Daryl C. Robinson

Glenn Research Center, Cleveland, Ohio

Nanotube Tunneling as a Consequence

of Probable Discrete Trajectories

NASA/TM—2001-211133

October 2001
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NASA/TM—2001-211133

October 2001

National Aeronautics and

Space Administration

Glenn Research Center

Prepared for the

2001 International Conference on Computational Nanoscience

sponsored by the Applied Computational Research Society

Hilton Head Island, South Carolina, March 19–21, 2001

Daryl C. Robinson

Glenn Research Center, Cleveland, Ohio

Nanotube Tunneling as a Consequence

of Probable Discrete Trajectories
Available from

NASA Center for Aerospace Information

7121 Standard Drive

Hanover, MD 21076

National Technical Information Service

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Springfield, VA 22100

This report contains preliminary

findings, subject to revision as

analysis proceeds.

Available electronically at http://gltrs.grc.nasa.gov/GLTRS
NASA/TM—2001-211133 1 

Nanotube Tunneling as a Consequence of Probable Discrete Trajectories 

Daryl C. Robinson 

National Aeronautics and Space Administration 

Glenn Research Center 

Cleveland, Ohio 44135 

ABSTRACT 

It has been recently reported that the electrical charge in 

a semiconductive carbon nanotube is not evenly distributed, 

but is divided into charge “islands [1,2].” A clear 

understanding of tunneling phenomena can be useful to 

elucidate the mechanism for electrical conduction in 

nanotubes. This paper represents the first attempt to shed 

light on the aforementioned phenomenon through viewing 

tunneling as a natural consequence of “discrete 

trajectories.” The relevance of this analysis is that it may 

provide further insight into the higher rate of tunneling 

processes, which makes tunneling devices attractive [3]. In 

a situation involving particles impinging on a classically 

impenetrable barrier, the result of quantum mechanics that 

the probability of detecting transmitted particles falls off 

exponentially is derived without wave theory. This paper 

should provide a basis for calculating the charge profile 

over the length of the tube so that nanoscale devices’ 

conductive properties may be fully exploited. 

1 INTRODUCTION 

This introduction serves as an outline for the paper. The 

idea of the “warp tunnel” is essential for subsequent 

calculations in this paper. Though it sounds futuristic, the 

idea is straightforward and sound. The warp tunnel is 

introduced in Figure 1. Its quantitative properties are 

explored in Figure 2, which attempts to provide insight into 

the higher rate of tunneling processes. Figure 3 hints at the 

explanation of the charge island phenomenon, and also 

provides a basis for understanding subsequent calculations. 

To show the potency of this analytical model, we calculate 

the transmission probability of a beam of electrons 

impinging on a classically impenetrable barrier, the 

scenario shown in Figure 4. Having successfully obtained 

the result, we look within the analysis to obtain the 

explanation for charge islands. 

2 BACKGROUND AND MOTIVATION 

We begin with consideration of Figure 4. Electrons 

emitted from a source S impinge on a potential barrier. To 

penetrate the barrier, they must tunnel a distance a while 

bridging the distance r. We wish to find p(L), or the 

probability of bridging the distance r while traveling only 

L = r – a. In Figure 5, we visualize distance r as consisting 

of warp tunnels of differing constituents. Uncharged 

particles are shown as white while electrons are black. 

Although the number of electrons in each warp tunnel 

increases from top to bottom in this figure, such an 

arrangement is solely for conceptual simplicity. The 

number of electrons pictured in each warp tunnel 

symbolizes the number of electrons involved in the 

momentum transfer to the outgoing electron in each warp 

tunnel. It is assumed that the greater the involvement of 

uncharged particles in the momentum transfer, the faster the 

transfer will occur, owing to the greater size of the 

uncharged particles. 

Figure 6 shows that the uncharged particles move but 

little in a chain reaction or momentum transfer, but the 

electrons, being smaller, must move more to propagate the 

chain reaction. When particle A, in Figure 6, is inserted into 

the warp tunnel in (I), its representative exits almost 

immediately. The effective distance traveled by A is on the 

order of its diameter. However, when A is inserted into 

warp tunnel (II), its effective distance traveled is greater. 

2.1 Model Verification Calculation 

To formalize the calculation, it is assumed that the mean 

free path of an electron is d, while that of an uncharged 

particle is 0. In warp tunnel (a) of Figure 5, it is presumed 

that an incoming electron will travel d once it enters the 

tunnel, collide with an uncharged particle, which will then 

transfer that momentum via other uncharged particles to an 

outgoing electron, which will exit the warp tunnel almost 

immediately after the first electron entered it. As depicted 

in Figure 3, such a scenario becomes increasingly 

improbable as the length of the warp tunnel increases. In 

warp tunnel (b) of Figure 5, since the mean free path of 

uncharged particles is 0 and the current flows to the left, we 

may assume that the electron inside the tunnel is flushed to 

the right of its adjacent uncharged particle on its left. That 

electron then having a mean free path of d has to its 

immediate right a distance d of intervening space separating 

it from the next uncharged particle in the tunnel. So the 

incoming electron travels a distance d once it enters the 

tube, transfers its momentum to the uncharged particles, 

which cause the middle electron to travel through a distance 

d, which was to its right. It collides with uncharged 

particles, which cause an electron to be ejected from the 

warp tunnel. The amount of time it takes to eject an electron 

after one has entered in warp tunnel (b) is about twice that 

in warp tunnel (a). A convenient abstraction is to regard an 

electron entering a tunnel as appearing for a few instants, 

disappearing, and reappearing at a point further downstream 

the tunnel. The more “appearances” an electron makes in a 

tunnel, the longer its trip through the tunnel. Here enters the 
NASA/TM—2001-211133 2 

notion of a “discrete trajectory.” The electron does not 

travel continuously through the tube, but successively 

appears at subsequent points in the tube and disappears 

between those points. 

In Figure 5, i - 1 electron mean free path domains are 

evenly distributed throughout the warp tunnel. This 

assumption is true on average. For a warp tunnel with 

L = i d, we calculate p(L) as follows. To choose i - 1 

domains out of the interval [-r,0] as free path positions for 

the i - 1 electrons, we recognize that the number of points 

from which to choose is n = r/d. The number of ways of 

obtaining a warp tunnel with L = i d, is n (n – 1) (n – 2) 

...(n – (i – 2)) = n!/(n – (i – 1))! We use Stirling’s Formula 

to simplify. 

p(L) ∝ 1/(n – (i – 1))! 

p(L) ∝ e– [(n – i + 2) + (n – i + 1⁄2) Ln(n – i + 2)] . 

L = i d ⇒ i = L/d, n – i = (r - L)/d. 

p(L) ∝ G’ ek L. 

The probability of detecting electrons on the other side of 

the barrier requires L = r – a, and is given by: 

p(L) ∝ e– [a/d + (a/d + 1⁄2) Ln(a/d + 2)] . 

If a/d >> 1, p(L) → 0 very quickly, but if a/d ≈ 1, 

p(L) = G e-k’ a, 

which falls off exponentially with increasing barrier width 

a. G and the k are independent of r. 

3 MODEL EXPLOITATION 

The calculation of the charge separation is lengthy and 

involved. Consequently, only general remarks, an outline 

of the calculation and reporting of those results appear here. 

The idea behind charge islands is closely associated with 

that of mean free path. A free-floating electron travels by a 

certain average amount before it collides with another 

particle, possibly dislodging an electron and imparting it 

with its motion. The electron need not directly bump into 

another electron though. It may transfer its motion by 

bumping into several intervening neutral particles. 

Tunneling viewed in this light is to induce the motion of a 

distinct, yet otherwise indistinguishable, particle indirectly 

via intervening, and possibly different, particles—much in 

the same way as making a combination shot in pool. 

Charge islands are formed at regular intervals due to the 

chain-reaction creation of charged particles a certain mean 

distance from an immediately upstream impulse due to the 

indirect motion of charged particles there. We may think of 

a length L nanotube as a warp tunnel containing i electron 

islands. These islands will be evenly spaced on average. In 

a nanoscale metallic conductor, the spacing of the charge 

islands is so close that the charge appears to be uniformly 

distributed. However, in a nanoscale semiconductor, the 

spacing of charge islands increases, revealing charge 

granularity. 

3.1 Spacing Calculation Outline 

Since charge is segregated into islands, to model this 

circuit, an appropriate technique seems to be that of the 

series RC circuit. A capacitor plate represents a charge 

island; the spacing between the plates of the capacitor 

represents the neutral gaps between adjacent charge islands. 

The capacitors are modeled as having equal capacitance 

while R represents the internal resistance of the circuit. 

Ordinarily, we would not expect a capacitor to conduct a 

DC current, but if the plate spacing is sufficiently close a 

tunneling current will flow in the same way that electrons 

will pass through the barrier shown in Figure 4 if a is small 

enough. When tunneling is viewed in this new light, a 

tunneling current can readily transmit through 40 nm 

barriers. 

Consider the warp tunnel in the top of Figure 1. 

Ignoring its coloring, let the particle entering the tunnel be a 

photon emitted by an electron, not displayed, to its left. 

Assume that the warp tunnel spans a barrier of width a. 

The particle emerging from the right side of the warp tunnel 

may be a photon that will be absorbed by an electron, not 

displayed, on the right side of the barrier. The outgoing 

photon thereby imparts that electron with the motion 

originally possessed by the other electron on the left side of 

the barrier. In these calculations, it is assumed that the 

photon has a width, denoted lt, directly proportional to its 

energy or frequency. The proportionality constant is derived 

from physical constants. 

Assuming the warp tunnel in Figure 1 is full, the 

entering photon need only move into the tunnel an amount 

lt when a representative photon will move out of the tunnel. 

The electron, which originally emitted the photon, has 

effectively bridged the distance across the barrier by 

“traveling” only a distance lt << a. The effective length of 

the barrier is only lt(f), or the width of the photon of 

frequency f perpetuating the motion. This analysis explains 

both the optical and quantum mechanical barrier 

penetration phenomena shown in Figure 7. In general, it is 

not the selfsame electron that impinges on the barrier in 

Figure 7.a that emerges on the other side. Likewise, it is 

not the selfsame photon that enters the prism on the left that 

is detected beyond the prism on the right. 

This model allows one to conclude that the effective 

length of the barrier may be given by the width, lt(f), of a 

photon within the barrier. To find the probability 

distribution function, f(L), of the length of the barrier, we 

use the Planck distribution function, which gives the 

thermal average number of photons in a single mode of 

frequency. For the temperature variable, we use room 

temperature. The majority of the frequencies are smaller 
NASA/TM—2001-211133 3 

than that of visible light; hence, f(L) ∝ 1/L, where L is a 

function of frequency. 

We return now to our classical RC circuit model. When 

the power is turned on in a classical DC RC circuit, we 

obtain a transient solution: a current that exponentially 

decays in time. However, if the capacitor plate spacing is 

not fixed, but is exponentially decreasing with increasing 

time, we find that the current through the capacitor is 

essentially constant and inversely exponentially depends on 

the original plate spacing. The probability distribution 

function, g(L), of the length L between the capacitor plates, 

or the plate spacing, must give g(L) ∝ 1/L to explain a plate 

spacing that exponentially decays with time. This 

remarkable and highly convincing result shows that 

harmony may be obtained between standard tunneling 

current formulae and that predicted by DC RC circuit 

analysis by assuming that the tunneled current follows a 

discrete trajectory across the barrier. 

Detailed calculations were performed to obtain the 

coefficient of time in the decreasing exponential factor 

involved in the plate spacing of the capacitor. This value, 

along with the internal resistance of the circuit, is needed to 

resolve the formula for the tunneling current through the 

nanotube circuit or the tunneling resistance of the nanotube. 

This model reasonably predicts the charge separation 

contingent on one measurement, which was not reported in 

the literature discussing this phenomenon: the internal 

resistance of the circuit when the semiconducting nanotube 

is short-circuited by a metallic nanotube or another 

extremely low resistance shunt. So, if the power supply 

were shut off, and the two nanotube terminals, gold in one 

test and platinum in another, were short-circuited by an 

extremely low resistance shunt and a highly sensitive 

ohmmeter were placed in series, one lead at the positive 

terminal of the power supply and one at the negative 

terminal of the power supply, the resistance, R, that the 

ohmmeter would read for the two scenarios reported in 

[2]—keeping all things constant such as temperature, the 

length of the tubes, the length and composition of the leads 

to the tubes, and identical final charge spacing—is needed. 

An equivalence is established which allows calculation 

of the charge separation based on traditional circuit 

parameters yielded readily from macroscopic measurements 

and the literature. We equate the tunneling resistance of the 

entire nanotube with no islands at t = 0 when the power 

supply is switched on to that which would result from an 

arbitrary number n of charge islands which are manifested 

at t > 0. Solving for n yields the number of charge islands, 

which the model says should be evenly spaced. Knowledge 

of the length of the nanotube gives the separation of the 

charge islands. The charge separation for various R has 

been calculated and is presented below: 

4 RESULTS 

Results are given in the form: (R in Ohms, charge 

island spacing in nm). For the 650 nm tube, (0.0001270, 

37.7), (0.0001316, 42.6), and (0.0001292, 40). For the 750 

nm tube, (0.0001370, 34.4), (0.0001408, 37.82), and 

(0.0001389, 36). These resistances seem in range. They 

are just what one would expect of the resistances of the 

leads in these experiments. 

5 REMARKS 

A general observation about tunneling is that as the 

mass or equivalent energy of a tunneled particle decreases, 

the greater its transmission probability. When we apply this 

analysis to the optical experiment in Figure 7, we see that d 

is on the order of the wavelength of the incident light. 

Observe that lower frequency electromagnetic radiation has 

greater penetrating ability, which also may be related to the 

size of its photons. As a final comment, a standing wave 

may be thought of as existing inside the potential barrier in 

Figure 4. Analogously, this “island,” phenomenon may 

also exist on a macroscopic scale and may be manifested 

with light in microwave ovens, which often do not heat 

food uniformly, but leave hot and cold spots in food. 

6 CONCLUSIONS 

A theoretical framework was introduced to better 

understand tunneling. It was successfully applied to 

determine the transmission probability of a stream of 

electrons incident on a potential barrier. An appeal was 

made to the established model to help explain the reported 

phenomenon of charge separation in carbon nanotubes. 

Calculations have been performed which show that 

while to calculate the capacitance above the nanoscale, the 

plate spacing d(t) of a capacitor should be a fixed constant, 

independent of time, at the nanoscale, d(t) rapidly decreases 

as a function of time in accordance with the probability 

distribution function—predicted by this theory—of the 

lengths of the discrete trajectories of electrons bridging the 

gap of the capacitor. When this calculation is performed, 

an equation—identical in form to that of Giaever—yielding 

the current through the circuit as an inverse exponential 

function of the original plate spacing of the capacitor is 

obtained. This equation is then used to obtain the spacing 

of the charge islands. The fact that the classical equations 

for a series RC circuit can still be utilized on the nanoscale 

by assuming the plate spacing of the capacitor is shrinking 

by a function of time which is predicted by this theory is 

strong proof alone that this theory is sound. 

General remarks about tunneling with light and 

electrons were made with the implication that the 

phenomenon of tunneling is more ubiquitous than perhaps 

generally thought. The theoretical model seems accurate 

and was inspired by a more general computer program 

written by the author. 
NASA/TM—2001-211133 4 
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10

Nanotube Tunneling as a Consequence of Probable Discrete Trajectories

Daryl C. Robinson

Nanotube; Tunneling; Discrete; Trajectory; Charge

Unclassified -Unlimited

Subject Category: 33 Distribution: Nonstandard

Prepared for the 2001 International Conference on Computational Nanoscience sponsored by the Applied Computational Research Society, Hilton Head Island, South Carolina, March 19–21, 2001. Responsible person, Daryl C.

Robinson, organization code 5610, 216–433–3553.

It has been recently reported that the electrical charge in a semiconductive carbon nanotube is not evenly distributed, but

is divided into charge “islands.” A clear understanding of tunneling phenomena can be useful to elucidate the mechanism

for electrical conduction in nanotubes. This paper represents the first attempt to shed light on the aforementioned

phenomenon through viewing tunneling as a natural consequence of “discrete trajectories.” The relevance of this analysis

is that it may provide further insight into the higher rate of tunneling processes, which makes tunneling devices attractive.

In a situation involving particles impinging on a classically impenetrable barrier, the result of quantum mechanics that the

probability of detecting transmitted particles falls off exponentially is derived without wave theory. This paper should

provide a basis for calculating the charge profile over the length of the tube so that nanoscale devices’ conductive

properties may be fully exploited.