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It is a fact of common experience that if acircular disk (for example, a penny) isspun upon a table, then ultimately itcomes to rest quite abruptly, the final stageof motion being characterized by a shudderand a whirring sound of rapidly increasingfrequency. As the disk rolls on its rim, thepoint P of rolling contact describes a circlewith angular velocity . In the classical(non-dissipative) theory 1, is constant andthe motion persists forever, in stark conflictwith observation. Here I show that viscousdissipation in the thin layer of air betweenthe disk and the table is sufficient to

account for the observed abruptness of thesettling process, during which, paradoxical-ly, increases without limit. I analyse thenature of this finite-time singularity, andshow how it must be resolved.

Let be the angle between the plane of the disk and the table. In the classicaldescription, and with the notation definedin Fig. 1, the points P and O are instanta-neously at rest in the disk, and the motion istherefore instantaneously one of rotationabout line PO with angular velocity ,say. The angular momentum of the disk istherefore h A e (t ), where A 1_4Ma

2 is the

moment of inertia of the disk of mass M about its diameter; e (t ) is a unit vectorin the direction PO ; e z , e d are unit vectorsin the directions Oz , OC , respectively (seeFig. 1). In a frame of reference rotat-ing with angular velocity d ez , the disk rotates about its axis OC with angularvelocity d d e d ; hence therolling condition is d cos . Theabsolute angular velocity of the disk is thus (ed cos ez ), and so

e sin .Eulers equation for the motion of a

rigid body is here given by d h /d t h G, where G Mga ez e is the grav-itational torque relative to P ( indicatesthe vector product ). This immediately givesthe result 2sin 4g /a , or, when issmall,

2

4g /a (1)The energy of the motion E is the sumof the kinetic energy 1_2A

2 1_2Mga sin ,

and the potential energy Mga sin , soE 3_2Mga sin 3_2Mga (2)

In the classical theory, , and E are allconstant, and the motion continues indefi-nitely. As observed above, this is utterly unrealistic.

Let us then consider one of the obviousmechanisms of energy dissipation, namely

that associated with the viscosityof the surrounding air. When

is small, the dominantcontribution to the

viscous dissipationcomes from the

layer of airbetween thedisk and thetable, which issubjected to

strong shearwhen is large.

We may estimatethe rate of dissipation

of energy in this layer asfollows. Let (r , be polar

coordinates with origin at O . Forsmall , the gap h (r, ,t ) between the

disk and the table is given by h (r, ,t ) (a r cos ), where t .We now concede that is a slowly varying

function of time t : we assume that | | ,and make the adiabatic assumption thatequation (1) continues to hold. Because theair moves a distance of order a in a time2 / , the horizontal velocity u H in the layerhas order of magnitude r sin ; and as thisvelocity satisfies the no-slip condition onz 0 and on z h ( O ( a )), the verticalshear | u H / z | is of the order (r / a )|sin |.The rate of viscous dissipation of energy is given by integrating ( u H/ z )

2 over thevolume V of the layer of air: this easily gives ga 2/ 2, using equation (1). The factthat as 0 should be noted.

The energy E now satisfies dE /dt (neglecting all other dissipation mecha-nisms). Hence, with E given by equation(2), it follows that

3_2Mga d /dt ga

2/ 2 (3)This integrates to give

3 2 (t 0 t )/ t 1 (4)where t 1 M / a , and t 0 is a constantof integration determined by the initialcondition: if 0 when t 0, thent 0 (

30/2 )t 1. What is striking here is that,

according to equation (4), does indeed goto zero at the finite time t t 0. The corre-sponding behaviour of is (t 0 t )

1/6,

which is certainly singular as t t 0.Of course, such a singularity cannot berealized in practice: nature abhors a singu-larity, and some physical effect must inter-vene to prevent its occurrence. Here it is notdifficult to identify this effect: the verticalacceleration | h | |a | cannot exceed g inmagnitude (as the normal reaction at P must remain positive). From equation (4),this implies that the above theory breaksdown at a time before t 0, where

t 0 t (2a /9g )3/5(2 /t 1)

1/5 (5)A toy, appropriately called Eulers

disk 2, is commercially available (Fig. 2; Tan-gent Toys, Sausalito, California). For thisdisk, M 400 g, and a 3.75 cm. Withthese values and with 1.78 10 4 gcm 1 s, t 1 M / a 0.8 10

6 s, and, if wetake 0 0.1( 6), we find t 0 100 s. Thisis indeed the order of magnitude (to within

20%) of the observed settling time inmany repetitions of the spinning of the disk (with quite variable and ill-controlled initialconditions), that is, there is no doubt thatdissipation associated with air friction issufficient to account for the observedbehaviour. The value of given by equation(5) is 10 2 s for the disk values given above;that is, the behaviour described by equation(4) persists until within 10 2 s of the singu-larity time t 0. At this stage, 0.5 10

2,h 0 a 0.2 mm, 500 Hz (and theadiabatic approximation is still well

brief communications

NATURE |VOL 404 |20 APRIL 2000 |www.nature.com 833

Eulers disk and its finite-time singularity Air viscosity makes the rolling speed of a disk go up as its energy goes down.

Figure 2 Eulers disk is a chrome-

plated steel disk with one edge machined to a smooth radius. If it

were not for friction and vibration, the disk would spin for ever.

Photo courtesy of Tangent Toys. See ht tp://www.tangenttoy.com/.

a

e

e z

r

c z

z= 0

P

O

e d

a cos

Figure 1 A heavy disk rolls on a horizontal table. The point of

rolling contactP moves on a circle with angular velocity . Owingto dissipative effects, the angle decreases to zero within a finitetime and increases in proportion to 1/2.

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distance above the device surface, we scanthe tip of an atomic-force microscope(AFM), on which a positive tip voltage, V t,is applied. The AFM tip acts as a local gateand induces a local potential barrier (Fig.1b) when it is above the tube. This canreduce the hole current through the nano-tube, which is recorded as a function of thetip position (Fig. 1c). The main point of SGPI is that the current reduction dependson the original local potential of the tube.In this way, SGPI maps out the local poten-

tial profile of the nanotube.The top images in Fig. 1d,e show regularAFM images of two different samples withtheir corresponding SGPI measurements atdifferent bias voltages underneath. Themost remarkable feature of these images isthat they show a sequence of current dipsalong the tubes. The dips appear to be con-fined to the region between the electrodes.Surprisingly, they are rather evenly spaced,with a distance of about 40 nm for the sam-ple in Fig. 1d, and 36 nm for the sample inFig. 1e. These observations indicate that theedge of the valence band does not vary in a

smooth monotonic way. Instead, they point

to a potential that is significantly modulat-ed, creating a sequence of barriers for thehole carriers (Fig. 1b). Similar SGPI mea-surements on metallic tubes did not show any contrast.

The effect of increased bias voltage is

illustrated in the lower panels of Fig. 1d,e.The emphasis of the dot pattern appears toshift towards the electrode with the lowerpotential. Existing dots vanish (particularly clear for very large bias; see Fig. 1e) and new dots appear (bottom right of Fig. 1d; bot-tom left of Fig. 1e) that were not present forlow bias. These trends may be due to theeffective gate voltage near the right and leftelectrodes being different at higher biasvoltages. Contrast also diminishes when thetube potential approaches the tip voltage.

The electronic properties of semicon-ducting nanotubes have been proposed to

be sensitive to perturbations by local disor-der 2,68. Our results confirm the occurrenceof such electronic disorder by direct spatialimages. The microscopic origin of this dis-order is still unclear, however. The mostlikely causes are localized charges near thenanotube, or mechanical deformations.Detailed height measurements by AFM didnot reveal any correlation between heightand electronic features.

Our results shed a new light on otherreported transport data. Step features havebeen found in currentvoltage ( I V ) curvesof TUBEFET devices (ref. 3, and S. J. T. et

al., unpublished results). Our findings may explain these observations because anincreasing bias voltage can bring down thepotential barriers one at a time, leading tostep-like features. Reported asymmetries inI V curves1,3,4 can now be corroborated by the asymmetries in the potential profilealong tubes at low bias. In conductionexperiments at low temperatures (ref. 8,and Z. Yao et al., unpublished results), phe-nomena related to multiple metallic islandshave been observed, which can be explainedby the barrier sequence seen in our SGPIimages. Near the tube on top of the elec-trodes, no contrast could be found in theSGPI images, even for large tip voltages (upto 3 V). This indicates that Schottky barri-ers do not exist at the metal interface, as wassuggested earlier 4.

New scanning techniques that give adirect view of the potential landscape, suchas the one presented here, provide apromising starting point for a better under-standing of the electronic structure of nanotube devices. It should, for example, befeasible to study the effect of deliberatebending of nanotubes, different substrateand electrode materials, and the differentgeometry of devices such as intramolecularkinked-nanotube diodes 5 and nanotubecrossings.Sander J. Tans*, Cees Dekker**Delft University of Technology, Department of

834 NATURE |VOL 404 |20 APRIL 2000 |www.nature.com

Molecular transistors

Potential modulationsalong carbon nanotubes

T rue molecular-scale transistors havebeen realized using semiconductingcarbon nanotubes 15, but no directmeasurements of the underlying electronicstructure of these have been made. Here weuse a new scanning-probe technique toinvestigate the potential profile of these

devices. Surprisingly, we find that thepotential does not vary in a smooth,monotonic way, but instead shows markedmodulations with a typical period of about40 nm. Our results have direct relevance formodelling this promising class of moleculardevices.

The principle of our scanning-gatepotential-imaging (SGPI) technique is asfollows. An individual semiconducting sin-gle-wall carbon nanotube is connected totwo metal electrodes, and this transistor isswitched to a conducting state by a negativegate voltage, V g (Fig. 1a). A current flows

when a bias voltage, V b, is applied. At a close

brief communications

Figure 1 Scanning-gate potential imaging (SGPI) along a semiconducting carbon nanotube.a, SGPI measurement set-up. Variable biasvoltages,V b, and a gate voltage,V g, of 6 V are applied to the TUBEFET device. An atomic-force microscope (AFM) tip at 500 mV isscanned at a constant height of about 10 nm above the surface by retracing each line taken in regular tapping mode AFM while setting acertain height offset and the cantilever amplitude to zero.b, Potential landscape of the device. In the conducting state, the valence bandedge is horizontal and pinned to the edge of the Fermi level of the electrode1,9. The tip voltage creates a potential dip (yellow) which pro-vides a probe for the local potential. SGPI measurements ( d,e ) show that the band edge of the nanotube is not smooth but strongly mod-ulated. c, Corresponding SGPI measurement. The device current (colour) is displayed as a function of tip position. Current is reducedwhen the AFM-tip-induced barrier aligns with minima in the original potential profile. The spatial resolution of the SGPI measurements,which we estimate to be of the order of 10 nm, is determined by the tipsample distance and the tip radius.d, AFM image of the firstsample and the corresponding SGPI images forV b values of 10, 100, 500 and 750 mV (top to bottom). The sample consists ofan individual single-wall carbon nanotube (horizontal line) on top of two 25-nm-high platinum electrodes (on the left and right) that arespaced by 650 nm. e, AFM image of a second sample and the corresponding SGPI images forV b values of 400, 500, 700 and 1,000 mV(top to bottom). The sample consists of an individual single-wall carbon nanotube on top of two 750-nm-spaced gold electrodes. Theelectrodes of this sample are embedded in the SiO2 substrate to create a flat surface

5 (see a ).

satisfied). This is as near to a singularity asthis simple toy can approach. The effect isnevertheless striking in practice.H. K. MoffattIsaac Newton Institute for Mathematical Sciences,

20 Clarkson Road, Cambridge CB3 0EH, UK e-mail: hkm2@newton.cam.ac.uk 1. Pars, L. A. Treatise on Analytical Dynamics (Heinemann,

London, 1965).2. Euler, L. Theoria Motus Corporum Solidorum Seu Rigidorum

(Greifswald, 1765).

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