Tunneling

Tunneling is the ability for electrons or other subatomic particles, to tunnel through a barrier rather than go over the barrier. If you face a wall and you want to get a ball from one side to the other without destroying the wall, you have the throw the ball over the wall. In general electrons behave exactly the same way. If they encounter a barrier they have to get over the barrier to move on. However, if the barrier is thin enough and the bias conditions are right, electrons can also tunnel through the barrier without having to go over. This is much easier to understand in a wave picture rather than a particle picture.

Challenges

The real challenge to make single-electronics an industrial reality is manufacturing precision. In order to build these devices one would need nano- and sub-nanometer control in 3-dimensions. Although one can produce single-electron devices on a small scale it is unclear if we will ever be able to produce them on a large scale. There is also the problem that single-electron transistors have very little gain. They therefore have to be used in combination with other gain elements, for example traditional CMOS.

Dig Deeper

Anybody who would like to dig deeper I recommend my book "Computational Single-Electronics" published by Springer (the introductory chapter is reproduced below). Or if you want to actually simulate single-electron devices and circuits you should take a look at SIMON, a simulator I wrote, which to this date is considered the gold standard of single-electron simulation.

The following excerpt is taken from my book "Computational Single-Electronics" published by Springer.

Single-Electronics – Made Easy

Single-electronics implies the possibility to control the movement and position of a single electron or a small number of electrons. It is interesting to see how strong an influence a single electron with the minute charge of 1.6·10^-19 As can have, given the right circumstances. Consider an uncharged small metallic sphere with a radius of 1 nm, something quite possible being produced today.

Fig. 1: An electron approaching a small uncharged metallic sphere will feel a small attractive force caused by its own image charge in the sphere. Once the sphere is charged by a single electron, following electrons will feel a strong repelling Coulomb force.

If such a small sphere is charged with a single electron (Fig. 1) the electric field on the surface of the sphere in vacuum will become about 1.4 GV/m (about 14 MV/cm). A remarkable large repelling force for any other electron which wants to approach the sphere. This phenomenon makes it possible to separate a single electron in a solid state structure. To be more accurate, we have not isolated a single electron, because many other electrons are present in the electron cloud of a metallic grain. But we have added precisely one single electron to the electrically neutral grain. Meaning we have control over single electrons and can manipulate them with single electron precision.

It turns out that the capacitance C and the associated charging energy E_C = e²/2C for a single electron with charge e are the correct measures to quantitatively understand single-electron transfer and related effects. Thus, if the involved capacitances are small enough charging energies will be dominating.

The simplest circuit which exhibits single-electron charging effects is the single-electron box (see Fig. 2).

Fig. 2: The electron-box can be filled with a precise number of  excess electrons.

The single-electron box is not just easy to understand but it is also relatively simple to manufacture and measure in the laboratory. A metal granule is only on one side connected by a tunnel junction. On this side electrons can tunnel in and out. Imagine for instance a metal grain embedded in oxide, as shown in Fig. 3.

Fig. 3: Realization of an electron box - metal grain embedded in oxide. Tunneling is only possible through the thin top layer of oxide.

The top oxide layer is thin enough for electrons to tunnel through. To transfer one electron onto the granule, the Coulomb energy E_C = e²/2C has to be 'payed'. Neglecting thermal and other forms of energy, the only energy source available is the bias voltage V_b. As long as the bias voltage is small enough, smaller than a threshold V_th = e/C, no electron can tunnel, because not enough energy is available to charge the island. This behavior is called the Coulomb blockade. Raising the bias voltage will populate the granule with one, then two and so on electrons, leading to a staircase-like characteristic shown on the right side in Fig. 2.

To better understand single electron transfer one must understand the movement of electric charge through a conductor. An electric current can flow through a conductor because some electrons are free to move through the lattice of atomic nuclei. The current is determined by the charge transferred through the conductor per time interval. Surprisingly this transferred charge can have practically any value, in particular, a fraction of the charge of a single electron. Hence, it is not quantized. This, at first glance counterintuitive fact, is a consequence of the displacement of the electron cloud against the lattice of atoms. This shift can be changed continuously and thus the transferred charge is a continuous quantity (see left side of Fig. 4).

Fig. 4: The top shows, that the electron cloud shift against the lattice of atoms is not quantized and thus charge transfer in a good conductor is continuous. The bottom shows an accumulation of electrons at a tunnel junction. A tunnel junction can only be penetrated by whole electrons and thus charge transfer through tunnel junctions is quantized.

A tunnel junction on the other hand restricts current flow to electrons penetrating the barrier. That is the current through a tunnel junction is quantized. Thus, if a tunnel junction interrupts an ordinary conductor, electric charge will move through the system by both a continuous and a discrete process. Since only discrete electrons can tunnel through junctions, charge will accumulate at the surface of the electrode against the isolating layer, until a high enough bias has built up across the tunnel junction (see right side of Fig. 4). Then one electron will be transferred by tunneling. K. Likharev \cite has coined the term `dripping tap' as an analogy of this process. In other words, if a single tunnel junction is biased with a constant current I, the so called single-electron tunneling oscillations will appear with frequency f = I/e, where e is the charge of an electron (see Fig. 5).

Fig. 5: Current biased tunnel junction showing Coulomb oscillations. On the left side is the circuit diagram, in the middle the 'tripping tap' analogy, and on the right side is the conduction and displacement current as functions of time.

Charge continuously accumulates on the tunnel junction like on a capacitor until it is energetically favorable for an electron to tunnel. This discharges the tunnel junction by an elementary charge e. Similar effects are observed in superconductors. There, charge carriers are Cooper pairs, and the characteristic frequency becomes f = I/2e, related to the so called Bloch oscillations.

It is easily understandable, that these single-electron phenomena, such as single-electron tunneling oscillations and Coulomb blockade, only matter, if the Coulomb energy is bigger than the thermal energy. Otherwise thermal fluctuations will disturb the motion of electrons and will smear out the quantization effects. The necessary condition is

E_C = e²/2C > k_B T,

where k_B is Boltzmann's constant and T is the absolute temperature. This means that the capacitance C has to be smaller than 12 aF for the observation of charging effects at the temperature of liquid nitrogen (77 K) and smaller than 3 aF for charging effects to appear at room temperature (300 K). This requires grains with a diameter smaller than 15 nm and 5 nm respectively. To use charging effects for deterministic logic, the operating temperature has to be considerably lower than this limit (by a factor of about 50), in order to suppress the thermal tail of electrons overcoming the Coulomb blockade and causing errors. This will require granules below 1 nm diameter for room temperature operation. A second condition for the observation of charging effects is that quantum fluctuations of the number of electrons on an island must be negligible. Electrons need to be well localized on the islands. If electrons would not be localized on islands one would not observe charging effects, since islands would not be separate quantum dots but rather one big uniform quantum space. The charging of one island with an integer number of the elementary charge would be impossible, because one electron is shared by more than one island. The Coulomb blockade would vanish, since no longer would a lower limit of the charge, an island could be charged with, exist. This leads to the requirement that all tunnel junctions must be opaque enough for electrons in order to confine them on islands. The `opacity' of a tunnel junction is given by its tunnel resistance R_T which must fulfill the following condition for observing discrete charging effects

R_T > h/e² ≈ 25813 Ω,

where h is Planck's constant. This should be understood as an order-of-magnitude measure, rather than an exact threshold.

Therefore, these effects are experimentally verifiable only for very small high-resistance tunnel junctions, meaning small granules with small capacitances and/or very low temperatures. Advanced fabrication techniques, such as the production of granular films with particle sizes down to 1 nm, or the usage of STM and AFM equipment together with deeper physical understanding allow today the study of many charging effects at room temperature.

Based on the Coulomb blockade several interesting devices are possible, such as precise current standards, very sensitive electrometers, logic gates, and memories with ultra low power consumption and down-scalability to atomic dimensions. It is not clear if single-electronics will make a commercial breakthrough. Particularly since CMOS technology is so well established and has potential for improvement. However, single-electronics has already increased our physical understanding and it seems likely that certain niche applications will be realized. A widespread use looks promising only for memory and storage applications. Looked at it differently, a sub 0.1 μm NMOS-FET has at any given time only few hundred electrons below its gate. So the single electron limit is not that far away.

A Historical Look Back

It took several Nobel prizes and Nobel prize worthy discoveries and insights for the field of single-electronics to emerge. One could argue that everything started end of the 19th century when Joseph John Thomson discovered the electron which was then called corpuscle. He received for his discovery the Nobel prize in 1906. Shortly after, Robert Millikan showed with his oil drop experiment that charge is discrete and single valued. He could manage to localize single electrons on oil drops and figure out how to measure their elementary charge. He received the Nobel prize in 1923. To replicate this remarkable feat of single electron manipulation in the solid state took several decades until in the late 1980s the field of single-electronics, as we know it today, took off. But before that could happen quantum mechanics and in particular the understanding of quantum mechanical tunneling had to be developed.

In 1923 Louis de Broglie introduced a new fundamental hypothesis that particles may also have the characteristics of waves. Erwin Schrödinger expressed this hypothesis 1926 in a definite form which is now known as the Schrödinger wave equation. A large portion of quantum mechanics can be reduced to finding a solution to the Schrödinger equation. The Nobel prize in physics for 1933 had been awarded to Erwin Schrödinger and Paul Dirac. The continuous nonzero nature of the wave equation solution, the wavefunction which represents an electron or particle, implies an ability to penetrate classically forbidden regions and a probability of tunneling from one classically allowed region to another. Fowler and Nordheim, building on the work of Schottky, explained in 1928 the main features of electron emission from cold metals by high external electric fields on the basis of tunneling through a triangular potential barrier. Conclusive experimental evidence for tunneling was found by Leo Esaki in 1957 and by Ivar Giaever in 1960. Esaki's tunnel diode had a large impact on the physics of semiconductors, leading to important developments such as the tunneling spectroscopy, and to increased understanding of tunneling phenomena in solids. Leo Esaki, Ivar Giaever, and Brian Josephson received 1973 the Nobel prize for their work about tunneling in semiconductors, superconductors, and theoretical predictions of the properties of a supercurrent through a tunnel barrier, respectively. The concept of resonant tunneling in double barriers was first introduced by R. Davis and H. Hosack.

In 1951 C. Gorter explained correctly a current suppression at low bias voltage as the cause of Coulomb repulsion. This phenomenon is known today as Coulomb blockade. About ten years later C. Neugebauer and M. Webb, H. Zeller and I. Giaever, and J. Lambe and R. Jaklevic, studying granular films, saw the same current suppression at low bias voltages. But it took another two decades until in 1985 Dimitri Averin and Konstantin Likharev formulated the ‘orthodox' theory of single-electron tunneling, which quantitatively describes important charging effects such as the Coulomb blockade and single-electron tunneling oscillations. Their work was based on the formulation for a specific case of Kulik and Shekhter.

Another crucial development to bring single-electronics where we are today happened before the orthodox theory of single-electron tunneling was developed. This development is again based on the increased understanding of quantum mechanical tunneling. Gerd Binnig and Heinrich Rohrer invented in 1981 the Scanning Tunneling Microscope (STM) in the IBM research lab in Zürich Switzerland, and received five years later in 1986 the Nobel prize in physics for their invention. The STM is not only an indispensable measurement tool to probe the electronic and atomic structure of mater, but can also be used to manufacture artificial structures on the atomic scale. This has been used in several variations to build single-electron devices in the nanoscale and opens the possibility for future manufacturing of room temperature single-electron devices if similar methods can be made faster and can be parallelized for higher throughput.

However, STM based manufacturing methods were not the first one with which single-electron devices were fabricated. Dolan developed the double shadow evaporation process with which Fulton and Dolan built the first single-electron transistor and observed single-electron charging effects. This technique and its variations are still today the most prevalent one to manufacture single-electron devices in metallic material systems (mainly Al/Al₂O₃).

Once the fundamental physical understanding was achieved and at least for a laboratory setting practical manufacturing methods were known, engineering single-electron devices and circuits became more important. One tried to understand the technological implications for digital and analog circuits and a quest for meaningful applications started. With this, the need for analysis tools grew rapidly. The Monte Carlo method seems almost ideal for the simulation of single-electron devices, since one can 'follow' a representative portion of individual electrons which make up the behavior of a structure and calculate their interaction with other electrons and the surrounding. The first who employed the idea of stochastic sampling was probably Buffon, a French mathematician, who in 1768 experimentally determined the value of π by casting a needle on a ruled grid. The theoretical and computational foundation of the Monte Carlo method was laid during and after the Manhattan project. John von Neumann established the mathematical basis for probability density functions, inverse cumulative distribution functions, and pseudorandom number generators. It was however Stanislaw Ulam who, working on modeling various neutron transport problems, realized the importance of the digital computer in the implementation of the Monte Carlo method and who suggested it to John von Neumann. Enrico Fermi was so intrigued by this technique, that he designed a Monte Carlo mechanical trolley, the Fermiac, which he used to follow neutrons along their trajectories. John von Neumann implemented the first Monte Carlo algorithm on a computer, the ENIAC. From then on many applications to this method, particularly in particle physics, were found. In the early days of the Monte Carlo method computational resources limited its application severely and the method was often referred to as the 'method of last resort'. Today with the unbroken exponential growth of processing speed and memory capacity and particularly with the availability of affordable parallel computers and clusters, the Monte Carlo method gained significantly in importance.

In the case of single-electron devices it was N. Bakhvalov et al. who first followed a Monte Carlo approach. E. Ben-Jacob et al. suggested a Master Equation method, as an appropriate technique applicable to single-electron devices. The Monte Carlo method and the direct solution of the Master Equation are today the mainstay of simulation of single-electron devices and circuits. In the last years a third method, the macro-modeling of single-electron devices in SPICE, is employed more frequently. It was first suggested and applied by Fujishima et al.. These first papers which outlined a Monte Carlo method and the direct solution of a Master Equation initiated at several places the development of more general simulation tools and from time to time simulation results were published. However, implementation details, limitations and assumptions of employed simulators were not always obvious. S. Roy explained in more detail his Monte Carlo analysis tools  which were developed from the study of linear arrays of tunnel junctions. The group at New York State University in Stony Brook, headed by Likharev, developed two very influential and publicly available programs, MOSES and SENECA. SENECA focuses especially on cotunneling and how to simulate it correctly and efficiently. It was an important tool to study error processes in the single-electron pump. Several other groups developed around the same time similar tools among them the one I developed.