Essay About Current Flow And Small Object

Electrical Resistance
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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
current—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 = Ϻ A/L (Ohm’s law)
where the conductivity Пє 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
2 Prologue: an atomistic view of electrical resistance
Gate
Insulator
Channel
Insulator
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