diffuse scatter-ing and is called the electron surface scattering coefficient. For purelyspecular scattering (p = 0), the theoretical resistivity can be expressedas

ρo

ρFS(p = 0, λ) = 3

4πs

∫s

ds

2π∫0

dφ

π∫0

dθ sin θ cos2 θ

[1 − exp

(−L OP

λ

)]

(1.1)

where ρo is the bulk resistivity of the material, λ is the electron meanfree path of the material, s is the cross-sectional surface area of thewire, point O is on the surface of a cross-sectional area of the wire,and P is a point located on the wire surface. The vector OP connectspoint O to point P and has the length LOP. The vector OP′ is theprojection of the vector OP onto surface s. The angle φ is the radialangle for OP′, and θ is the azimuthal angle of vector OP. Equation(1.1) is integrated over all points on the surface s. In essence, the meanfree path for electrons is reduced due to electron surface scattering onthe wire surface which causes an increase in resistivity for the wire ascompared to the bulk material.

Copyri

ghted

Mate

rial

8 N a n o w i r e s

h

(A)

(B )

P

(111) Z

Y

O

X

A

DC

B

w

X

Y

α

θ

φs

α

φ1

φ2φ3

φ4

φ

O

FIGURE 1.2 Illustration of a wire with a trapezoidal cross-section.

The equation for the electrical resistivity of a trapezoidal wireis given in Eq. (1.2). To derive this equation from Eq. (1.1), we firstconsider a point O in the cross section and electrons passing throughit in the direction of OP, where P is on the top wire surface, as shownin Fig. 1.2. The integration of Eq. (1.1) is over all locations throughoutthe wire cross section s, with LOP being the distance from the locationof point O in the cross section to the point P on the wire surface inthe direction of the azimuthal angle θ and the radial angle φ. Fromthis calculation, the resultant electron mean free path in the wire isreduced to λ[1 − exp( −LOP

λ)] because of electron scattering on the wire

surface.25 In Fig. 1.2, the dimensions 2a , 2b, and h are the top width,bottom width, and height for the trapezoidal wire, respectively. Thebottom angle of the trapezoid wire is given by

α = arctan(

hb − a

)(1.2)

The cross-section area of the trapezoid wire is written as

s = (2a + 2b)h2

= (a + b)h (1.3)

By examining the in-plane cross section of trapezoid ABCD (the sur-face s) and the projection of OP (OP′) from the electron traveling path,

Copyri

ghted

Mate

rial


the path length LOP for an electron scattering at point P on the wiretop surface (P′ on the segment AB) can be written as

LOP AB = LOP ′

sin θ= h − y

sin θ sin φ(1.4a)

Similarly, the path length for an electron scattering at point P on thewire for P′ on segment BC can be written as

LOP BC = LOP′

sin θ= (b + x − y/tan α) sin α

sin θ sin(φ − α)(1.4b)

The path length for an electron scattering at point P on the wirefor P′ on segment CD can be written as

LOP CD = LOP′

sin θ= −y

sin θ sin φ(1.4c)

Finally, the path length for an electron scattering at point P on thewire for P′ on segment DA can be written as

LOP DA = LOP′

sin θ= (b − x − y/tan α) sin α

sin θ sin(φ + α)(1.4d)

As shown in the same figure, the integral limits for the radialangles φ are determined from the geometry as

φ1 =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

arctan(

h − ya − x

)(for x < a )

π

2+ arctan

(x − ah − y

)(for a ≤ x < b)

(1.5a)

φ2 = π

2+ arctan

(a + xh − y

)(1.5b)

φ3 = 3π

2− arctan

(b + x

y

)(1.5c)

φ4 = 3π

2+ arctan

(b − x

y

)(1.5d)

φ 5 = −π

2+ arctan

(b − x

y

)(1.5e)

Copyri

ghted

Mate

rial

10 N a n o w i r e s

The limits of integration for radial angle φ are [φ 1, φ 2], [φ 2, φ 3],[φ 3, φ 4], and [φ 5, φ 1] for the traveling paths of LOP AB, LOP BC,LOP CD, and LOP DA, respectively. Substituting Eq. (1.5) intoEq. (1.1), the solution of a trapezoid wire resistivity is

ρ0

ρFS(p = 0, λ) = 3

2π (a + b)h

∫ a

0dx

∫ h

0dy

∫ π2 +arctan( a+x

h−y )

arctan( h−ya−x )

dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[ −(h − y)λ sin θ sin φ

]}

+ 32π (a + b)h

∫ a

0dx

∫ h

0dy

∫ 3π2 −arctan( b+x

y )

π2 +arctan( a+x

h−y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[−(b + x − y/tan α) sin α

λ sin θ sin(φ − α)

]}

+ 32π (a + b)h

∫ a

0dx

∫ h

0dy

∫ 3π2 +arctan( b−x

y )

3π2 −arctan( b+x

y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[y

λ sin θ sin φ

]}

+ 32π (a + b)h

∫ a

0dx

∫ h

0dy

∫ arctan( h−ya−x )

− π2 +arctan( b−x

y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[−(b − x − y/tan α) sin α

λ sin θ sin(φ + α)

]}

+ 32π (a + b)h

∫ b

adx

∫ (b−x) tan α

0dy


h−y )

π2 +arctan( x−a

h−y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp


]}

+ 32π (a + b)h

∫ b

adx

∫ (b−x) tan α

0dy


y )

π2 +arctan( a+x

h−y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp



]}

Copyri

ghted

Mate

rial


+ 32π (a + b)h

∫ b

adx

∫ (b−x) tan α

0dy


y )

3π2 −arctan( b+x

y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[y

λ sin θ sin φ

]}

+ 32π (a + b)h

∫ b

adx

∫ (b−x) tan α

0dy

∫ π2 +arctan( x−a

h−y )


y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp



]}

(1.6)

The last four terms of Eq. (1.6) drop away when a = b for the caseof wires with a rectangular cross section (where α = 90◦). In this case,the first four terms in Eq. (1.6) can be simplified to

ρ0

ρFS(p = 0, λ) = 3

4πah

∫ a

0dx

∫ h

0dy


h−y )

arctan( h−ya−x )

dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp


]}

+ 34πah

∫ a

0dx

∫ h

0dy

∫ 3π2 −arctan( a+x

y )

π2 +arctan( a+x

h−y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[(a + x)

λ sin θ cos φ

]}

+ 34πah

∫ a

0dx

∫ h

0dy

∫ 3π2 +arctan( a−x

y )

3π2 −arctan( a+x

y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[y

λ sin θ sin φ

]}

+ 34πah

∫ a

0dx

∫ h

0dy

∫ arctan( h−ya−x )

− π2 +arctan( a−x

y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[ −(a − x)λ sin θ cos φ

]}

(1.7)

Copyri

ghted

Mate

rial


Because of symmetry of the cross section, the equation for re-sistivity of a rectangular nanowire with electron surface scatteringeffects is26,27

ρ0

ρFS(p = 0, λ) = 6

4πwh

w∫0

dx

h∫0

dy

arctan[(w−x)/y]∫− arctan(x/y)

dφ

π∫0

dθ sin θ cos 2θ

[1 − exp

( −yλ cos φ sin θ

)]

+ 64πwh

w∫0

dx

h∫0

dy

arctan[(h−y)/(w−x)]∫− arctan[y/(w−x)]

dφ

π∫0

dθ sin θ cos 2θ

[1 − exp

( − (w − x)λ cos φ sin θ

)]

(1.8)

The theoretical equation for a square nanowire can be obtained byletting w = h in the integral equation. For the case of triangular cross-sectional wires, the first four terms in Eq. (1.6) cancel because a = 0.The fifth integral term in Eq. (1.6) is also equal to zero because theintegral limits for φ are same. Therefore, the solution of the electricalresistivity for a triangular cross-section wire is

ρ0

ρFS(p = 0, λ) = 3

2πbh

∫ b

0dx

∫ (b−x) tan α

0dy


y )

π2 +arctan( a+x

h−y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp



]}

+ 32πbh

∫ b

0dx

∫ (b−x) tan α

0dy


y )

3π2 −arctan( b+x

y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp

[y

λ sin θ sin φ

]}

+ 32πbh

∫ b

0dx

∫ (b−x) tan α

0dy

∫ π2 +arctan( x−a

h−y )


y )dφ

∫ π

0dθ sin θ cos2 θ

{1 − exp



]}

(1.9)

Copyri

ghted

Mate

rial


Finally, for a circular cross-section, the equation for resistivity withelectron surface scattering effects is25

ρ0

ρFS(p = 0, λ) =

(1 − 3

2πa2

) a∫0

rdr

2π∫0

dφ

π∫0

dθ sin θ cos 2θ exp[− (a − r )

l sin θ

](1.10)

All of the foregoing equations are purely diffuse electron surfacescattering. For the case of partially diffusive scattering (p �= 0), theexact solution for the resistivity requires summing over all possiblemultiple scattering paths. The resistivity is determined with the fol-lowing series of expansions:25−28

ρ0

ρFS(p, λ) = (1 − p2)

∞∑k=1

[kpk−1 · ρ0

ρ

(p = 0,

λ

k

)](1.11)

Note that the series solution is obtained when the specularly scat-tered electrons travel equal distances between successive scatteringevents.25 For trapezoidal wires, the requirement of equal path lengthsbetween successive specular scattering events is not completely sat-isfied. Therefore, Eq. (1.11) is an approximate solution to model theresistivity for trapezoidal and rectangular wires when it is a partiallydiffuse scattering. For rectangular wires, the equation for resistivitywith surface scattering effects for p �= 0 can be approximated usingSondheimer’s approach24:

ρFS = ρ0

[1 + 3

8λ0 (1 − p)

(1t

+ 1w

)](1.12)

where t and w are the thickness and the width of a nanowire, re-spectively. The error for Eq. (1.12) is estimated to be approximately5 percent or less for larger nanowires (dimensions larger than 200 nm)as compared to the exact solution calculated with Eq. (1.6).28,29

1.2.2 Grain Boundary ScatteringMayadas-Shatzkes (MS) developed a model resistivity in grain bound-ary scattering of conduction electrons.30 The probability for the elec-trons to be reflected at the grain boundary is denoted by the reflectioncoefficient R. Because the reflected electrons reduce the electrical cur-rent, grain boundary scattering results in a higher resistivity. The effect

Copyri

ghted

Mate

rial


of grain boundary scattering on the resistivity has the form

ρMS = ρ0

1 − 3α

2+ 3α2 − 3α3 ln

(1 + 1

α

) (1.13)

where α = λod

R1−R and d is the mean grain size.

At the nanoscale, both surface and grain boundary scatteringcontribute to the electrical resistivity of interconnects. FollowingMatthiessen’s rule, the combined model for the size dependent re-sistivity of a rectangular wire can be approximated with28

ρ = ρ0

⎡⎢⎢⎣3

8λ0 (1 − p)

(1t

+ 1w

)+ 1

1 − 3α

2+ 3α2 − 3α3 ln

(1 + 1

α

)⎤⎥⎥⎦

(1.14)

These equations represent a summary of the theoretical modelscommonly used to study the size effects on the electrical resistivity ofmetallic nanowires. In the following section, we present a summary ofelectrical resistivity properties, including measured p and R values,for commonly used materials for nanowire interconnects, namely, Cu,Ag, and Au.

1.2.3 Electrical Resistivity of Cu, Ag, andAu Metallic Nanowires

Figures 1.3 and 1.4 illustrate the change in electrical resistivity forsquare Cu, Ag, and Au nanowires with dimensions ranging from10 nm to 250 nm. These graphs were calculated by calculatingEq. (1.6) (the exact solution using Chamber’s method) and Eq. (1.12)(the approximate solution using Sondheimer’s method). The materialparameters for the metals are (i) Cu: λo = 40 nm,28 ρo = 1.712 μ�-cm31; (ii) Ag: λo = 51 nm,32 ρo = 1.629 μ�-cm31; and (iii) Au: λo = 38nm,33 ρo = 2.255 μ�-cm.31 As can be seen in these graphs, the overallelectrical resistivity increases as the nanowire dimensions decrease,and the overall electrical resistivity decreases as p increases. The nu-merical results indicate that the approximate solution has an error of3 percent. As can also be seen, Cu has the lowest resistivity for thethree materials. Although Ag had the lowest bulk resistivity, becauseof its electron mean free path, it has a larger theoretical resistivity thanCu. The Au nanowires have the greatest theoretical resistivity amongthe three metals.

Many researchers have measured the overall electrical resistiv-ity of nanowires from which they have obtained electron surface

Copyri

ghted

Mate

rial


100 1000 100001

2

3

4

5

6

7

8

9R

esis

tivity

(μΩ

. cm

)

Cross-sectional area (nm2)

Copperp=0p=0.2p=0.4p=0.6p=0.8p=1.0

100 1000 100001

2

3

4

5

6

7

8

9

Res

istiv

ity (

μΩ .

cm)


Silverp=0p=0.2p=0.4p=0.6p=0.8p=1.0

100 1000 100001

2

3

4

5

6

7

8

9

Res

istiv

ity (

μΩ .

cm)


Goldp=0p=0.2p=0.4p=0.6p=0.8p=1.0

FIGURE 1.3 Wire resistivity as a function of the cross-sectional area forcopper, silver, and gold square nanowires: calculated with approximatesolution.

scattering and grain boundary reflectivity coefficients. For example,Fig. 1.5 shows the electrical resistivity measurements for rectangularCu nanowires.22 These wires were fabricated with e-beam lithographyand metal evaporation. Three curves are shown, one for electron sur-face scattering (–), grain boundary scattering (. . .), and the combinedresistivity) (—). The measured resistivity increases as the nanowiresdecrease in width. For these systems, the obtained p and R valueswere 0.50 and 0.34, respectively.

A summary of published literature values for the electrical resistiv-ity properties is presented in Table 1.1. The summary includes resultsfor experimentally measured electrical resistivity (ρ), electron surfacescattering coefficients (p), and grain boundary reflectivity values (R)of Cu, Ag, and Au nanowires synthesized by various methods andwith varying dimensions. As can be seen, the experimentally mea-sured values for p and R vary significantly for nanowires. Because

Copyri

ghted

Mate

rial


100 1000 100001.0

2.0

3.0

4.0

5.0

6.0

7.0R

esis

tivity

(μΩ

⋅ cm

)


Copperp=0p=0.2p=0.4p=0.6p=0.8p=1.0

100 1000 100001.0

2.0

3.0

4.0

5.0

6.0

7.0

Res

istiv

ity (

μΩ ⋅

cm)


Silverp=0p=0.2p=0.4p=0.6p=0.8p=1.0

100 1000 100001.0

2.0

3.0

4.0

5.0

6.0

7.0

Res

istiv

ity (

μΩ ⋅

cm)


Goldp=0p=0.2p=0.4p=0.6p=0.8p=1.0

FIGURE 1.4 Wire resistivity as a function of the cross-sectional area forcopper, silver, and gold square nanowires calculated with the FS model.The material parameters for the calculations are (i) Cu: λo = 40 nm,ρo = 1.712 μ�-cm; (ii) Ag: λo = 51 nm, ρo = 1.629 μ�-cm; and(iii) Au: λo = 38 nm, ρo = 2.255 μ�-cm.

there is wide variability in fabrication methods and dimensions of thenanowires, the variation in results suggest that it would be prudentfor researchers to measure the resistivity of a sample set of nanowiresas part of the standard process monitoring when fabricating nanoelec-tronics.

1.3 Failure Properties of Metallic Nanowires

1.3.1 Failure of Nanowires from Joule HeatingPublished research on the width dependence of the temperature coef-ficient of resistance (αR) for metallic nanowires is not widely available.

Copyri

ghted

Mate

rial


2.0100 150 200 250 300

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

Res

istiv

ity ρ

(10

–8 μ

Ω ⋅

cm)

Width w (nm)

Experimental data

Combined modelp=0.03, R=0.48

Surface scattering

Grain boundary scattering

FIGURE 1.5 The measured resistivity of Cu nanowires versus the wire width atroom temperature (closed squares) compared with the combined model (solidcurve), FS model (dashed curve), and MS model (dotted curve). (Note:l = 2.04 μm and t = 50 nm.)

This material parameter is important because nanoscale interconnectsexperience current densities much larger than microscale intercon-nects. As a result, the subsequent Joule heating can cause a significantincrease in the nanowire resistance, which results in a positive feed-back where an increase in resistance leads to an increase in heating.The result is that material diffusion can occur from the Joule heating.This material diffusion would cause local discontinuities in the uni-formity of the nanowire geometries. These geometric discontinuitiescan cause localized hot spots in the nanowire, which may also furtheraccelerate material diffusion. Therefore, an accurate determination ofαR is important to model the temperature profile in a nanowire inorder to design nanoelectronic systems.

A summary of experimentally measured values for αR is given inTable 1.2. As can be seen, the general trend for Cu and Au nanowiresis that αR decreases as the nanowire dimensions decrease. This is nottrue in the case of Ag. However, we were not able to find additionalpublished results for Ag to make any observation on size effects as thenanowires decrease in size. The reason for the apparently reduced αR

in Cu (or Au) nanowires is not well understood. Two possible explana-tions have been previously proposed: heating from the SEM electronbeam during measurements of thermal properties, and a strain effectdue to the mismatch in the coefficients of thermal expansion αTE.39 Forexample, the αTE for Cu is 17 × 10−6 K−1 at 300 K (see Ref. 31) and fora SiN substrate underneath the Cu nanowire, the αTE is 4.5 × 10−6 K−1

Copyri

ghted

Mate

rial

Resistivity(μΩ-cm) P∗ R∗∗ Dimensions Synthesis Technique Comments Ref.

Cu

5.5–6.4 0.5 0.34 w = 90–330 nm, t = 50 nm E-beam evaporation 4-point 22

2.45–4.6 0.4–0.6 0.5 w = 40–800 nm, t = 230 nm Electrochemical I = 10 μA 27, 34

2.4–4.7 0.0 — w = 43–810 nm, t = 230 nm — Autoprober 34

2.3–3.2 — 0.57–0.62 w = 95–305 nm, t = 130 nm Electroplating 4-point 35

1.9–3.1 0.0 0.12–0.21 w = 80–2,000 nm, t = 250 nm Electrochemical I < 500 μA 29

Ag1.95–3.67 0 0.3 w = 50–1,000 nm, t = 300 nm Electroplating 4-point 26

2.00–4.11 0.5 — d = 15–200 nm Electrochemical I < 100 μA 36

Au6.8–11.0 0.5 0.9 w = 20–60 nm, t = 40 nm Evaporation 4-point 37

3.7–5.4 0–0.5 0.45 d = 80–300 nm Electrochemical 2-point 38

p∗: Electron surface reflection coefficientR∗∗: Electron grain boundary reflection coefficient

TABLE 1.1 Electrical Resistivity of Cu, Ag, and Au Nanowires Measured at Room Temperature

18

Copyri

ghted

Mate

rial

TCR (× 10−3K−1) Geometries or Dimensions Synthesis Technique Experimental Method Reference

3.9, 4.1, 4.3 Bulk — — 31, 34, 42

3.82 Film of 330-nm thickness on SiN/Sisubstrate

Sputtered Four-point probe direct currenttechnique

45, 46

3.55 Film of 330-nm thickness on SiN/Si substrate E-beam evaporation

3.6 Nanowires, l = 200 μm, t = 230 nm, andw = 310–810 nm

Standardelectrochemicaldeposition

Measured using an autoprober 34, 47

Cu 3.2 Nanowires, l = 200 μm, t = 230 nm, andw = 110 nm

2.9 Nanowires, l = 200 μm, t = 230 nm, andw = 88 nm

2.5 Nanowires, l = 200 μm, t = 230 nm, andw = 44 nm

1.34–1.54 Nanowires, l = 2.04 μm, t = 50 nm,w = 90–330 nm

E-beam evaporation Four-point probe measurement inUHV condition

22

3.8 Bulk — — 42

Ag 4.3 Nanowires with diameters of 15, 30, 100,and 200 nm

Electrochemicaldeposition

Two-point measurement using alock-in amplifier

36

3.4, 3.9 Bulk — — 31, 42

3.0–3.4 Film of t = 10–20 nm Thermal evaporation DC electrical sheet resistancemeasurement by four-probe vander Pauw method

48

Au2.0 Wires of l = 10 μm, w = 1 μm, and

t = 36 nmThermal evaporation Four-point measurement 49

1.10–1.34 Nanowires, l = 2.1 μm, t = 50 nm, w = 64,95, and 145 nm

E-beam evaporation Four-point probe measurement inUHV condition

50

TABLE 1.2 Temperature Coefficient of Resistance for Cu, Ag, and Au

19

Copyri

ghted

Mate

rial


at 300 K.40 In our own research, we have measured the temperature-dependent resistivity of Cu nanowires for various wire widths in orderto calculate the corresponding temperature coefficient of resistance,αR.22 We found that heating from the electron beam for Cu nanowireswas negligible because the change in temperature as a result ofthe electron beam was much smaller than the mean temperature ofthe wire. When the thermal strain was studied, it was found that thestrain in a nanowire due to a thermal mismatch at the nanowire andsubstrate interface is the same for all samples because the nanowireswere tested at the same temperatures.41 As a result, thermal strainsare not believed to cause the width dependence of αR. It should benoted that the strains on the edges of the nanowires may also par-tially relax, which results in an inhomogenous strain distribution inthe nanowires. However, this effect would cause increasing αR valuesfor smaller wire widths,42 which is opposite to the trend observed ex-perimentally. Another cause for the reduced temperature coefficient ofresistance for smaller wires may be surface diffuse scattering.42,43 Be-cause the ratio of surface area to volume increases as the wire widthdecreases, the influence of surface diffuse scattering becomes a sig-nificant effect and would decrease the value for αR. However, theseresearch findings are not conclusive, and further research is neededto fully understand how size and surface may influence αR.

Similarly, researchers have also found that the mean temperatureat current density to failure, calculated with αR, is lower than the bulkmelting temperature of the material. Conversely, the current densityto failure increases as the cross-sectional area decreases for a nanowire.This increase in failure current density has been attributed to the largesurface-to-volume ratio, which allows improved heat transfer fromthe nanowire to the local environment.34,44 Other researchers havefound that heat transfer from the nanowire to the substrate is the causefor this increase in failure current density as size decreases. However,the average temperatures of the Cu nanowires before failure have beenfound to be lower than the melting temperature (1356 K) for the bulkmetal.31 Researchers have hypothesized that there may be two causesfor enhanced surface diffusion (i.e., lower failure temperatures). Onemay be a result of “hot spots” within the wire, that is, localized pointsof higher resistance than the surrounding materials due to impuritiesor defects. These hot spots locally have greater Joule heating and thuslocalized higher temperatures, which in turn increase the resistivity,thus forming a feedback for increased Joule heating that acceleratesdiffusion locally. A second cause for the enhanced surface diffusionmay be due to surface effects. For example, Karabacak et al. recentlypublished premelting temperature of 673 to 773 K for copper nanorodarrays with a diameter of approximately 100 nm. This premeltingtemperature range is lower than the bulk melting temperature and is

Copyri

ghted

Mate

rial


believed to be a result of the nanowire size. The smaller size of thenanowires allows a large percentage of the Cu atoms to be positionedat the surface of a nanowire, and these surface atoms are more re-sponsive to diffusion. Therefore, the surface atoms may detach morereadily from their equilibrium positions and start to diffuse at a lowertemperature.21

In summary, the nature of a decrease in αR as nanowire sizes de-crease may be due to electron surface diffuse scattering. The size ofa nanowire appears to influence the current density to failure by in-creasing the failure point as the size decreases. However, the meantemperature at failure appears to be lower than the bulk melting tem-perature and may be a result of surface effects. As with the resistivityof nanowire, there is a size dependence to αR that needs further re-search. If this material property is important to a researcher, then it isrecommended that αR be characterized for more accurate identifica-tion of material properties. It should also be noted that αR is necessaryto measure the activation energy for electromigration failure in a wire.In the following section, we present a summary of activation energiesfor Cu, Ag, and Au nanowires and how these energies are used topredict a material lifetime.

1.3.2 Electromigration of Cu, Ag, and Au NanowiresElectromigration is a major type of failure mechanism in electronicdevices that affects the reliability assessment of very large-scale inte-grated (VLSI) microelectronic devices.51 Electromigration is a materialdiffusion that results from momentum transfer from the moving elec-trons, called “electron wind,” to atoms in a wire when an electricalcurrent is present.52 The material atoms will move in the direction ofthe electron flow once activation energy is achieved. As a result, metalions will accumulate at individual grain boundaries forming hillocksand voids where atoms vacate a region at a greater rate than the ar-rival of atoms. Voids in the metallic nanowires will locally increasethe current density, and therefore increase the local Joule heating andthe electrical resistance. The higher temperature and current densitywill accelerate the atomic diffusion and eventually result in a rapidincrease in resistance before failure. Furthermore, electromigrationmay also affect the reliability of electronic devices as the hillocks mayshort-connect adjacent wires.53 Determining the activation energy forelectromigration failure is a critical concern to modeling the stabilityof electrical resistivity for a nanowire.

The failure criterion to determine mean time to failure t f , which isalso used to determine the activation energy for electromigration, istypically defined as a 10 percent54 or 20 percent55 relative increase inresistance. Black developed an empirical model to estimate the mean

Copyri

ghted

Mate

rial

Activation Diffusion Geometry or Synthesis ExperimentalEnergy (eV) Pathways Dimensions Technique Conditions Reference

Cu

2.1 Lattice

— — — 591.2 Grain boundary

0.7 Surface

1.2 Grain boundary Nanowires of l = 500 μm,w = 100 and 150 nm, andt = 200 nm

Electrochemicalplating

J = 1.5–3.7 × 1010 A-m−2,T = 300◦C

60

0.94, 1.06 Surface or grainboundary

Nanowires of l = 2.04 μm,w = 90 and 141 nm, andt = 50 nm

E-beamevaporation

Ultrahigh vacuum, J =8.5–10.9 × 1011 A-m−2,T = 107–161◦C

61

0.81 Surface w = 0.25–1.0 μm E-beamevaporation

Vacuum, J = 2.5 × 1010

A·m−2, T = 375◦C62

0.28 Surface l = 3,000 μm,t = 425 nm, w = 1 μm

Chemical vapordeposition

J = 2.7–3.3 × 1011 A-m−2,T = 300◦C

63

Ag

0.95 Grain boundary Film of t = 170 nm Chemical vapordeposition

J = 0.68 × 1010 A-m−2,T = 225◦C

64

0.3 Surface Film of t = 170 nm Chemical vapordeposition

J = 0.68 × 1010 A-m−2,T = 150–215◦C

64

Au

0.98 — Film of t = 170 nm Chemical vapordeposition

J = 0.63 × 1010 A-m−2,T = 260–380◦C

64

0.8 Grain boundary Film of t = 52.5 and500 nm

E-beamevaporation

J = 1.8–2.5 × 1010 A-m−2,T = 120–425◦C

65, 66

0.675 — l = 800 μm, t = 1.4 μm,w = 3 μm

— J = 0.2–6 × 1010 A-m−2,T = 160–300◦C

54

0.59 Surface orinterface

l = 450 μm, t = 1 μm,w = 2 μm

Electrochemicalplating

Ambient, J = 2.0 × 1010

A-m−2, T = 341–391◦C67

0.5 Surface Nanowires of l = 2.1 μm,w = 95 nm, and t = 50 nm

E-beamevaporation

Ultrahigh vacuum, J =1.16–1.37 × 1012 A-m−2,T = 98–222◦C

50

TABLE 1.3 Activation Energies for Cu, Ag, and Au

22

Copyri

ghted

Mate

rial


time to failure t f of a wire, where Black’s law is expressed as56

t f = AJ n

· exp(

Ea

K B · T

)(1.15)

Here, A is a constant based on the cross-sectional area of the wire,J is the current density, Ea is the activation energy, kB is theBoltzmann constant, and T is the mean temperature of the wire inkelvins. The current exponent, n, can range in value from 1 to 3; how-ever, it is typically set to n = 2.57 Determining the activation energy forelectromigration failure is critical for predicting the lifetime becauset f is dependent on the activation energy exponentially.

Three diffusion pathways—lattice, grain boundary, and surface/interface—are possible for a material during electromigration. Theactivation energies will vary depending on the diffusion pathways.Lattice diffusion is the slowest pathway and has the largest activa-tion energy. Grain boundary diffusion has a lower Ea value, whereassurfaces are the most rapid diffusion pathways with the lowest acti-vation energy. The activation energy for electromigration failure willdepend on the dominant diffusion pathway, which depends on suchfactors as the material, wire dimensions, crystalline structure, andsurface properties. As can be seen in Table 1.3, the activation energiesvary greatly for all three metals. In general, the measured electro-migration activation energies have been found to range between thegrain boundary and surface diffusion activation energies. Because thefabrication process will influence the crystalline or grain structure ofmetallic nanowires, there is generally a wide range of reported activa-tion energies for nanowires. For example, the reported values for Ea

of Cu wires range from 0.5 eV to 2 eV.58 Researchers may find that aconservative estimation to predict a nanowire’s lifetime with electro-migration may be approximated by using surface diffusion activationenergies when they are using metallic nanowires. However, as men-tioned previously, measured activation energies vary significantly anddirect measurement of the activation energies may be necessary fordesigning successful nanoelectronics.

1.4 SummaryWe have presented a comprehensive review of electrical propertiesof metal nanowires that scientist and engineers may find useful fortheir design needs of nanoelectronics. As can be seen, the propertiesof nanowires change significantly at the nanoscale as compared tothe bulk properties. A significant difference is that their resistivity ingeneral is larger than the bulk resistivity and increases as the sizesdecrease. The temperature coefficient of resistance at the nanoscale

Copyri

ghted

Mate

rial


is in general lower than the bulk value and is an advantage to re-duce Joule heating as the current densities in the nanowires (>108

A/m2) are generally much larger than wires at the microscale (on theorder of 106A/m2). The lower melting temperatures for current den-sity to failure imply that there may be enhanced surface diffusion atthe nanoscale that may affect the long-term stability of the nanowiresand, as a result, the reliability of the nanoscale design. Finally, a re-view of electromigration studies for metallic nanowires also indicatesthat surface diffusion may become a significant diffusion pathway fornanowires. In this case, researchers may find that a conservative es-timate of the nanowire lifetime may be made using surface diffusionactivation energies with Black’s law.

References1. J. Song, W. Dongmok, M. C. McAlpine, R. S. Friedman, W. Yue, and C. M.

Lieber, “Scalable interconnection and integration of nanowire devices withoutregistration,” Nano Letters, vol. 4, pp. 915-919, 2004.

2. S. Perisanu, P. Vincent, A. Ayari, M. Choueib, S. T. Purcell, M. Bechelany, andD. Cornu, “High Q factor for mechanical resonances of batch-fabricated SiCnanowires,” Applied Physics Letters, vol. 90, 43113, 2007.

3. Z. L. Wang, G. Rui Ping, P. Zheng Wei, and D. Zu Rong, “Nano-scale mechan-ics of nanotubes, nanowires, and nanobelts,” Advanced Engineering Materials,vol. 3, pp. 657-661, 2001.

4. A. Shik, H. E. Ruda, and I. G. Currie, “Electromechanical and electro-opticalproperties of nanowires,” Journal of Applied Physics, vol. 98, 94306, 2005.

5. A. Husain, J. Hone, H. W. C. Postma, X. M. H. Huang, T. Drake, M. Barbic,A. Scherer, and M. L. Roukes, “Nanowire-based very-high-frequency elec-tromechanical resonator,” Applied Physics Letters, vol. 83, pp. 1240-1242, 2003.

6. R. Moradian and A. Fathalian, “Ferromagnetic semiconductor single-wallcarbon nanotubes,” Nanotechnology, vol. 17, pp. 1835-1842, 2006.

7. X. Ma, Y. Cai, N. Lun, and S. Wen, “Ferromagnetic Metal Nanoclusters En-capsulated in Carbon Nanotubes,” Nanotube-based Devices, vol. 772, pp. 87-92,2003.

8. E. Borowiak-Palen, E. Mendoza, A. Bachmatiuk, M. H. Rummeli, T. Gemming,J. Nogues, V. Skumryev, R. J. Kalenczuk, T. Pichler, and S. R. P. Silva, “Ironfilled single-wall carbon nanotubes, a novel ferromagnetic medium,” ChemicalPhysics Letters, vol. 421, pp. 129-133, 2006.

9. Y.-L. Mao, X.-H. Yan, and Y. Xiao, “First-principles study of transition-metal-dope single walled carbon nanotubes,” Nanotechnology, vol. 16, pp. 3092-3096,2005.

10. M. Weissmann, G. Garcia, M. Kiwi, and R. Ramirez, “Theoretical study ofcarbon-coated iron nanowires,” Physical Review B, vol. 70, 2004.

11. Y. C. Lee, B. A. Parviz, J. A. Chiou, and S. C. Chen, “Packaging for micro-electromechanical and nanoelectromechanical systems,” IEEE Transactions onAdvanced Packaging, vol. 26, pp. 217-226, 2003.

12. C. P. Poole, Jr., and Frank Owen, Introduction to Nanotechnology, Wiley, NewYork, 2003.

13. T. Hanrath and B. A. Korgel, “Influence of surface states on electron trans-port through intrinsic Ge nanowires,” Journal of Physical Chemistry B, vol. 109,pp. 5518-5524, 2005.

14. C. M. Lilley and H. Qiaojian, “Surface contamination effects on resistance ofgold nanowires,” Applied Physics Letters, vol. 89, 203114, 2006.

Copyri

ghted

Mate

rial


15. S. Cuenot, C. Fretigny, S. Demoustier-Champagne, and B. Nysten, “Surface ten-sion effect on the mechanical properties of nanomaterials measured by atomicforce microscopy,” Physical Review B, vol. 69, 2004.

16. G. Y. Jing, H. L. Duan, X. M. Sun, Z. S. Zhang, J. Xu., Y. D. Li, J. X. Wang,and D. P. Yu, “Surface effects on elastic properties of silver nanowires: Contactatomic-force microscopy,” Physical Review B, vol. 73, 2006.

17. F. D. Novaes, Antonio J. R. da Silva, E. Z. da Silva, and A. Fazzio, “Effect ofimpurities in the large Au-Au distances in gold nanowires,” Physical ReviewLetters, vol. 90, 036101, 2003.

18. J. He and C. M. Lilley, “Surface effect on the elastic behavior of static bendingnanowires,” Nano Letters, vol. 8, pp. 1798-1802, 2008.

19. T. Last, S. Hacia, S. F. Fischer, and U. Kunze, “Aging phenomena in sin-gle nanopatterned permalloy wires,” Physica B: Condensed Matter, vol. 384,pp. 9-11, 2006.

20. E. R. Weber, “Understanding defects in semiconductors as key to advancingdevice technology,” Physica B, vol. 340-342, pp. 1-14, 2003.

21. T. Karabacak, J. S. DeLuca, P. I. Wang, G. A. Ten Eyck, D. Ye, G. C. Wang, andT. M. Lu, “Low temperature melting of copper nanorod arrays,” Journal ofApplied Physics, vol. 99, 064304, 2006.

22. Q. Huang, C. M. Lilley, M. Bode, and R. Divan, “Surface and size effects onthe electrical properties of Cu nanowires,” Journal of Applied Physics, vol. 104,023709, 2008.

23. K. Fuchs, “The conductivity of thin metallic films according to the electrontheory of metals,” Proceedings of the Cambridge Philosophical Society, vol. 34,pp. 100-108, 1938.

24. E. H. Sondheimer, “The mean free path of electrons in metals,” Advances inPhysics, vol. 1, pp. 1-42, 1952.

25. R. G. Chambers, “The conductivity of thin wires in a magnetic field,” Proceed-ings of the Royal Society of London. Series A, Mathematical and Physical Sciences,vol. 202, pp. 378-394, 1950.

26. D. Josell, C. Burkhard, Y. Li, Y. W. Cheng, R. R. Keller, C. A. Witt, D. R. Kelley,J. E. Bonevich, B. C. Baker, and T. P. Moffat, “Electrical properties of super-filled sub-micrometer silver metallizations,” Journal of Applied Physics, vol. 96,pp. 759-768, 2004.

27. W. Steinhogl, G. Schindler, G. Steinlesberger, and M. Engelhardt, “Size-dependent resistivity of metallic wires in the mesoscopic range,” PhysicalReview B, vol. 66, 075414, 2002.

28. W. Steinhoegl, G. Schindler, G. Steinlesberger, M. Traving, and M. Engelhardt,“Scaling laws for the resistivity increase of sub-100 nm interconnects,” in IEEEInternational Conference on Simulation of Semiconductor Processes and Devices,September 2003, pp. 27-30.

29. S. Maitrejean, R. Gers, T. Mourier, A. Toffoli, and G. Passemard, “Experimentalmeasurements of electron scattering parameters in Cu narrow lines,” Micro-electronic Engineering, vol. 83, pp. 2396-2401, 2006.

30. A. F. Mayadas and M. Shatzkes, “Electrical-resistivity model for polycrystallinefilms: the case of arbitrary reflection at external surfaces,” Physical Review B,vol. 1, p. 1382, 1970.

31. D. R. Lide, Handbook of Chemistry and Physics. New York: CRC, 2005-2006.32. X. Y. Qin, L. D. Zhang, G. S. Cheng, X. J. Liu, and D. Jin, “The low-temperature

resistance and its density effects of bulk nanostructured silver,” Journal ofPhysics D—Applied Physics, vol. 31, pp. 24-31, 1998.

33. W. Zhang, S. H. Brongersma, O. Richard, B. Brijs, R. Palmans, L. Froyen, and K.Maex, “Influence of the electron mean free path on the resistivity of thin metalfilms,” Microelectronic Engineering, vol. 76, p. 146, 2004.

34. G. Schindler, G. Steinlesberger, M. Engelhardt, and W. Steinhogl, “Electricalcharacterization of copper interconnects with end-of-roadmap feature sizes,”Solid-State Electronics, vol. 47, pp. 1233-1236, 2003.

Copyri

ghted

Mate

rial


35. W. Wu, S. H. Brongersma, M. V. Hove, and K. Maex, “Influence of surface andgrain-boundary scattering on the resistivity of copper in reduced dimensions,”Applied Physics Letters, vol. 84, pp. 2838-2840, 2004.

36. A. Bid, A. Bora, and A. K. Raychaudhuri, “Temperature dependence of theresistance of metallic nanowires of diameter >= 15 nm: Applicability of Bloch-Gruneisen theorem,” Physical Review B, vol. 74, p. 035426, 2006.

37. C. Durkan and M. E. Welland, “Size effects in the electrical resistivity of poly-crystalline nanowires,” Physical Review B, vol. 61, pp. 14215-14218, 2000.

38. S. Karim, W. Ensinger, T. W. Cornelius, and R. Neumann, “Investigation ofsize effects in the electrical resistivity of single electrochemically fabricatedgold nanowires,” Physica E: Low-dimensional Systems and Nanostructures, vol. 40,p. 3173, 2008.

39. E. J. Menke, M. A. Thompson, C. Xiang, L. C. Yang, and R. M. Penner, “Litho-graphically patterned nanowire electrodeposition,” Nature Materials, vol. 5,pp. 914-919, 2006.

40. A. Kwatera and A. Sawka, “Preparation of amorphous composites of siliconnitride and carbon layers on silica glass by chemical vapor deposition method,”Journal of Non-crystalline Solids, vol. 265, pp. 120-124, 2000.

41. B. S. Verma and S. K. Sharma, “Effect of thermal strains on the temperaturecoefficient of resistance,” Thin Solid Films, vol. 5, p. R44, 1970.

42. F. Warkusz, “The size effect and the temperature coefficient of resistance inthin films,” Journal of Physics D: Applied Physics, vol. 11, pp. 689-694, 1978.

43. J. B. Thompson, “Absolute thermoelectric power and temperature coefficientof resistance of thin continuous metal films,” Thin Solid Films, vol. 18, p. 77,1973.

44. C. Durkan and M. E. Welland, “Analysis of failure mechanisms in electricallystressed gold nanowires,” Ultramicroscopy, vol. 82, pp. 125-133, 2000.

45. C. Ciofi, V. Damlo, and B. Neri, “Copper interconnection lines: SARF charac-terization and lifetime test,” in Proceedings of the 7th European Symposium onReliability of Electron Devices, Failure Physics and Analysis, p. 1747, 1996.

46. P. Bruschi, C. Ciofi, V. Dattilo, A. Diligenti, A. Nannini, and B. Neri, “Coppermetallizations for integrated circuits: TEM analysis and electrical characteri-zation,” Journal of Electronic Materials, vol. 26, pp. L17-L20, 1997.

47. G. Steinlesberger, M. Engelhardt, G. Schindler, J. Kretz, W. Steinhogl, andE. Bertagnolli, “Processing technology for the investigation of sub-50 nm cop-per damascene interconnects,” Solid-State Electronics, vol. 47, pp. 1237-1241,2003.

48. L. Lecko and R. Hrach, “Electrical and optical properties of very thin goldfilms,” International Journal of Electronics, vol. 77, pp. 989-991, 1994.

49. B. Stahlmecke and G. Dumpich, “Resistance behaviour and morphologicalchanges during electromigration in gold wires,” Journal of Physics—CondensedMatter, vol. 19, p. 046210, 2007.

50. Q. Huang, C. M. Lilley, R. Divan, and M. Bode, “Electrical failure analysis ofAu nanowires,” Nanotechnology, IEEE Transactions on, vol. 7, pp. 688-692, 2008.

51. ASTM, Standard Test Method for Estimating Electromigration Median Time-To-Failure and Sigma of Integrated Circuit Metallizations [Metric], ASTM Interna-tional, pp. F1260M-96, 2003.

52. P. S. Ho and T. Kwok, “Electromigration in metals,” Reports on Progress inPhysics, vol. 52, pp. 301-348, 1989.

53. J. Lienig, “Invited Talk: Introduction to electromigration-aware physical de-sign,” in Proceedings of the 2006 International Symposium on Physical Design, SanJose, CA, 2006, pp. 39-46.

54. K. Croes, R. Dreesen, J. Manca, W. De Ceuninck, L. De Schepper, L. Tiele-mans, and P. Van der Wel, “High-resolution in-situ study of gold electromi-gration: test time reduction,” Microelectronics Reliability, vol. 41, pp. 1439-1442,2001.

Copyri

ghted

Mate

rial


55. C. Pennetta, E. Alfinito, L. Reggiani, F. Fantini, I. DeMunari, and A. Scorzoni,“Biased resistor network model for electromigration failure and related phe-nomena in metallic lines,” Physical Review B, vol. 70, p. 174305, 2004.

56. J. R. Black, “Electromigration—a brief survey and some recent results,”IEEETransactions on Electron Devices, vol. 16, p. 338, 1969.

57. J. R. Lloyd, “Black’s law revisited—nucleation and growth in electromigrationfailure,” Microelectronics Reliability, vol. 47, p. 1468, 2007.

58. C. K. Hu, R. Rosenberg, and K. Y. Lee, “Electromigration path in Cu thin-filmlines,” Applied Physics Letters, vol. 74, pp. 2945-2947, 1999.

59. J. R. Lloyd, J. Clemens, and R. Snede, “Copper metallization reliability,”Microelectronics Reliability, vol. 39, p. 1595, 1999.

60. A. Roy, R. Kumar, C. M. Tan, T. K. S. Wong, and C. H. Tung, “Electromigration indamascene copper interconnects of line width down to 100 nm,” SemiconductorScience and Technology, vol. 21, pp. 1369-1372, 2006.

61. Q. Huang, C. M. Lilley, and R. Divan, “An in situ investigation of electromi-gration in Cu nanowires,” Nanotechnology, vol. 20, p. 075706, 2008.

62. C. K. Hu, K. Y. Lee, L. Gignac, and R. Carruthers, “Electromigration in 0.25 μmwide Cu line on W,” Thin Solid Films, vol. 308-309, p. 443, 1997.

63. L. Arnaud, R. Gonella, G. Tartavel, J. Torres, C. Gounelle, Y. Gobil, andY. Morand, “Electromigration failure modes in damascene copper intercon-nects,” Microelectronics and Reliability, vol. 38, pp. 1029-1034, 1998.

64. R. E. Hummel and H. J. Geier, “Activation energy for electrotransport in thinsilver and gold films,” Thin Solid Films, vol. 25, p. 335, 1975.

65. K. L. Tai and M. Ohring, “Grain-boundary electromigration in thin films II.Tracer measurements in pure Au,” Journal of Applied Physics, vol. 48, pp. 36-45,1977.

66. B. J. Klein, “Electromigration in thin gold films,” Journal of Physics F: MetalPhysics, vol. 3, pp. 691-696, 1973.

67. S. Kilgore, C. Gaw, H. Henry, D. Hill, and D. Schroder, “Electromigration ofElectroplated Gold Interconnects,” Materials Research Society Symposium Pro-ceedings, vol. 863, p. B8.30, 2005.

Copyri

ghted

Mate

rial

Documents

Contents - Professional · PDF fileContents List of Authors ... Kabir Mirpuri, Jerzy Szpunar, ... Zhiyong Fan University of California at Berkeley, Berkeley, CA