Quantum effects explain changes in nanometric circuit electron flows
December 13, 2017
By José Tadeu Arantes | Agência FAPESP – Transistors capable of functioning with an electrical current consisting of the passage of a single electron at each point in time are on the horizon for research in the field of information technology. By associating the 0/1 binary with electron transit or non-transit, the device can drastically improve space usage and reduce power consumption in future computers. This option is not yet economically feasible, but it already exists in the laboratory.
An experiment with a device of this kind performed in 2015 at ETH Zürich, the Swiss Federal Institute of Technology in Zurich, raised theoretical issues that have now been resolved by a group of researchers comprising 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 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,” Dias da Silva told Agência FAPESP.
Circuits with merely a quantum dot – a very small region in the range of a few tens of nanometers to which the electrons are confined – have been studied since the 1990s. However, the device in question contains not just a quantum dot but also a cavity – a slightly larger region with a curved edge that functions as a mirror. The electrons leave the quantum dot, rebound from the curved surface of the cavity and are temporarily imprisoned.
Figure A shows the source, drain, quantum dot and cavity. Electrons can transit directly from the source to the drain via the dot, or they can go from the source to the cavity, from which they are reflected, and will then follow a complex path to the drain.
When the cavity is weakly coupled to the dot, the conductance curve displays the shape shown in Figure B, with a peak in conductance values each time an electron passes. When the cavity is strongly coupled, the peaks become troughs, as seen in Figure C.
“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,” Dias da Silva said. “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 behavior shown in Figure B, obtained only with the quantum dot (excluding the cavity), can be easily understood based on the concept of energy quantization.
“Given the quantum nature of energy, the energy levels accessible to electrons aren’t continuous but discrete,” Dias da Silva said. “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 electron’s passage leads to a conductance peak, after which the value of conductance falls again due to the electrostatic barrier effect, also known as the Coulomb blockade, named as a tribute to French physicist Charles-Augustin de Coulomb (1736-1806), a pioneering researcher in electrostatics.
“Because energy is quantized, the variation in potential enables other alignments to be obtained and other doors to be opened,” Dias da Silva said. “The graph 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 situation is more complicated when the cavity is included, as besides occurring with the opening of doors, an interference effect also occurs due to the wave-like behavior of an electron. All things considered, the phenomenon is similar to what happens when mechanical waves propagate across the surface of the water in a swimming pool: There is interference between incoming and outgoing waves, with constructive or destructive 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 Figure C. Our paper shows this consistently,” Dias da Silva said.
“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.”
Low temperatures and commercial use
Dias da Silva highlighted the technology used by the group at ETH Zürich to make the device. “They’ve perfected lithographic techniques to the point where they can define structures with a few nanometers of precision,” he said. “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.”
Everything is done at very low temperatures, i.e., below 4 Kelvin. Such temperatures, obtained by refrigeration with liquid helium, have become standard in experiments of this kind.
“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,” Dias da Silva said.
Operation at very low temperatures is one of the obstacles to the commercial use of the device, but should not prevent its use at the frontier of industrial research. Thus, the study in question, although it is essentially theoretical, is not entirely without potential applications. This question is not a matter of quantum computing, but of leveraging quantum effects in the context 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,” Dias da Silva said.
The article “Conductance and Kondo interference beyond proportional coupling” (https://doi.org/10.1103/PhysRevLett.119.116801) by Luis G. G. V. Dias da Silva, Caio H. Lewenkopf, Edson Vernek, Gerson J. Ferreira and Sergio E. Ulloa can be retrieved at the following address: journals.aps.org/prl/abstract/10.1103/PhysRevLett.119.116801.
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