Lab Week 3 Module α2 3.014 Materials Laboratory Instructor: F. Stellacci Nov. 13th – Nov. 17th, 2006 Lab Week 3 – Module α2 Materials as “particle in a box” models: Synthesis & optical study of CdSe quantum dots Instructor: Francesco Stellacci OBJECTIVES  Introduce the particle-wave duality principle  Introduce the concept of quantum mechanical particles  Electrons/photons interactions  Introduce quantization and “particles in a box”  Study the relationship between nanoparticle size and optical properties of CdSe quantum dots  Gain experience in wet chemical synthesis and optical characterization methods Questions At the end of this laboratory experience you should be able to answer the following questions: 1) What is the concept of “box” in quantum mechanics? 2) Why an electron in a box can appear as a colored object? 3) Why the size of a CdSe nanoparticle can determine the optical properties of such a particle?. 1Lab Week 3 Module α2 3.014 Materials Laboratory Instructor: F. Stellacci Nov. 13th – Nov. 17th, 2006 BACKGROUND Particle in a box: a way for visualizing quantum mechanics One of the most immediate consequences of the physical principles of quantum mechanics is the presence of discrete energy levels in real systems. The goal of modules β1, β2, and β3 is to provide students with a visual proof of this event, in order to make the understanding of the coming lectures in 3.012 easier. Let us first refresh some key concepts of modern physics: Particle Wave duality Every physical entity behaves at the same time as a particle and as a wave. Thus one has to describe physical objects as particle and waves at the same time. Light for example can be described as a wave or as a flux of particles called photons. Photons are particles whose energy can be directly related to their wave properties via the relationλυhchE ==, where is the Planck constant; and Jsh 626184.6=υ is the oscillation frequency, while λ is the wavelength. One concept to keep in mind in order to understand the following discussion is that photons interact with other particles is a very peculiar way, in fact these particle appear (via an emission mechanism) and disappear (via an absorption mechanism) commonly. In particular, the interaction of a photon with an electron leads to the disappearance of the photon and to an energy gain for the electron that equals the energy of the photon that disappeared. Similarly photons can “appear” when a particle loses energy. Because of this property we can use photons as very simple and versatile means of providing energy to other particles. Because of this property we can use photons as very simple and versatile means of providing energy to other particles. In particular photon-electrons interactions are the cause of colors in materials. Color theory is a complex science, indeed the color of a material is determined by the various types of interactions that photon and electron have (absorption, reflection, scattering, non-linear absorption and conversion, and emission). In this lab we will try to understand the two main forms of interactions (that happen to be the easiest to understand ☺) absorption and emission. Absorption is the disappearance of a photon of a given energy (i.e. color) with the simultaneous gain of the same amount of energy for one electron in a material. The probability of this event is called the absorption cross section. 2Lab Week 3 Module α2 3.014 Materials Laboratory Instructor: F. Stellacci Nov. 13th – Nov. 17th, 2006 Emission is the appearance of a photon of a given energy (i.e. color) with the simultaneous loss of the same amount of energy for one electron in a material. The probability of this event is called the emission (or fluorescence) quantum yield. In order for both these phenomena to happen the electron need has to be able to “exist” in two different states whose energy difference equals the energy of the absorbed/emitted photons. (This may sound complex, but it is not, also is really nice to see, I promise!) Particle in an infinite well Let us consider now a particle confined in one dimension (x) in between two infinite energy barriers. A physical way of representing such situation is to define a potential E 0 d x ⎪⎩⎪⎨⎧>∞<<<∞=dxdxxxV 000)( (in any potential the origin can be chosen freely, thus the 0 value of V in the region where the particle is confined was chosen for the sake of making the calculation easier. Any other number would be equally acceptable, leading to the same results.) A particle (in this specific case an electron) is described by its wavefunction (t,r)ψ, a function defined in all places in space (r) and time (t). The values of this function are complex numbers and are related to the probability of finding that particle in that given position of space (r) at that given time t. In our case we will concentrate on a one dimensional problem, thus we will substitute r with x, moreover we will assume that the wavefunction can be divided in two parts one that depends on time but not on space and the other that depends on space and not time, ()())(, txtxξϕψ= 3Lab Week 3 Module α2 3.014 Materials Laboratory Instructor: F. Stellacci Nov. 13th – Nov. 17th, 2006 In this case the spatially varying part of the equation has to obey the time independent Schrödinger equation: () () ()() () ()DinxExxVxxmDinEVm1)(23)(222222ϕϕϕϕϕϕ=+∂∂−=+∇−hhrrrr where π2h=h , V is the potential energy of the system and E is the total energy of the particle. The existence, for x>d and x<0, of an infinite potential (i.e. a very large barrier) prevents the particle from ever being in those regions of space. Being the wavefunction a measure of the probability of finding the particle in a region of space, we have to impose ()0=xϕ for x>d and x<0. For 0<x<d the time independent Schrödinger equation becomes (being V(x)=0) () ()xExxmϕϕ=∂∂−2222h This is a classical wave equation whose general solution is ()ikxikxBeAex−+=ϕ with A and B being complex number. In order to prove that this is the correct solution and to find the possible values of k it is enough to substitute this function back in the Schrödinger equation ()() ()()EmkBeAeEBeAemkBeAeEBeAexmikxikxikxikxikxikxikxikx=+=+>−−−+=+∂∂−−−−−2222222222hhh To summarize, by