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<h3>Preface</h3>
<p> In this series of posts, inspired by a lecture course given by Dr. Phil Buckle, we will develop an understanding of the concepts that underpin the exciting field of <em>low dimensional semiconductor devices.</em> While some of this treatise is mathematical, it is my hope that this work will bring value to readers of all abilities. </p>
<p>http://i.imgur.com/t9ZQief.jpg</p>
<h2>Quantum Confinement</h2>
<p>We begin by considering the notion of quantum confinement. </p>
<p>http://i.imgur.com/49l5CpK.png</p>
<p><strong>FIG. 1:</strong> Different degrees of quantum confinement</p>
<p>2D confinment, 1D confinement and 0D confinement characterise quantum wells, quantum wires, and quantum dots, respectively. This can be a confusing concept, so it is important to remember that the dimensionality of this description refers to the dimension the system is <em>confined to. </em>This convention is used extensively throughout the literature, so it is important that this is understood. </p>
<h2>Surface to Volume Ratio</h2>
<ul>
<li>This is extremal for nanostructures. At smaller length scales, one would be considering just a system of atoms. This gives rise to the term <em>extreme functional material</em>, to which nanostructures are often referred to.</li>
<li>The pros of nanostructure materials are that they make good catalysts, due to the high surface to volume ratio, as well as being useful for application in plasmonic and coatings.</li>
<li>The cons are that properties of the material may be dominated by undesirable surface effects, for example, carrier loss in optical systems. </li>
</ul>
<h3>Quantum Well Confinement</h3>
<p>For a quantum well of length l and width a, where l>>a, the surface to volume ratio is given by</p>
<p>http://i.imgur.com/InKhUrL.png</p>
<h3>Quantum Wire Confinement</h3>
<p>For a quantum wire of length l and radius r, where l>>>>r, the surface to volume ratio is given by</p>
<p>http://i.imgur.com/6oXmEiP.png</p>
<h3>Quantum Dot Confinement</h3>
<p>If we model a quantum dot as a cube of dimension a , then the surface to volume ratio is given by </p>
<p>http://i.imgur.com/8M2zW9X.png</p>
<p>while if we model a quantum dot as a sphere of radius r, the surface to volume ratio is given by </p>
<p>http://i.imgur.com/Q0VIdGH.png</p>
<p><strong>In each case, the inverse power law with nanoscale dimension is critically controlling the surface to volume ratio.</strong></p>
<h2><strong>Number of Atoms</strong></h2>
<p>So how many atoms do we expect? </p>
<p>Let's consider a 10 nm, gold nanoparticle. Now, this has an <a href="https://en.wikipedia.org/wiki/Cubic_crystal_system">FCC lattice structure</a> (4 atoms/unit cell), with a unit cell size of ~ 0.408 nm. We determine the number of unit cells, N by dividing taking the ratio of the sphere volume to the unit cell volume </p>
<p>http://i.imgur.com/VYAHVtA.png</p>
<p>which leads us to a value of ~247,000 atoms.</p>
<p>Now, as the size of the nanoparticle decreases, the number of consituent atoms decreases dramatically: a 5 nm gold nanoparticle, for example, constitutes ~31,000 atoms. </p>
<p><strong>The majority of the atoms in a given nanostructure reside at the surface. </strong></p>
<h2><strong>Length Scales</strong></h2>
<p>Planck was the first to put forward the relation between the momentum of a particle and its corresponding wavelength, understood through wave-particle duality</p>
<p>http://i.imgur.com/SDIsTNL.png</p>
<p>The de Broglie wavelength is what we refer to when we talk about a quantum wavefunction. A free electron has a de Broglie wavelength of ~0.24 nm.</p>
<p>As a fun example, if we consider a person of mass 70 kg, walking at 1 m/s directly into a wall - their de Broglie wavelength is ~9.5 x10^-36 m. This means they will walk right into the wall (ouch) rather than tunnelling through it!</p>
<p>The de Broglie wavelength of a semiconductor electron, however is ~73 nm, and for a semiconductor hole is ~18 nm. <em>This is good news. </em></p>
<p>In the next post, we will discuss this further. </p>
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