Lecture 1 – Introductory concepts

Introduction

mésos” in ancient Greek means middle, intermediate and the word mesoscopic is coined by Van Kampen in 1981 to define the physics between the microscopic and the macroscopic length scales. Modern solid state physics problems are much related to the behaviour of electrons at length scales in between the macroscopic and the microscopic limits. These limits are defined by a certain correlation length \(\eta\), in most of the times comparable to a microscopic lengths. However, \(\eta\) can become very large at low temperatures and in the vicinity of a second order phase transition. Under these conditions, a new physics opens up and lead to a plethora of novel phenomena such as quantum Hall effect,  universal conductance fluctuations, Schubnikov – de Haas oscillations, superconducting quantum interference etc.

In this course, we assume that the student has a basic understanding about solid state physics concepts, quantum mechanics and other relevant physics. As this is an introductory course, the main motivation of this course is to get the student familiar with the concepts of the modern solid state physics. The course will focus on the theoretical aspects of the topic, however, experimental realization of the mesoscopic devices and experiments will be explained as well.

I also would like to stress out that this course is not just about some obscure theories that no one will ever test or feel the need to apply in ant sorts of applications in the near future. On the contrary, the topics that will be discussed here are the most important discoveries of the “nanotechnology” and their applications in real-life problems has already been achieved. Thus, for any graduate (or an advanced undergraduate student) who is willing to pursue “nano” electronics, this course will provide an essential basis.

References

There are three textbooks that I use when I prepare the lessons:

  1. Introduction to Mesoscopic Physics, Yoseph Imry
  2. Mesoscopic Electronics in Solid State Nanostructures, Thomas Heinzel
  3. Mesoscopic Physics and Electronics, T. Ando, Y. Arakawa, K. Furuya, S. Komiyama, H. Nakashima

I will not cite this work in-line however most of the information that I present will be a mixture from these three books.

What Characterizes the Mesoscopic Regime?

There are several length scales that can be used to define a mesoscopic system and the length scale is determined by the particular quantity under study. Let’s think of a conductive crystalline material. An accurate description of the system will involve a quantum mechanical treatment of all the electrons and the evolution of wavefunction in the potential created by the ion cores. Moreover defects and impurities within the crystal will effect the electronic transport. However this system is practically impossible to deal with. Thus we have methods to simplify. Yet, even the simplified picture will be too complex for what we are trying to achieve here. Rather lets assume a much more simple picture where we can gain an understanding of how mesoscopic phenomena emerges. Lets take a slab of material with two leads connected to the ends of the longer side of length \(L\) and investigate the motion of a single electron under the influence of an applied external electric field. Another assumption we will make is that all the ion cores and the other electrons in the system are a source of occasional scattering to the electron under investigation. Thus we can write an expression to describe the motion of an electron under the external electric field in the presence of occasional scatterrers: \(m\ddot x -m\dot x / \tau = qE\). At this stage we are after a general expression that governs the motion of the electrons and as the average time between collisions is \(\tau\), we can also say that after some transient time, the electron will reach a terminal average velocity which we name as the drift velocity \(v_d\). Thus the expression we wrote above will be simplified to  \(v_d  = \tau eE/m\). Here, \( \tau e/m\) is defined as the mobility and denoted by \(\mu\). This expression has the essence of the Ohm’s law. Here, we can extract another useful quantity, diffusion length, \(l_d=v_d \tau\). This is the average length an electron can travel through the material without experiencing any collisions. Now for the sake of completeness, I will derive the relation between the electric field and the current density. A side not, I refrained from using the vectors in the equations to keep everything clean and simple, however in a more serious approach the electric field, instantaneous velocity and the current density should be taken as vector quantities. Now, it is straightforward to show that the current density \(J= nev_d\). If we swap \(v_d\) in the previous expression with the relation with the electric field we obtain \( J = ne\mu E\). This is the Ohm’s law. Here \(\sigma= ne\mu\) is the electrical conductivity of the material.

This is a macroscopic treatment of the classical conductivity and named as the Drude model. Now we are at a point to pose this question, what happens when \(l_d \approx L\)? This requires a quantum treatment of the problem as now all the assumptions we made regarding the nature of the electron and the scatterrers are no longer valid.

A Reminder of Quantum Mechanics and Solid State Physics

I asserted that when \(l_d \approx L\) we need a quantum treatment of the problem, yet I haven’t stated why this is a necessity. To have a fundamental understanding regarding how quantum phenomena emerges we should take a look at some of the early problems that the classical mechanics have faced.

1. Fraunhofer Lines

 The spectrum given (c.f. https://en.wikipedia.org/wiki/Fraunhofer_lines) is the spectrum of the visible light from the sun. Notice that there are sharp lines missing in the spectrum. Each missing line corresponds to absorption of a certain element in the sun, earth’s atmosphere and in the space and quantum understanding of atoms is required for a successful explanation of these missing lines.

2. Black Body Radiation

“Black body” is an object whose radiation only depends on the temperature of the object. The spectrum given (c.f. https://en.wikipedia.org/wiki/Black-body_radiation) here shows the actual spectra for different temperatures and the calculation by the classical theory. This “ultraviolet catastrophe” can be resolved by quantization (discretization) of energy in multiples of the Planck’s constant \(h= 6.626 x 10^{-34} \) J.s and the frequency of the emitted electromagnetic wave  \(f \). A rigorous discussion of how Raleigh-Jeans law fails and the derivation of the Planck’s law is not within the context of this course. If you are interested in you may refer to relevant articles in Wikipedia.

3. Photoelectric Effect & Wave-particle duality of the light

Another important discovery that leads to the quantum mechanics is explanation of the photoelectric effect (PE) by Einstein. PE is emission of electrons from a metal plate under electromagnetic radiation, first observed by Hertz. A successful explanation requires quantization of the electromagnetic waves, i.e. giving electromagnetic waves a particle nature. This explanation shows that just like the energy released in black body the electromagnetic waves are quantized.

4. Problems with the Rutherford Atom

Roughly around the same time, experiments by J.J. Thompson and several years later by Rutherford illuminated the structure of the atom. The major problem here was that the positively charged nucleus at the center of the atom with negatively charged particle rotating around it should be unstable according to the classical mechanics as the charged particle orbits around the nucleus performs an accelerated motion. Thus, it should radiate its energy to spiral into the nucleus. Solution to this problem has been proposed by Niels Bohr. The idea he had was quantizing the angular momentum of the electrons orbiting around the nucleus: \(mvr=n\hbar\) where \(n=1,2,3,…\). This quantization ensures that a specific energy is required to excite an electron to a higher energy state or to reduce a lower energy state.

These four solutions to the fundamental problems cannot be explained using the classical mechanics and indicates a need for a new theory that is more fundamental than the classical mechanics. In our next lesson we will study the basic quantum mechanics that will be useful in our discussion of mesoscopic phenomena.