Introduction

Since its birth, electronics always strived for miniaturization, costantly trying to scale down the current technology and, whenever physical limits oare approached, searching for new technologies enabling it to achieve an even greater reduction in the dimension of the circuits.

At the present day, one of such limits is near. The dimensions of the transistors for the CMOS technology have been scaled down so much that the scaling process will have to stop in a few years, and completely new approaches will have to be developed to remain on track with Moore’s law.

As a result of the downscaling afforded by present technologies, we have finally reached a landmark step: we are moving from the macroscopic to the mesoscopic realm. We are leaving the field of the microelectronics and entering that of the nanoelectronics.

This thesis focuses specifically on some aspects of transport in meso- scopic systems.

In this new framework we need to abandon the notions of classical me- chanics, since we have stepped into, the world quantum world. We can still describe a system in terms of conductance, for example, but Ohm’s law does not hold any longer in most cases. Resistence becomes a global property

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of the entire structure we are studying and can not be defined as a local property.

In the first chapter of this work we will introduce the field of mesoscopic transport. After a brief introduction about heterostructures, one of the most common material systems used for the fabrication of mesoscopic devices, we will discuss the hypoteses leading to the approximation we use to study the physics of mesoscopic systems.

In the final part of chapter 1 we will introduce the notion of conductance in mesoscopic systems: from where it arises, how can we evaluate it. We will also briefly discuss shot noise, since it, too, is related to the transmission matrix.

Then we will introduce the technique of conductance spetroscopy, which is used to map current throughout the device under study. We will relate the transmission matrix introduced earlier for conductance and shot noise to a new concept: the Green’s function of the system.

The Green’s functions represent the topic of the second chapter.

We will start by introducing the concept of a Green’s function from a mathematical point of view, and then explain how it is interpreted when ap- plied to solid state physic. We will then introduce the tight-binding model,

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or rather the way we discretize our problem so that it can be solved numeri- cally, and we will proceed to the calculation of the Hamiltonian matrix that will be used in our simulations.

Chapter 3 will introduce practical ways to use the Green’s function tech- nique to solve complex problems on real structures. We will introduce the recursive Green’s function approach, which allows us to subdivide the trans- port problem into simpler ones.

We will complete the discussion of the practical methods to use Green’s functions by introducing the way we apply the probe on the system in order to perform our simulatef experiments of conductance spectroscopy. This will allow us to discuss an interesting approach proposed by Metalidis and Bruno.

In the fourth chapter we will finally move from the realm of theory to that of practice. A number of possible algorithms will be proposed, and examined, so that their computational complexity, their Cost or “speed eval- uation” function, can be compared in the final chapter. We will also discuss other approaches we can possibly follow in order to further increase the speed of our code.

Since before analyzing a realistic structure, characterized by soft-wall potential and defects, it is customary to analyze the same system in an ideal case with no defects and with hard-wall potentials, our research on how to

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improve performances did includes also the ideal case, along with the real one.

The simulation of an ideal system does not provide quantitatively reli- able results, nevertheless it can give us a good “feeling” of its operation. It has also one major advantage: it is faster (potentially much faster, depending on the specifics of the structures under analysis).

Finally, in the fifth chapter we will compare the algorithms discussed in the fourth chapter, and decide which one to use in the final version of the code. Information about the code will than be provided in terms of a flow chart highlighting all the major points.

Finally, we will present the results of the application of the code to some test cases, discussing the speed-up achieved with respect to an existing reference code.

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