While the electronics industry wonders what will happen when transistors become so small that quantum effects become important, researchers are building new transistors that actively exploit the quantum properties of electrons. Michel Devoret and Christian Glattli report.
The invention of the transistor by John Bardeen and William Shockley in 1948 triggered a new era in electronics. Originally designed simply to emulate the vacuum tube, scientists soon found that this solid-state device could offer much more. The great potential of the transistor for speed, miniaturization and reliability has been fully exploited since well controlled materials such as pure single-crystal silicon became available. According to the latest “road-map” for the microelectronics industry, microchips containing one billion transistors and operating with a clock cycle of a billionth of a second will be on the market just a few years into the new millennium.
As transistors continue to shrink, a question naturally arises: will the quantum nature of electrons and atoms become important in determining how the devices are built? In other words, what will happen when a transistor is reduced to the size of a few atoms or a single molecule?
Researchers seeking to answer these questions have devised the so-called single-electron tunnelling transistor – a device that exploits the quantum effect of tunnelling to control and measure the movement of single electrons. Experiments have shown that charge does not flow continuously in these devices but in a quantized way. Indeed, single-electron transistors are so sensitive to charge that they can be used as extremely precise electrometers.
At the beginning
The most common transistor in today’s microchips is the metal-oxide-semiconductor field-effect transistor (MOSFET). Its operation is surprisingly simple: not much quantum mechanics is required to understand how it works, even though the size of a typical device is now just a few thousand atoms placed side by side.
Two conducting electrodes, called the source and drain, are connected together by a channel of material in which the density of free electrons can be varied – in practice a semiconductor (figure 1). A voltage is applied to the semiconducting channel through the “gate”, a third electrode that is separated from the channel by a thin insulating layer. When the gate voltage is zero, the channel does not contain any conduction electrons and is insulating. But as the voltage is increased, the electric field at the gate attracts electrons from the source and drain, and the channel becomes conducting.
This field effect leads to an amplification mechanism in which the gate voltage can control the current flowing between the source and drain when a bias voltage is applied across these two electrodes (figure 2). The source-drain current is determined by the conductance of the channel, which in turn depends on two factors: the density of the conduction electrons and their mobility. The mobility of electrons depends on how often the electrons collide with crystal imperfections, and is essentially independent of the gate voltage. In contrast, the density of electrons is controlled directly by the gate voltage.
The transistor therefore works like a tap controlling the flow of water between two tanks, where the opening of the tap is set by the pressure of the water in a third tank. The difference is that electrons in the channel behave as a compressible fluid with a local density that depends strongly on the electric potential at that point. In other words, the electric field produced by the gate does not generate a “hard wall” for electrons inside the channel, but a smoothly varying potential that is modified by the presence of electrons (figure 1d-f ).
Note that we have made no reference to the wave-like properties of electrons, nor to the fact that the channel is made from individual atoms. The only quantum property that comes into play is the Pauli exclusion principle, which dictates that each possible state in the channel can be occupied by only one electron. This means that only a certain number of electrons can accumulate in the channel, and this sets a limit on the current flow.
However, the quantum properties of electrons and atoms will play a more important role as transistors are made smaller. For example, the wave-like nature of electrons will influence the way in which they travel through the channel. When the width of the channel becomes comparable to the wavelength of electrons (around 100 nm), electron propagation becomes more sensitive to the atomic disorder in the device, which is created in the fabrication process. This will pose a major problem if the reduction in size is not accompanied by an improvement in the atomic structure of the fabricated devices.
The technological constraints of moving towards the atomic scale may force us to adopt a new physical principle for achieving the transistor’s function. Alternatively, a new principle might be found that can provide functions that are not possible with current devices.
Towards single-electron devices
In 1985 Dmitri Averin and Konstantin Likharev, then working at the University of Moscow, proposed the idea of a new three-terminal device called a single-electron tunnelling (SET) transistor. Two years later Theodore Fulton and Gerald Dolan at Bell Labs in the US fabricated such a device and demonstrated how it operates.
Unlike field-effect transistors, single-electron devices are based on an intrinsically quantum phenomenon: the tunnel effect. This is observed when two metallic electrodes are separated by an insulating barrier about 1 nm thick – in other words, just 10 atoms in a row. Electrons at the Fermi energy can “tunnel” through the insulator, even though in classical terms their energy would be too low to overcome the potential barrier.
The electrical behaviour of the tunnel junction depends on how effectively the barrier transmits electron waves, which decreases exponentially with its thickness, and on the number of electron-wave modes that impinge on the barrier, which is given by the area of the tunnel junction divided by the square of the electron wavelength. A single-electron transistor exploits the fact that the transfer of charge through the barrier becomes quantized when the junction is made sufficiently resistive.
This quantization process is shown particularly clearly in a simple system known as a single-electron box (figure 3). If a voltage source charges a capacitor, Cg, through an ordinary resistor, the charge on the capacitor is strictly proportional to the voltage and shows no sign of charge quantization. But if the resistance is replaced by a tunnel junction, the metallic area between the capacitor plate and one side of the junction forms a conducting “island” surrounded by insulating materials. In this case the transfer of charge onto the island becomes quantized as the voltage increases, leading to the so-called Coulomb staircase (figure 3c). This effect was first observed by Philippe Lafarge and collaborators in our laboratory in 1991.
This Coulomb staircase is only seen under certain conditions. Firstly, the energy of the electrons due to thermal fluctuations must be significantly smaller than the Coulomb energy, which is the energy needed to transfer a single electron onto the island when the applied voltage is zero. This Coulomb energy is given by e2/2C, where e is the charge of an electron and C is the total capacitance of the gate capacitor, Cg, and the tunnel junctions. Secondly, the tunnel effect itself should be weak enough to prevent the charge of the tunnelling electrons from becoming delocalized over the two electrodes of the junction, as happens in chemical bonds. According to recent work by theorists at the universities of Freiburg and Karlsruhe in Germany, this means that the conductance of the tunnel junction should be much less than the quantum of conductance, 2e2/h, where h is Planck’s constant.
When both these conditions are met, the steps observed in the charge are somewhat analogous to the quantization of charge on oil droplets observed by Millikan in 1911. In a single-electron box, however, the charge on the island is not random but is controlled by the applied voltage. As the temperature or the conductance of the barrier is increased, the steps become rounded and eventually merge into the straight line typical of an ordinary resistor.
A single-electron transistor
The SET transistor can be viewed as an electron box that has two separate junctions for the entrance and exit of single electrons (figure 4). It can also be viewed as a field-effect transistor in which the channel is replaced by two tunnel junctions forming a metallic island. The voltage applied to the gate electrode affects the amount of energy needed to change the number of electrons on the island.
The SET transistor comes in two versions that have been nicknamed “metallic” and “semiconducting”. These names are slightly misleading, however, since the principle of both devices is based on the use of insulating tunnel barriers to separate conducting electrodes.
In the original metallic version fabricated by Fulton and Dolan, a metallic material such as a thin aluminium film is used to make all of the electrodes. The metal is first evaporated through a shadow mask to form the source, drain and gate electrodes. The tunnel junctions are then formed by introducing oxygen into the chamber so that the metal becomes coated by a thin layer of its natural oxide. Finally, a second layer of the metal – shifted from the first by rotating the sample – is evaporated to form the island.
In the semiconducting versions, the source, drain and island are usually obtained by “cutting” regions in a two-dimensional electron gas formed at the interface between two layers of semiconductors such as gallium aluminium arsenide and gallium arsenide. In this case the conducting regions are defined by metallic electrodes patterned on the top semiconducting layer. Negative voltages applied to these electrodes deplete the electron gas just beneath them, and the depleted regions can be made sufficiently narrow to allow tunnelling between the source, island and drain. Moreover, the electrode that shapes the island can be used as the gate electrode.
In this semiconducting version of the SET, the island is often referred to as a quantum dot, since the electrons in the dot are confined in all three directions. In the last few years researchers at the Delft University of Technology in the Netherlands and at NTT in Japan have shown that quantum dots can behave like artificial atoms. Indeed, it has been possible to construct a new periodic table that describes dots containing different numbers of electrons (see “Quantum dots” by Leo Kouwenhoven and Charles Marcus Physics World June pp35-39).
Operation of a SET transistor
So how does a SET transistor work? The key point is that charge passes through the island in quantized units. For an electron to hop onto the island, its energy must equal the Coulomb energy e2/2C. When both the gate and bias voltages are zero, electrons do not have enough energy to enter the island and current does not flow. As the bias voltage between the source and drain is increased, an electron can pass through the island when the energy in the system reaches the Coulomb energy. This effect is known as the Coulomb blockade, and the critical voltage needed to transfer an electron onto the island, equal to e/C, is called the Coulomb gap voltage.
Now imagine that the bias voltage is kept below the Coulomb gap voltage. If the gate voltage is increased, the energy of the initial system (with no electrons on the island) gradually increases, while the energy of the system with one excess electron on the island gradually decreases. At the gate voltage corresponding to the point of maximum slope on the Coulomb staircase, both of these configurations equally qualify as the lowest energy states of the system. This lifts the Coulomb blockade, allowing electrons to tunnel into and out of the island.
The Coulomb blockade is lifted when the gate capacitance is charged with exactly minus half an electron, which is not as surprising as it may seem. The island is surrounded by insulators, which means that the charge on it must be quantized in units of e, but the gate is a metallic electrode connected to a plentiful supply of electrons. The charge on the gate capacitor merely represents a displacement of electrons relative to a background of positive ions.
If we further increase the gate voltage so that the gate capacitor becomes charged with –e, the island again has only one stable configuration separated from the next-lowest-energy states by the Coulomb energy. The Coulomb blockade is set up again, but the island now contains a single excess electron. The conductance of the SET transistor therefore oscillates between minima for gate charges that are integer multiples of e, and maxima for half-integer multiples of e (figure 5).
Accurate measures of charge
Such a rapid variation in conductance makes the single-electron transistor an ideal device for high-precision electrometry. In this type of application the SET has two gate electrodes, and the bias voltage is kept close to the Coulomb blockade voltage to enhance the sensitivity of the current to changes in the gate voltage.
The voltage of the first gate is initially tuned to a point where the variation in current reaches a maximum. By adjusting the gate voltage around this point, the device can measure the charge of a capacitor-like system connected to the second gate electrode. A fraction of this measured charge is shared by the second gate capacitor, and a variation in charge of ¼e is enough to change the current by about half the maximum current that can flow through the transistor at the Coulomb blockade voltage. The variation in current can be as large as 10 billion electrons per second, which means that these devices can achieve a charge sensitivity that outperforms other instruments by several orders of magnitude. Indeed, a collaboration between researchers at Yale University in the US and Chalmers University in Gothenburg, Sweden, recently showed that charge variations smaller than 10-5 e can be detected in a measurement period of just one second and with a bandwidth of several hundred megahertz.
SET transistors have already been used in mesoscopic physics experiments that have required extreme charge sensitivity. For example, earlier this year Robert Westervelt and co-workers at Harvard University in the US used this type of device to measure the rounding of steps observed in a Coulomb staircase.
Electrometers based on SET transistors could also be used to measure the delicate quantum superpositions of charge states in a superconducting island connected by a tunnel junction to a superconductor. Such an island can accommodate not only several charge states corresponding to different numbers of Cooper pairs, but also coherent quantum superpositions of these states. Superconducting islands could therefore provide a means for implementing the quantum bits needed for a quantum computer (see “Fundamentals of quantum information” by Anton Zeilinger Physics World March pp35-40). The feasibility of the idea has been shown by experiments at Delft University, the State University of New York at Stony Brook, the PTB Laboratory in Germany, the NEC Laboratory in Japan and in our lab.
As we have seen, the charge sensitivity of the SET is ultimately linked to the fact that electrons traverse the island one at a time. In 1990 Bart Geerligs, Valerie Anderegg, Peter Holweg and Hans Mooij at Delft, together with Hugues Pothier, Daniel Esteve, Cristian Urbina and one of us (MD) at CEA Saclay showed that electrons can be counted one by one by creating devices that combine several SET transistors. And in 1996 John Martinis and colleagues at the National Institute for Standards and Technology in Boulder, Colorado, showed that a device called the electron pump can count electrons with an accuracy of 15 parts in a billion. The same group is now attempting to measure the charge of the electron with an accuracy better than 1 part per 10 million by combining an electron pump with a specially calibrated capacitor. Other metrology labs are aiming to use arrays of single-electron transistors to establish a standard for electric current.
The precision with which electrons can be counted is ultimately limited by the quantum delocalization of charge that occurs when the tunnel-junction conductance becomes comparable with the conductance quantum, 2e2/h. However, the current through a SET transistor increases with the conductance of the junctions, so it is important to understand how the single-electron effects and Coulomb blockade disappear when the tunnel conductance is increased beyond 2e2/h. In 1991 Konstantin Matveev, now at Duke University in North Carolina, drew a parallel between the suppression of the Coulomb blockade and the Kondo effect, in which magnetic impurities in metals are screened by conduction electrons. Experiments by Philippe Joyez and co-workers in our lab, and by Leonid Kuzmin and collaborators at Chalmers University have confirmed that quantum fluctuations in the charge of the tunnelling electrons reduce the Coulomb energy.
Towards room temperature
Until recently single-electron transistors had to be kept at temperatures of a few hundred millikelvin to maintain the thermal energy of the electrons below the Coulomb energy of the device. Most early devices had Coulomb energies of a few hundred microelectronvolts because they were fabricated using conventional electron-beam lithography, and the size and capacitance of the island were relatively large. For a SET transistor to work at room temperature, the capacitance of the island must be less than 10-17 F and therefore its size must be smaller than 10 nm.
This year two experiments have demonstrated that SET transistors can work at room temperature. Lei Zhuang and Lingjie Guo of the University of Minnesota, and Stephen Chou of Princeton University in the US fabricated a SET transistor in a similar way to a field-effect transistor with a channel just 16 nm wide. The fabrication process generated variations in the channel that act as tunnel junctions defining several different islands, and the behaviour of the device is dominated by the smallest island.
As expected from theory, the conductance of the device shows a series of peaks as a function of gate voltage. The Coulomb energy of the device is around 100 meV, which is large enough to reveal single-electron effects at room temperature. However, the Coulomb energy is too small to provide a large Coulomb gap, so the transistor is not sensitive enough to be used as an electrometer. Interestingly, the wavelength of electrons is comparable with the size of the dot, which means that their confinement energy makes a significant contribution to the Coulomb energy.
Meanwhile, Jun-ichi Shirakashi and colleagues at the Electrotechnical Laboratory and the Tokyo Institute of Technology fabricated a metallic SET from a very thin layer of niobium. Tunnel junctions were created by oxidizing areas of the niobium with the tip of a scanning electron microscope. The tip had almost atomic resolution, allowing the researchers to make very narrow oxide barriers between the island and the source and drain, and a wider barrier between the island and gate.
The team measured the current through the device as a function of both bias and gate voltages, and the results closely match theoretical predictions. The Coulomb energy of the device is 250 meV, which means that single-electron effects can readily be seen at room temperature. However, tunnel barriers fabricated in this way are highly resistive, which means that the current is about 100 times smaller than in devices operating at low temperatures. This problem also limits the usefulness of the device for electrometry.
Perspectives on the future
Researchers have long considered whether SET transistors could be used for digital electronics. Although the current varies periodically with gate voltage – in contrast to the threshold behaviour of the field-effect transistor – a SET could still form a compact and efficient memory device. However, even the latest SET transistors suffer from “offset charges”, which means that the gate voltage needed to achieve maximum current varies randomly from device to device. Such fluctuations make it impossible to build complex circuits.
One way to overcome this problem might be to combine the island, two tunnel junctions and the gate capacitor that comprise a single-electron transistor in a single molecule – after all, the intrinsically quantum behaviour of a SET transistor should not be affected at the molecular scale. In principle, the reproducibility of such futuristic transistors would be determined by chemistry, and not by the accuracy of the fabrication process. Last year a team led by Cees Dekker at Delft and Richard Smalley at Rice University in the US made a crucial step in this direction by observing Coulomb blockade in an island consisting of a single carbon nanotube.
It is not yet clear whether electronics based on individual molecules and single-electron effects will replace conventional circuits based on scaled-down versions of field-effect transistors. Only one thing is certain: if the pace of miniaturization continues unabated, the quantum properties of electrons will become crucial in determining the design of electronic devices before the end of the next decade.