ÐÓ°ÉÔ­´´

(see Graphic)
At its most basic level, a computer is little more than a box of
interconnected switches. Its speed of operation, which in turn governs the
amount of information it can handle, is determined by the speed at which an
electrical signal can be transferred between the switches and the speed of
the switches themselves. The search for ever faster computers has led to an
inexorable decrease in the size of electronic components; modern techniques
allow the routine fabrication of devices and interconnections on scales less
than one millionth of a metre, smaller than the wavelength of light. Even at
this level, though, size is unimportant in terms of the physics controlling
devices. The operation of a transistor a millimetre in size is unchanged if
it is scaled down a hundredfold.

But even the ancient Greeks knew that things cannot be made smaller
indefinitely. Eventually, if components are made sufficiently small, it is
no longer valid to think of the electrons which carry the electric current
in Newtonian terms, as a mass of tiny billiard balls. If we consider a
device so small that it contains only one or two electrons, we have to treat
the electrons as quantum mechanical particles which behave in
counter-intuitive ways. Far from being a problem, this raises the
possibility of using properties such as the quantum character of the
electrons to develop a whole new regime of electronics based on a completely
different physics. The vast amount of research devoted to this aim over the
past decade or so, driven by the fabrication technology, has also forced
physicists to rethink the way they look at the conduction of electricity.

In 1827 the German scientist Georg Simon Ohm made the first step in
understanding electrical conduction in solids with the formulation of his
famous Ohm’s law. The law, familiar to anyone who has studied physics,
expresses the linear relation between the current flowing in a conductor and
the voltage applied to it – the ratio of the voltage to the current being
called the resistance. Alternatively, the linearity can also be expressed
in terms of the conductance – the ratio of the current to the voltage.

So familiar is the law that it seems almost common sense, yet in many ways
it is very subtle indeed. We know that the electrical current is due to the
flow of electrons and the voltage is due to the presence of an electric
field. It follows that a steady current is a steady flow of electrons.
However, the definition of an electric field means that it exerts a force on
any charged particle such as an electron, so Newton’s laws of motion would
predict that the electrons would accelerate. How then is it possible for a
steady current to flow?

The answer to this question was supplied by another German physicist, Paul
Drude, in the early years of the century. The picture he introduced was of
an electron being accelerated by the applied electric field and then
undergoing a collision which ‘scrambles’ its motion before it is accelerated
again to another collision and so on. The net result is that the electrons
acquire, on average, a drift velocity of a few millimetres per second
superimposed on their much higher speed of random motion of a few hundred
kilometres per second. It is the average drift which is the current, so the
resistivity of a material depends principally on the number of electrons
free to move and how far they can go on average before undergoing a
collision which alters their momentum. This distance is called the elastic
mean free path and it is a measure of the quality of a conductor. In a
typical metal, say copper at room temperature, it is only 30 nanometres. An
electron will undergo many millions of collisions while travelling the
length of an ordinary piece of wire.

Drude’s theory treated the electrons as if they were classical particles,
simply obeying Newton’s laws. But we now know that classical mechanics is
usually insufficient to describe electrons. For example, it is completely
impossible to understand the physics of electrons in an atom without
invoking quantum mechanics, in which the electron is considered to be a
wave analogous to light or water waves. The important parameter here is the
wavelength of the electron wave, which depends on its energy. In a typical
metal this is only a few times larger than the spacing of atoms in the
crystal and is much smaller than the electron mean free path. In such
situations, where the electron wavelength is very small compared to other
lengths in the system, the classical description of the electron motion is a
good approximation. This is why Drude’s theory worked so well.

To see quantum effects, or for the classical approximation to break down, it
is necessary to do two things. First, the size of a device must be made
small enough to be comparable to the electron wavelength. Secondly, the
material must be made pure enough to remove the effect of collisions. The
first condition is almost impossible to achieve in a metal where the
electron wavelength is very small, but it is far more plausible in a
semiconductor in which the electrons may have much smaller momentum. Since
the electron wavelength depends inversely on momentum, the wavelength may be
considerably larger than in a metal. With modern technology it is possible
to produce devices which are smaller than the elastic mean free path and
comparable with the electron wavelength.

But the second condition – to remove the effect of collisions – must also be
met. To see how this can be done, it is necessary to consider what causes
the collisions in the first place. Collisions are caused by imperfections in
the periodicity of the crystal. These can be due either to vibrations of the
atoms, just like masses on springs, or to permanent imperfections such as
dislocations or impurities. It is relatively easy to remove the vibrations
by lowering the temperature. To remove the impurities and dislocations
requires both high-quality crystal growth and very pure material.

One of the best ways to achieve these dual goals is to grow a very pure
crystal with alternate layers of two semiconductors. This so-called
semiconducting heterostructure is made using molecular beam
epitaxy (see ‘A quantum leap to smaller chips’, New ÐÓ°ÉÔ­´´, 12 January
1991). The epitaxial growth technique creates almost perfect crystals by the
deposition of atoms one at a time. Paradoxically, it is necessary to add
impurity atoms to the heterostructure to provide the electrons. Without
them, the material would not be able to conduct electricity. But the
impurity atoms are physically separated from the electrons, and the
probability of the two colliding is much reduced. Mean free paths of a
tenth of a millimetre are possible, thousands of times longer than in copper
at room temperature. So if the device is smaller than this, the electrons
can move ballistically, that is, in straight lines without collisions, and
an electrical signal may be transmitted much more quickly.

Another feature of the electrons in a heterostructure is that they move
strictly in a plane and for this reason the electrons are said to
form a two-dimensional electron gas or 2DEG. This allows the energy of a
ballistic electron to be conserved, because in a 2DEG the frictional process
that is equivalent to air resistance for a projectile in air is negligible.
Horst Stormer and his colleagues at AT&T Bell Laboratories, New Jersey, have
used this property to develop a ballistic switch.

Just like light

If an electron passes through a region where its potential energy is
increased, then its kinetic energy will be reduced and it will slow down. It
is relatively easy to introduce regions of higher potential energy
electrostatically by applying voltages to metallic layers, or ‘gates’,
fabricated on the surface of the heterostructure in which the 2DEG is
embedded. The slowing of the electron under one of these gates is analogous
to the slowing of a ray of light as it passes into glass. The electron
trajectories are equivalent to the rays in ray optics, and by suitable
choices of the shape of the gates, electron prisms and lenses can be
constructed – as Stormer and his colleagues have shown. Of course, it is
very easy to change the ‘refractive index’ in this system because the speed
of the electron depends on the magnitude of the electrostatic potential.
Such prisms and lenses have been used in the laboratory to deflect electron
beams, creating very fast switches. The major drawback to commercialisation
is that they require very low operating temperatures – around one degree
above absolute zero.

To a physicist, such ballistic switching devices are an excellent example of
how we can use Newton’s laws to describe the motion of the electrons. But
they also lead to the question: ‘What is the resistance of a wire in which
the electrons undergo no collisions?’ Bearing in mind Drude’s model, many
scientists believed that the resistance would be zero. They turned out to be
wrong – for reasons intimately related to the quantum mechanical character
of the electrons. The unravelling of the unexpected results has led to a
new way of thinking about the conduction of electricity in small devices.

In the mid-1980s, Mike Pepper and his group at the University of Cambridge
pioneered fabri-cation techniques which could be used to form just about any
shape of conductor from a 2DEG, including narrow wires. Both the length and
width of the wire are much shorter than the mean free path. In 1988, such
devices were first studied simultaneously at Cambridge and also in the
Netherlands in a collaboration between Delft University of Technology and
Philips Research. Far from having zero resistance, or equivalently
infinite conductance, the conductance of the wire jumped from one constant
value to another as the width of the wire was increased
(see Figure).FIG-mg18774503.GIF

This was surprising enough, but the values of the conductance on these
plateaus were all equal to an integer multiplied by a constant. That
constant was precisely equal to
2e²/h where e is the charge of the electron
and h is Planck’s constant. This was a staggering result. It meant that the
resistance of a wire in which the electrons had no collisions was not zero
but took on discrete (‘quantised’) values which depended only on fundamental
constants of nature and was independent of the length of the wire and the
material from which it was made. Such wires became known as quantum point
contacts (QPC).

Quantised conductance is best understood by using ideas developed in the
mid-1980s at IBM’s Thomas J. Watson Research Center at Yorktown Heights in
New York state, by Joe Imry, Rolf Landauer and, most notably, Marcus
Buttiker. The central concept is one familiar to quantum mechanics – that
the measurement perturbs the system. Although the wire itself contains
nothing that impedes the electrons, any external apparatus such as an
ammeter, current source or voltmeter, must be connected to the wire via
electrical contacts which do contain lots of defects and impurities with
which the electrons can collide. These contacts act as ‘reservoirs’ which
will accept electrons impinging upon them and, also, will inject electrons
into the wire. So the idea is that the wire itself really does have little
or no resistance but that as soon as any electrical leads are attached, as
is always necessary in a measurement, collisions are introduced and a
non-zero resistance is measured. This explains why the resistance is
non-zero and also why the resistance is independent of the length of the
wire.

But it does not explain why the resistance takes on discrete values. This is
a direct consequence of the quantum mechanical character of the electrons.
With wires of such small dimensions, comparable with the electron
wavelength, the electron motion is strictly one-dimensional, with electrons
forming standing waves across the width of the channel . By
increasing the width of the wire, several one-dimensional (1D) states will
result, each corresponding to a particular standing-wave in the channel.
These 1D states can be considered as independent provided that there is no
possibility of an electron transferring from one state to another along the
length of the wire – which will be the case if the electrons are travelling
ballistically. In each 1D state, the ability to carry current, or the
conductance, depends on the number of electrons and how fast they are
travelling.

Generally, both the number of electrons and their speed depend on the
material through which the electrons are travelling. But in 1D, and only in
1D, the properties of the material cancel out when the number and the speed
are multiplied together. So the conductance of a 1D state depends only on
fundamental constants. The conductance of the wire will then be the
conductance of a 1D state multiplied by the number of 1D states in the wire.
This number is determined by the number of electron half-wavelengths that
can be fitted in the width of the wire. Therefore, the conductance of the
wire depends on its width and not on any property of the material.

The potential application of QPCs to digital electronics is this: in
principle, simply by applying a known voltage to a gate, one would be able
to regulate the resistance of the device between rigidly defined quantised
values, to act as a logic element in a computer or a voltage-controlled
resistor. There are, unfortunately, some very serious problems, most notably
the stability of the devices, and the requirement for temperatures near to
absolute zero. This seems to be fundamental, so it is highly unlikely we
shall ever see a commercial device based on a QPC even in a very specialised
electronic application.

Single electrons

In the devices described above, and indeed in Drude’s basic theory, the
electrons are treated successfully as independent, non-interacting
particles. At first sight, this is most surprising because, in a vacuum,
electrostatic forces due to the electron charge mean that electrons strongly
repel each other. However, a solid is made from neutral atoms, and for each
negatively charged electron there is a positively charged ion. The
positively charged ions compensate most of the effect of the electron
charge. This screening allows electrostatic forces between electrons to be
largely ignored in a normal solid. However, the new microfabrication
technology has enabled construction of a new class of device in which these
forces cannot be ignored.

These microscopic devices, made from metals or semiconductors, were
pioneered in the late 1980s by Hans Mooij’s group in Delft University of
Technology, Michel Devoret in Saclay France, and others. They have a very
small capacitance (c) and are sensitive to the presence of a single
electron. Capacitance is the ability of a system to store electrical
charge. Mooij’s capacitors have a capacitance so small that the addition of
a single electron causes an appreciable rise in the voltage across the
device. The single-electron transistor is a charge-sensitive device which
can detect the presence of an excess charge equivalent to that of one
electron.

Another similar and equally fascinating structure is the ‘electron
turnstile’. This relies on the fact that the electron possesses a discrete
charge (e), so that any movement of charge through a circuit involves at
least this amount of charge. If an electron is to pass between the plates of
a microscopic capacitor, the energy provided by the voltage difference
across the capacitor must exceed the minimum charging energy (e to the
power of 2/2c). (The charging energy is inversely proportional to the
capacitance, so it is negligible in the larger capacitors found, for
example, in a transistor radio.) By adjusting the voltage drop across the
microscopic capacitor it is possible to control the passage of electrons one
at a time. Konstantin Likharev at the Massachusetts Institute of Technology
has suggested that this might be the basis for electronics in which a ‘bit’
of information might be carried by a single electron – the ultimate in
digital technology.

The electron turnstile has another application that is interesting at the
most fundamental level. By applying a gate voltage that is oscillatory, the
current (I) through the turnstile is given by the charge on the electron (e)
multiplied by the frequency at which the voltage (V) is altered. Since the
application of a voltage step to the gate allows one electron through the
device, an alternating voltage will allow a steady flow of electrons, one by
one, and hence an electrical current. The current through the turnstile is
then just the charge on the electron multiplied by the frequency with which
the gate voltage is altered. Currents are very small – a frequency of 10 MHz
only gives a current of 1.6 picoamperes – but scientists involved with
measuring fundamental constants, including Tony Hartland at the National
Physical Laboratory, Middlesex, are excited that this might enable an
independent current standard where the ampere is defined in terms of
fundamental constants of nature. As well as being useful in its own right,
such a device might also resolve a puzzling anomaly in the determinations of
the ratio e²/h, where e²
is the square of the charge on the
electron, and h is Planck’s constant. High-precision calculations of the
fine structure constant, which is proportional to e ² and relates to the differing energy states of the atom, yield one result, while
measurement of quantised conductance in the quantum Hall effect, analogous
to that seen in a QPC, yields another. The two values disagree by a margin
much greater than the experimental errors in their measurement.

The development of ultra-small capacitance circuits as well as the other
ballistic and quantum devices has stretched existing theories of electrical
conduction beyond their limits. ÐÓ°ÉÔ­´´s can routinely fabricate
structures which rely for their operation on the quantum character of the
electrons or the fixed charge. As yet, manufacturers show little or no
interest. To break that barrier, physicists will need to find a way to make
these devices work at about room temperature and, just as important, with a
degree of reliability that has not been achieved to date.

On a more fundamental level, it is likely that the physics will continue to
be driven by technology. As fabrication techniques improve until perhaps one
can control small groups of atoms, or even single atoms, then the
distinction between physical manipulation and chemical reaction will be
blurred. There is the prospect of a truly molecular electronics.

Peter Main is a reader in experimental physics in the department of physics
at the University of Nottingham.

* * *

Making electrons move in mysterious ways

Molecular beam epitaxy is the growth of layers of material by firing beams
of atoms or molecules at a suitable substrate. The process takes place in an
ultra-high vacuum to remove impurities and collisions with air molecules.
Amazingly, provided the substrate temperature is carefully controlled, the
molecules stick to the surface but remain sufficiently mobile to form very
high-quality crystalline layers. It is, therefore, possible to grow crystals
atom by atom (typical rates are around 1 micrometre per hour) with complete
control of composition.

Two compound semiconductors, gallium arsenide (GaAs) and aluminium arsenide
(AlAs), are particularly attractive for this sort of layered growth. Not
only do they have the same crystal lattice structure, but the spacing
between atoms in both structures is also almost identical. Successive layers
of each, or intermediate alloys (AlGa)As, can be grown in a heterostructure,
introducing the minimum of atomic misalignment.

An electron moving in a semiconductor has a minimum kinetic energy, known as
the conduction band minimum. In (AlGa)As this minimum is higher than in
GaAs. The conduction band minimum can be thought of as the potential energy
of an electron in a given layer. However, the situation is usually more
complicated in a heterostructure because an individual electron feels the
electrostatic effect of all the other electrons, plus the positive charge
due to impurity atoms (‘dopants’) which have been introduced to increase
the electron density.

Because of the different conduction band minima, electrons are confined
within a plane in the GaAs layer, very close to its junction with the
(AlGa)As layer. They sit in the area of lowest potential energy, called the
triangular region
(left).FIG-mg18774501.GIF
The spatial extent of the region is tiny –
comparable with the wavelength of the electrons, which consequently form a
standing wave. All the mobile electrons in the heterostructure have this
standing wave character but are free to move in the other two directions.
Such electrons are called a two-dimensional electron gas or 2DEG. Their
motion is confined quantum mechanically to a plane perpendicular to the
growth direction of the heterostructure.

Further confinement of the electrons can be introduced by evaporating thin
metallic strips, or ‘gates’, to the top surface of the heterostructure. A
negative voltage applied to the gate raises the potential energy of the
electrons locally beneath the gate. Sufficient voltage can raise the
potential energy enough to reduce the electron concentration to zero.
However, if there is a gap in the metal strip forming the gate, then
immediately beneath the gap there will still be some electrons. These split
gates, first developed by Mike Pepper’s group at the University of
Cambridge, enable the creation of narrow conducting channels in the 2DEG
between the gates. If the channel is made narrow enough, standing waves may
be set up across the width of the channel to force the electrons into
one-dimensional motion along the wire.