Explaining Alterations in Nanometric Circuit Electron Flow by Using Quantum Dots

Transistors with the ability to operate with an electrical current including the transit of a single electron at every point in time are the next hot topic for study in the information technology field.

The device can relate the 0/1 binary with electron transit or non-transit to dramatically enhance the usage of space and minimize power consumption in futuristic computers. Although this feature is economically not possible yet, it already prevails in the lab.

In 2015, when an experiment was performed by using such a device at ETH Zürich, the Swiss Federal Institute of Technology in Zurich, there were theoretical problems. These problems have now been overcome by a research team comprising of Luis Gregório Dias da Silva (University of São Paulo, USP, Brazil), Caio Lewenkopf (Fluminense Federal University, Brazil), Edson Vernek (USP), Gerson Ferreira Júnior (Federal University of Uberlândia, Brazil), and Sergio Ulloa (Ohio University, USA).

The team has published a paper in the journal Physical Review Letters. The study was supported by the Sao Paulo Research Foundation, FAPESP.

The object studied was a nanometric circuit in which electron transmission from one part to another undergoes quantum effects owing to its very small scale. Among other things, this means the electrons in transit display both properties typical of particles and properties characteristic of waves.

Luis Gregório Dias da Silva (University of São Paulo, USP, Brazil).

From the 1990s, circuits with just one quantum dot have been investigated, where a quantum dot is a tiny region with dimensions of a few tens of nanometers in which the electrons are enclosed. Yet, such a device not only comprises one quantum dot but also a cavity, that is, a marginally larger region including a curved edge that acts as a mirror. The electrons escape from the quantum dot, bounce back from the cavity’s curved surface, and are imprisoned for a short time.

In the investigated nanometric circuit, electrons have the ability to directly pass from the source to the drain through the dot, or they can travel from the source to the cavity, from which they are reflected. Then, they follow a tedious path toward the drain.

If the cavity is weakly connected to the dot, the conductance exhibits a pattern where there is a peak in the conductance values every time an electron transits. If the cavity is strongly connected, the peaks transform into troughs.

The researchers in Switzerland couldn’t understand the transition from peaks to troughs, and this was the problem we set out to study and succeeded in solving,” stated Dias da Silva. “Our theoretical calculations for the two regimes—weak coupling and strong coupling—showed qualitative behaviors corresponding exactly to that observed in the experiment. So, we offer a very natural explanation for what the experiment detected.”

The functioning achieved only by using the quantum dot, and not the cavity, can be easily perceived by means of the energy quantization concept.

Given the quantum nature of energy, the energy levels accessible to electrons aren’t continuous but discrete,” stated Dias da Silva. “Variations in electrostatic potential enable these levels to be aligned with the energy of the electron that tries to cross the dot. When alignment occurs, it’s as if a door opened in the repelling wall, made up of negative charges, and the electron is highly likely to get through.”

The transit of the electron passage results in a conductance peak, following which there is again a dip in the conductance value owing to the electrostatic barrier effect, or the Coulomb blockade, named after the French physicist Charles-Augustin de Coulomb (1736-1806), a pioneering scientist in the field of electrostatics.

Because energy is quantized, the variation in potential enables other alignments to be obtained and other doors to be opened,” stated Dias da Silva. “The admeasurement of variation in conductance as a function of variation in potential therefore shows a succession of peaks separated by troughs. Each peak corresponds to the tunneling of an electron through the barrier.”

The circumstances become more complex when the cavity is included, as besides taking place when the doors open, an interference effect also takes place because of the wave-like action of an electron. Taking all aspects into account, the phenomenon is akin to the effect of propagation of mechanical waves across the water surface in a swimming pool: interference occurs between incoming and outgoing waves, with destructive or constructive effects.

The wave from the electron that rebounds from the surface of the cavity interferes with the wave from the electron that travels from the quantum dot to the drain. The interference can be constructive or destructive. Destructive interference produces the troughs in the values of conductance of the circuit. Our paper shows this consistently, the study embodies a theoretical step forward by extending the scope of application of the mathematical expression available previously, known as the Meir-Wingreen formula, to calculate electrical conductance in quantum systems. This equation, first established in 1992 by physicists Yigal Meir and Ned Wingreen, was applied only to the simplest situation involving a system without a cavity. The introduction of a cavity greatly increases the number of possible transitions from source to drain. We extend or generalize the Meir-Wingreen formula so as to cover more complex instances of the phenomenon. This generalization enables us to theoretically explain the experimental results obtained by the Swiss group.

Luis Gregório Dias da Silva (University of São Paulo, USP, Brazil).

Low temperatures and commercial use

Dias da Silva underscored the technique adopted by the team at ETH Zürich to develop the device. “They’ve perfected lithographic techniques to the point where they can define structures with a few nanometers of precision,” stated Dias da Silva. “Moreover, they make the contacts work. The key element in the circuit is the cavity, which is about 1 micron in length, or roughly one-hundredth the diameter of a human hair. In addition, the sample semiconductor [gallium arsenide] on which the structure is built is of first-rate quality.”

Each process is conducted at extremely low temperatures of less than 4 K. Lower temperatures of this order, achieved by performing refrigeration using liquid helium, have become standard in experiments of this nature.

If the temperature is very low, in the range of a few tens of thousandths of a Kelvin, something a little more exotic may happen: I refer to the Kondo effect [first described by Japanese theoretical physicist Jun Kondo], whose signature is increased conductance in some of the troughs,” stated Dias da Silva.

Performing experiments at extremely lower temperatures is one of the barriers in enabling commercial application of the device; however, it should not stop the application of the device in industrial research. Hence, the proposed study, though it is actually theoretical, is not completely without prospective applications. The study does not relate to quantum computing but leveraging quantum effects within a framework of classical circuits.

Classical circuits, which have various technological applications in day-to-day devices, are relatively complicated, but the laws that enable us to calculate currents in each part of the circuit are well-known and easy to apply. In the case of circuits in which quantum mechanics predominates, much investigation still needs to be done to find out how currents behave. These circuits will mainly be applied in electronics, but we still have much to learn in terms of basic physics.

Luis Gregório Dias da Silva (University of São Paulo, USP, Brazil).

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