Electrical Resistance

Let me start with a brief explanation since this is not a typical “prologue.” For one it

is too long, indeed as long as the average chapter. The reason for this is that I have

a very broad objective in mind, namely to review all the relevant concepts needed to

understand current flow through a very small object that has only one energy level in

the energy range of interest. Remarkably enough, this can be done without invoking

any significant background in quantum mechanics. What requires serious quantum

mechanics is to understand where the energy levels come from and to describe large

conductors with multiple energy levels. Before we get lost in these details (and we

have the whole book for it!) it is useful to understand the factors that influence the

currentвЂ"voltage relation of a really small object.

This “bottom-up” view is different from the standard “top-down” approach to electrical

resistance. We start in college by learning that the conductance G (inverse of

the resistance) of a large macroscopic conductor is directly proportional to its crosssectional

area A and inversely proportional to its length L:

G = ПÑ" A/L (Ohm’s law)

where the conductivity ПÑ" is a material property of the conductor. Years later in graduate

school we learn about the factors that determine the conductivity and if we stick around

long enough we eventually talk about what happens when the conductor is so small that

one cannot define its conductivity. I believe the reason for this “top-down” approach

is historical. Till recently, no one was sure how to describe the conductance of a really

small object, or if it even made sense to talk about the conductance of something really

small. To measure the conductance of anything we need to attach two large contact

pads to it, across which a battery can be connected. No one knew how to attach contact

pads to a small molecule till the late twentieth century, and so no one knew what the

conductance of a really small object was. But now that we are able to do so, the answers

look fairly simple, except for unusual things like the Kondo effect that are seen only for

a special range of parameters. Of course, it is quite likely that many new effects will be

discovered as we experiment more on small conductors and the description presented

here is certainly not intended to be the last word. But I think it should be the “first

1

2 Prologue: an atomistic view of electrical resistance

D

R

A

I

N

S

O

U

R

C

E

Gate

Insulator

Channel

Insulator

z

L

VG

I

VD

x

Fig. 1.1 Sketch of a nanoscale field effect transistor. The insulator should be thick enough to

ensure that no current flows into the gate terminal, but thin enough to ensure that the gate voltage

can control the electron density in the channel.

word” since the traditional top-down approach tends to obscure the simple physics of

very small conductors.

The generic structure I will often use is a simple version of a “nanotransistor” consisting

of a semiconducting channel separated by an insulator layer (typically silicon

dioxide) from the metallic gate (Fig. 1.1). The regions marked source and drain are

the two contact pads, which are assumed to be highly conducting. The resistance of

the channel determines the current that flows from the source to the drain when a voltage

VD is applied between them. The voltage VG on the gate is used to control the electron

density in the channel and hence its resistance. Such a voltage-controlled resistor is the

essence of any field effect transistor (FET) although the details differ from one version

to another. The channel length L has been progressively reduced from в?ј10 Ојm in 1960

to в?ј0.1 Ојm in 2000, allowing circuit designers to pack (100)2 = 10 000 times more

transistors (and hence that much more computing power) into a chip of given surface

area. This increase in packing density is at the heart of the computer revolution. How

much longer can the downscaling continue? No one really knows. However, one thing

seems certain. Regardless of what form future electronic devices take, we will have to

learn how to model and describe the electronic properties of device