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SOLID-STATE devices in which electrons are confined to two-dimensional planes have provided some of the most excitingscientific and technological breakthroughs of the last 50

years.From metal-oxide-silicon field effect transistors to high-mobility gallium-arsenide heterostructures, these deviceshave played a key role in the microelectronics revolution andare critical components in a wide array of products fromcomputers to compact-disk players. From a more parochialperspective, the study of electrons in two-dimensional sys-tems has also been responsible for two Nobel prizes in physics

to Klaus von Klitzing in 1985 and to Robert Laughlin,Horst Strmer and Daniel Tsui in 1998. This is testimonyto the basic as well as applied interest of such devices (seeHeiblum and Stern in further reading).

However, 1-D systems are also proving to be very exciting.For many years, studies of quasi 1-D systems, such as con-

ducting polymers, have provided a fascinating insight into thenature of electronic instabilities in one dimension. In addi-tion, 1-D devices such as electron waveguides in whichelectrons propagate through a narrow channel of material have been created.Experiments on these devices have shown,for example, that the conductance of ballistic 1-D systems

in which electrons travel the length of the channel withoutbeing scattered is quantized in units of e2/h, where eis thecharge on the electron and his the Planck constant.

These systems, however, have been limited by the fact thatthey are inherently complex and/or difficult to make. Whathas been lacking is the perfect model system for exploring one-dimensional transport a 1-D conductor that is cheap and

easy to make, can be individually manipulated and measured,and has little structural disorder. Single-wall carbon nano-tubes fit this bill remarkably well.These thin,hollow cylindersof carbon were discovered in 1993 by groups led by SumioIijima at the NEC Fundamental Research Laboratory inTsukuba, Japan, and by Donald Bethune at IBMs AlmadenResearch Center in California and were first mass-producedin 1995 by Rick Smalleys group at Rice University in Texas.Since then, this new type of 1-D conductor has been the focusof amazingly intense study. Here I will describe just a smallpart of that activity: the creation of tiny nanoelectronicdevices in which nanotubes are the active element.

As we will see, some nanotubes are semiconductors. They

can therefore be used to construct devices that are one-

dimensional analogues of metal-oxide-silicon field effecttransistors, in which the electrons move along the surface of athin two-dimensional layer. Other nanotubes, in contrast, arenearly perfect metallic conductors, and are a new laborat-ory for studying the motion of electrons in one dimension.Both semiconducting- and metallic-nanotube devices arelikely to have significant technological applications.

Electronic structure of nanotubesThe remarkable electrical properties of single-wall carbonnanotubes stem from the unusual electronic structure of

graphene the 2-D material from which they are made.

C A R B O N N A N O T U B E SNanotubes are ideal systems for studying the transport of electrons in one dimension,

and have commercial potential as nanoscale wires, transistors and sensors

Single-wallcarbon nanotubes

Paul L McEuen

1 Curling up with a nanotube

x

x

E

ky

kx

E

kx

EF

EF

E

y

x

y

metallic

semiconducting

Fermi

energy

Fermi

energy

y

(a) The lattice structure of graphene the two-dimensional material that is

rolled up to form a nanotube. The lattice is made up of a honeycomb of

carbon atoms. (b) The energy of the conducting states in graphene as a

function of the wavevector,k, of the electrons. The material does not conduct,

except along certain, special directions where cones of states exist. (c) If the

graphene is rolled up around theyaxis, the nanotube is a metal (upper figure),

but if it is rolled up around the x axis, the nanotube is a semiconductor (lower

figure). The band structure of the nanotube is then given by one-dimensional

slices through the two-dimensional band structure shown in (b). The

permitted wavevectors are quantized along the axis of the tube.

a

c

b


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P H Y S I C S W O R L D J U N E 2 0 0 032

Graphene is simply a single atomic layer of graphite, the ma-terial that makes up pencil lead. Graphene has a two-dimen-sional honeycomb structure, made up of sp2-bondedcarbon atoms (figure 1a). Its conducting properties are deter-mined by the nature of the electronic states near the Fermienergy,EF, which is the energy of the highest occupied elec-

tronic state at zero temperature.The energy of the electronicstates as a function of their wavevector, k, nearEF is shown infigure 1(b). This band structure, which is determined by theway in which electrons scatter from the atoms in the crystallattice, is quite unusual. It is not like that of a metal, which hasmany states that freely propagate through the crystal at EF.Nor is the band structure like that of a semiconductor, whichhas an energy gap with no electronic states near EF due to thebackscattering of electrons from the lattice.

The band structure of graphene is instead somewhere inbetween these extremes. In most directions, electrons movingat the Fermi energy are backscattered by atoms in the lattice,which gives the material an energy band gap like that of a

semiconductor. However, in other directions, the electronsthat scatter from different atoms in the lattice interfere de-structively, which suppresses the backscattering and leads tometallic behaviour. This suppression only happens in theydirection and in other directions that are 60o, 120o, 180o and240o fromy (figure 1b). Graphene is therefore called a semi-metal, since it is metallic in these special directions and semi-conducting in the others.

Looking more closely at figure 1(b), the band structure ofthe low-energy states appear to be a series of cones. At lowenergies,graphene resembles a two-dimensional world popu-lated by massless fermions.

To make a 1-D conductor from this 2-D world,we follow the

lead of string theorists and curl up one of the extra dimensions

to form a tube (figure 1c). The resultingperiodic boundary conditions on thewavefunction quantizes kn, the compo-nent ofkperpendicular to the axis of thetube: in the simplest case, kn= 2n/C,whereCis the circumference of the tubeand n is an integer. The component ofkalong the length of the tube, meanwhile,remains a continuous variable.

If the tube axis is chosen to point in they direction, the energy as a function ofk(i.e. the band structure) is a slice throughthe centre of the cone. The tube thenacts as a 1-D metal with a Fermi velocitythat is similar to most metals.However, ifthe tube axis points in different direc-tions, such as along the xaxis, then theband structure has a different conicsection. This typically results in a semi-conducting 1-D band structure, with anenergy gap between the filled hole statesand the empty electron states.

The bottom line is that a nanotube canbe either a metal or a semiconductor,depending on how the tube is rolled up.This remarkable theoretical predictionhas been verified using a number ofmeasurement techniques. Perhaps themost direct was carried out by Cees

Dekkers group at the Delft University of Technology in theNetherlands and by Charles Liebers group at HarvardUniversity in the US. The Delft and Harvard researchers usedscanning tunnelling microscopy to determine the atomicstructure of a particular tube out of the many types of tubethat are produced when a sample is grown before probing its

electronic properties with the microscope. Their measure-ments confirmed the relationship between the structure of ananotube and its electronic properties as outlined above.

Nanotubes: how they conductBefore we can measure the conducting properties of a nano-tube, we have to wire up the tube by attaching metallic elec-trodes to it. The electrodes,which can be connected to eithera single tube or a bundleof up to several hundred tubes, areusually made using electron-beam lithography. The tubes canbe attached to the electrodes in a number of different ways.One way is to make the electrodes and then drop the tubesonto them (figure 2a). Another is to deposit the tubes on a sub-

strate, locate them with a scanning electron microscope oratomic force microscope, and then attach leads to the tubesusing lithography (figure 2b). More advanced techniques arealso being developed to make device fabrication more repro-ducible and controlled.These include the possibility of grow-ing the tubes between electrodes (see the article by Dai onpage 43), or by attaching the tubes to the surface in a control-lable fashion using either electrostatic or chemical forces.

The source and drain electrodes so named in anal-ogy to standard semiconducting devices allow the conduct-ing properties of the nanotube to be measured.In addition,athird terminal called a gate is often used (figure 2c).Thegate and the tube act like the two plates of a capacitor, which

means that the gate can be used to electrostatically induce

2 Measuring the conductance of nanotubes

To find out how nanotubes conduct electricity, wehave to attach electrodes to them. Electron-beamlithography is normally used to fabricate theelectrodes, which are then connected to either a

single tube or a bundle of tubes. (a) An atomic forcemicroscope (AFM) image of a single nanotube

device made by researchers at the Delft Institute ofTechnology. The nanotube is the tiny red line runningfrom the bottom centre to the top left. (b) An AFMimage of a nanotube cross, made in my lab at the

University of California at Berkeley. The two nanotubes are the green lines that join the electrodes. In bothcases, the devices are fabricated on an conducting substrate covered with an insulating oxide layer. (c) Thesubstrate acts as a gate to allow the charge density of the nanotube to be varied.

source

drain

gate

C A R B O N N A N O T U B E S

a b

c

1 m

1 m


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carriers onto the tube. A negative bias on the gate inducespositive charges onto the tube, and a positive bias inducesnegative charges.

When the conductance of the tube is measured as a func-tion of the gate voltage (and hence as a function of the chargeper unit length of the tube), two types of behaviour are ob-served, corresponding to metal and semiconducting tubes.Individual metallic single-walled nanotubes were first studiedin 1997 by Dekkers group at Delft and by the authors groupat the University of California at Berkeley, both in conjunc-tion with Smalleys group at Rice.Semiconducting behaviourwas then reported by the Delft group in 1998.

Since then, many groups have made and measured theproperties of similar devices. Indeed,most major universitiesand industrial laboratories, such as IBM, now have at leastone group studying these materials for a variety of electronicapplications. Although the data presented in this article aretaken entirely from the Berkeley group led by Alex Zettl,Steven Louie,Marvin Cohen and me, they should be viewedas representative of the field. In most cases, similar resultshave also been obtained by other researchers.

Nanotube transistorsSemiconducting nanotubes can work as transistors.The tubecan be turned on i.e. made to conduct by applying anegative bias to the gate,and turned off with a positive bias(figure 3a ). A negative bias induces holes on the tube andmakes it conduct. Positive biases, on the other hand, depletethe holes and decrease the conductance. Indeed, the resist-ance of the off state can be more than a million times greaterthan the on state. This behaviour is analogous to that of ap-type metal-oxide-silicon field effect transistor (MOSFET),except that the nanotube replaces silicon as the material thathosts the charge carriers.

But why is the tube p-type? After all, one might expect an

isolated semiconducting nanotube to be an intrinsic semi-conductor in other words, the only excess electrons wouldbe those created by thermal fluctuations alone. However, it isnow believed that the metal electrodes as well as chemicalspecies adsorbed on the tube dope the tube to be p-type.In other words, they remove electrons from the tube, leavingthe remaining mobile holes responsible for conduction. In-deed, recent experiments by Hongjie Dais group at StanfordUniversity and by the group at Berkeley show that changing atubes chemical environment can change the level of doping,significantly changing the voltage at which the device turnson. More dramatically, tubes can even be doped n-type byexposing the tube to elements such as potassium that donate

electrons to the tube.The semiconducting device of the type shown in figure 3 is,

in many ways, truly remarkable. First, it is only one nano-metre wide.While much work has been done to create ultra-small semiconducting devices from bulk semiconductors,such devices have always been plagued by surface states electronic states that arise when a three-dimensional crystalis interrupted by a surface. These surface states generallydegrade the operating properties of the device, and control-ling them is one of the key technological challenges to deviceminiaturization. Nanotubes solve the surface-state problemin an elegant fashion. First, they are inherently two-dimen-sional materials, so the problem of a 3-D lattice meeting a

surface does not exist. Second, they avoid the problem of

edges because a cylinder has no edges!Looking more closely at the conductance of semiconduct-

ing nanotubes, we see that initially it rises linearly as the gatevoltage is reduced, conducting better as more and more holesare added from the electrode to the nanotube. The conduct-ance is limited only by any barriers that the holes see as theytraverse the tube. These barriers may be caused by structuraldefects in the tube, by atoms adsorbed on the tube, or bylocalized charges near the tube. The holes therefore see a

series of peaks and valleys in the potential landscape, throughwhich they must hop if the tube is to conduct (figure 3b).Theresistance of the tube will be dominated by the highest bar-riers in the tube.

Recent experiments by the Berkeley and Delft groups con-firm this simple picture. The researchers used the tip of ascanning probe microscope to identify the major scatteringsites, thus enabling a map of the barriers to conduction to beproduced (figure 3c).

At lower gate voltages, the conductance eventually stopsincreasing and becomes constant, because the contact resist-ance between the metallic electrodes and the tube can bequite high. Unfortunately, this contact resistance can vary by

several orders of magnitude between devices, probably due

3 Nanotubes as transistors


0.001

0.0000 5

gate voltage (V)

con

ductance(e2/h)

sourcesemiconducting

single-walled nanotube

drain

hole energy

e

(a) The conductance of a semiconducting carbon nanotube as a function of

gate voltage. The tube can be turned on by applying a negative voltage, and

turned off with a positive voltage. The device turns on at negative voltages

because holes are added to the tube. (b) The potential profile seen by these

holes due to disorder in the structure of the nanotube and imperfect contacts

between the electrodes and the tube. The holes must hop through the

barriers in this profile if the nanotube is to conduct. (c) The tip of a scanning

probe microscope can be used to map the barriers to conduction. The

horizontal line indicates the location of the nanotube and the vertical lines

indicate the contact boundaries. The conductance of the tube is measured as

the positively biased tip is scanned over the sample. The bright spots arewhere the tip decreased the conductance, with greater intensity

corresponding to a greater change in the conductance.

a

b

0.7

5m

c


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P H Y S I C S W O R L D J U N E 2 0 0 034

to mundane issues such as surface cleanliness. To improve theconsistency of nanotube transistors,many groups are there-fore trying to improve the quality of these contacts by devel-oping new cleaning and annealing procedures with somesignificant success.

These tiny MOSFET-like devices will probably just be thefirst in a host of new semiconducting-device structures basedon carbon nanotubes. Other devices, such as nanotube pn-

junction diodes and bipolar transistors, have been discussedtheoretically and are likely to be realized soon.

Nanotubes as one-dimensional metals

In dramatic contrast to semiconducting nanotubes, the con-ductance of some other nanotubes near room temperature isnot noticeably affected by the addition of a few carriers. Thisbehaviour is typical of metals,which have a large number ofcarriers and have conducting properties that are not signifi-cantly affected by the addition of a few more carriers. Theconductance of these metallic nanotubes is also much largerthan the semiconducting-nanotube devices, as expected. In-deed, a number of groups have made tubes with conduct-ances that are between 25% and 50% of the value of4e2/hthat has been predicted for perfectly conducting ballistic

nanotubes. This result indicates that electrons can travel fordistances of several microns down a tube before they are scat-tered. Several measurements support this conclusion, inclu-ding those carried out by our group using scanning probemicroscopy. These measurements also show that the contactresistance between the tube and the electrodes can be sub-stantial, just as it is with semiconducting tubes.

Further evidence for the near-perfect nature of these tubescomes from the way they behave at low temperatures. Theconductance is observed to oscillate as a function of gate volt-age (figure 4). These Coulomb oscillationsoccur each timean additional electron is added to the nanotube.In essence,thetube acts like a long box for electrons, often called a quantum

dot (see Kouwenhoven and Marcus in further reading). The

electronic and magnetic properties of these nanotube quan-tum dots reveal a great deal about the behaviour of electrons innanotubes.For example,the fact that the oscillations are quiteregular and periodic indicates that the electronic states areextended along the entire length of the tube. If,however, therewas significant scattering in the tube, the states would become

localized and the Coulomb oscillations would be less regular.Nanotube quantum dots that are as long as 10 m have beenfound to exhibit these well-ordered oscillations, again indica-ting that the mean free path can be very long.

The experiments described above indicate that electronscan travel for long distances in nanotubes without beingbackscattered. This is in striking contrast to the behaviourobserved in traditional metals like copper, in which scatteringlengths from lattice vibrations are typically only several nano-metres at room temperature. The main reason for this re-markable difference is that an electron in a 1-D system (like ananotube) can only scatter by completely reversing its direc-tion, whereas electrons in a 2-D or 3-D material can scatter by

simply changing direction through a tiny angle. Phonons long-wavelength lattice vibrations that scatter electrons inboth 2-D and 3-D materials at room temperature do nothave enough momentum to reverse the direction of a speed-ing electron in a 1-D nanotube. They therefore do not influ-ence its conductance, at least not at low voltages.

Recent experiments by Dekkers group at Delft haveshown that at high voltages (greater than 0.15 V), electronscan emit high-momentum phonons that can scatter elec-trons in 1-D nanotubes. This leads to a dramatic reductionin the conductance at high voltages, causing the current tosaturate at about 25 microamps for a single nanotube. Still,this is a remarkably macroscopic current to be carried by

such a nanoscopic system!The fact that a metallic nanotube acts like a near-perfect

1-D conductor at low voltages makes it an ideal system to testsome ideas about electrons in one dimension that have beenaround for half a century. Starting in the 1950s, a series ofpapers by Sin-Itiro Tomonaga, Joaquin Luttinger and laterDuncan Haldane made it clear that a 1-D electron systemshould behave very differently from its 2-D and 3-D counter-parts when the repulsive Coulomb interactions betweenneighbouring electrons are taken into account. Under ordin-ary conditions, a 2-D or 3-D metallic conductor behaves as aFermi liquid, even when the electrons interact with eachother via the Coulomb force. The electrons in such materials

fill the low-energy states up to the Fermi energy, creating what

4 Nanotubes as metals


0.4

0.2

0.0

0.0 0.9 1.8 2.7 3.6

285 K

129 K

40 K

16 K

0.4

0.2

0.0100 200 300

gate voltage (V)

conductance

(e2/h)

conductance

(e2/h)

temperature (K)

8 K

1.3 K

The conductance of a metallic nanotube at six different temperatures as a

function of gate voltage. At low temperatures the conductance oscillates as

individual electrons are added to the tube. This indicates that the nanotube

acts like a long and narrow quantum dot, with electronic states that extend over

the entire length of the tube. The average conductance of the tubes slowly

decreases as the temperature is lowered (see insert). The functional form is

consistent with the power-law behaviour predicted for tunnelling into aLuttinger liquid.

5 Electrons in one dimension

Fermi liquid

E

EF

Luttinger liquid

e eE

(a) An electron tunnelling from a metal electrode into a Fermi liquid leaves

other electrons in the Fermi sea relatively undisturbed. (b) An electron finds it

less easy, however, to tunnel into a Luttinger liquid, because collective

excitations in the electron liquid must be excited. Calculations show that the

Luttinger liquid has a tunnelling conductance that decreases in proportion to

(EEF)whereEis the energy of the electron, EF is the Fermi energy and is a

power. The excess energy of the tunnelling particle is provided by either an

applied temperature or voltage.

a b


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P H Y S I C S W O R L D J U N E 2 0 0 0 35

is known as a Fermi sea of electrons. The low-energy exci-tations (or quasiparticles) of this system act almost likecompletely free electrons, moving entirely independently ofone another. In other words, an excited state looks very muchlike a single extra electron above the Fermi sea.

In 1-D systems, on the other hand, the low-energy excita-

tions are collective excitations of the entire electron system.The electrons move in concert, rather than as independentparticles of a Fermi liquid. This system is referred to as aTomonagaLuttinger liquid (or, more simply, a Luttingerliquid) to emphasize its difference from the standard Fermi-liquid behaviour of 2-D and 3-D metals.

One way to test this prediction is to see if an electron cantunnel into the system from the outside world for examplefrom a metallic contact. If the low-energy excitations aresimple quasiparticles, then an electron will have no difficultytunnelling into the system (figure 5a). The tunnelling conduct-ance would not be expected to change with temperature orbias voltage.If, on the other hand, the low-energy excitations

are collective in nature, the other electrons in the tube mustmove in concert with the tunnelling electron to make room forit. The electron must literally make a splash when it jumpsinto the Luttinger liquid (figure 5b). If the energy, E, of thetunnelling electron is much higher than the Fermi energy,EF,then this splash is not a problem. As the electron tunnels inwith less and less excess energy, however, it has less and lessenergy to push the other electrons out of the way.

Calculations show that the Luttinger liquid has a tunnellingconductance that decreases in proportion to (EEF)

, whereis a particular power. The value of depends on the strengthof the Coulomb interaction between the electrons. It alsodepends on whether the electron tunnels into the middle of a

tube, the end of a tube, or between the ends of two tubes.

Theorists have been able to estimatethese powers fairly accurately for nano-tubes, resulting in very specific predic-tions that experimentalists can test.

Our group at Berkeley tested thesepredictions by measuring the tunnellingconductance into a nanotube from ametallic electrode as a function of thetemperature and bias. In this case, thepoor contacts worked to our advantage,serving as the tunnel barriers betweenthe tube and the electrode. The averageconductance decreases slowly as a func-tion of temperature (figure 4). The rela-tionship is described by a power law thatagrees well with theory. The group hasalso measured the powers for electronstunnelling into the middle and ends of atube, while our colleagues at Delft havedone the same for electrons tunnellingfrom the end of one nanotube into theend of another. All of these results agree

well with the theoretical predictions.These experiments clearly demon-

strate that interacting 1-D metals be-have very differently to 2-D and 3-Dmetals. This is perhaps not so surprising

to use a traffic analogy, carcar inter-actions are much more important on a

one-lane highway than they are in a 2-D parking lot, where acar can move more-or-less independently of the other cars.What is surprising, however, is how long it took before thesepredictions were tested in detail. While previous measure-ments of other systems had shown evidence for Luttingerbehaviour, nanotubes represent perhaps the clearest and most

straightforward realization of Luttinger-liquid physics to date.

New devices and geometriesWhile the above experiments demonstrate that many of thebasic properties of single-wall carbon nanotubes are nowunderstood, there is an almost limitless number of new geo-metries and topics waiting to be explored and all manner ofnew structures to be created. Indeed, researchers are devel-oping a host of new techniques that creatively combine litho-graphy, chemistry and nanoscale manipulation, for exampleby growing tubes on prefabricated structures or by pushingthem around with the tips of atomic force microscopes. Itis quite remarkable how far the field has come since the first

measurements were made in 1997 and this progress showsno sign of slowing.

For example,new devices can be created by the intersectionof two nanotubes, such as a metallic tube crossing over asemiconducting tube (figure 6). The metallic tube locallydepletes the holes in the underlying p-type semiconductingtube. This means that an electron traversing the semicon-ducting tube must overcome the barrier created by this metaltube. Biasing one end of the semiconducting tube relative tothe metal tube leads to rectifying behaviour. In other words,the barrier is overcome in one bias direction, but not in theother. This structure is just one of many possibilities for nano-tube devices waiting to be explored.

Meanwhile,Phaedon Avouris and co-workers at IBMs T J


6 Nanotubes as rectifiers

400

200

0

500 0 500 1000

voltage (mV)

forward

bias

reverse

bias

current(nA)

reverse bias

hole energy

location of

metal tube

e|V|

forward bias

e|V|

e

location of

metal tube

(a) A metallic nanotube crossing over a semiconducting nanotube creates a rectifier. (b) In other words,

a positive voltage causes current to flow in one direction, while a negative voltage stops the current flow

altogether. The metallic tube locally depletes the electrons in the underlying semiconducting tube,

creating a barrier, the height of which is fixed by the potential applied to the metallic tube. (c) A positive

voltage applied to the semiconducting tube gives the holes the necessary energy to overcome the

potential barrier, whereas a negative bias does not (d).

a b

c d

V= 0

V

I

semiconducting tube

metal tube


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