## Vibrations on a Linear Atomic Chain of length L Na

On a chain of N masses of length L, and connected by springs of constant K, denote the longitudinal displacement of the nth mass from its equilibrium position by un. The differential equation (Newton's Second Law) F ma for the nth mass is md2u dt2 + K(u +i-2u -un_i) 0. (2.4) A traveling wave solution to this equation is Here kna denotes kx 2px k, where k is referred to as the wave number. Substitution of this solution into the difference equation reveals the auxiliary condition mx2 4Ksin2(ka...

## Ws12 [Wnxi Wmx2 Wnx2 WmxiV2 and

5.2 Nano-symmetry, Di-atoms, and Ferromagnets 83 wa(1,2) Wn(Xl) Wm(X2) - Wn(*2) Wm(Xl) V2, (5.5) respectively, are correctly symmetric and anti-symmetric. Fermions The antisymmetric combination WA, equation (5.5), is found to apply to electrons, and to other particles, including protons and neutrons, which are called fermions. By looking at WA in the case m n, one finds WA 0. The wavefunction for two fermions in exactly the same state, is zero This is a statement of the Pauli exclusion...

## Single Electron Transistors

The single electron transistor (SET) is configured with a source and drain like a field-effect transistor (FET), except that the active region, whose potential is controlled by the gate electrode, is so small, and its capacitance is so small, that its capacitive charging energy enforces a single electron rule of occupancy. In the simplest terms, e2 2C > > kBT, and the role of the gate voltage Vg is to adjust the residual charge on the island to the magic value e 2. At this residual charge...

## Electron Spin V2 as a Qubit for a Quantum Computer Quantum Superposition Coherence

Nuclear and electron spins are candidates for a qubit in quantum computing, for reasons given in the introduction to this chapter. As a concrete example of a proton qubit state, the medical Magnetic Resonance Imaging (MRI) apparatus takes majority population of spin-down protons (in water) to a linear combination of down and up, by the application of a p 2 pulse. This generates the a b superposition state, and in this macroscopic case leads to a magnetization freely rotating and tilted at 90...

## Fe3O4 Magnetite and Fe3S4 Greigite Nanoparticles in Magnetotactic Bacteria

Magnetotactic bacteria such as Magnetospirillum magnetotacticum are known to contain linear arrays of antiferromagnetic nanocrystals of Fe3O4 (magnetite) or Fe3S4 (greigite). The arrays are oriented parallel to the long direction of the bacteria. Both minerals are antiferromagnetic, which means that there are two magnetic sublat-tices of opposed but unequal magnetization, leading to a net magnetization in the direction ofthe stronger moment. Figure 6.7 Image of magneto-tactic bacteria, showing...

## Exercises Chapter

Referring to Figure 1.2 if there are 10 million transistors uniformly distributed on a one centimeter square silicon chip, what is the linear size of each unit 2. A contemporary computer chip dissipates 54 Watts on an area of one centimeter square. Assuming that transistor elements in succeeding computer generations require constant power independent of their size (a hypothetical assumption), estimate the power that will be needed for a one centimeter square silicon chip in 5 years. Base your...

## Q

Quantization of angular momentum 49, 50 quantized conductance 209 quantum computing 118 algorithm advantage 225 conceptual questions 225 relation to encryption 225 quantum condition 51 quantum dot fluorescent 5 quantum dots (QDs) 3, 4, 9, 10, 34, 52, 68 coated 11 quantum interference ofelectrons in carbon nanotube FET 212, 213 quantum limit 2 quantum mechanics 1 quantum nanophysics, probabilistic 2 quantum number n 18, 50, 72, 267 quantum of conductance 208 quantum rules 49 quantum sensitivity...

## Viscous Forces Become Dominant for Small Particles in Fluid Media

The motion of a mass in a fluid, such as air or water, eventually changes from iner-tial to diffusive as the mass of the moving object is reduced. Newton's Laws (inertial) are a good starting point for the motions of artillery shells and baseballs, even though these masses move through a viscous medium, the atmosphere. The first corrections for air resistance are usually velocity-dependent drag forces. A completely different approach has to taken for the motion of a falling leaf or for the...

## The Origins of GMR Giant Magnetoresistance Spindependent Scattering of Electrons

Ferromagnetic metals have mobile electrons (carriers) that are released from the d states of the atoms and also mobile electrons that are released from the s states of the same atoms. The conductivity of the metal is the sum of the conductivities of these two types (although usually the conductivity of the s electrons is dominant, because of the lower effective mass, see equation (5.44)). The two types of carriers are suggested in Figure 5.22, with the left panel showing free-electron-like...

## Quantum Nature of the Nanoworld

The particles of matter (electrons, protons, neutrons) provide limits to the smallness of anything composed of chemical matter, which is itself composed of atoms. The rules that these particles obey are different from the rules of macroscopic matter. An understanding of the rules is useful to understand the structure of atoms and chemical matter. An understanding of these rules, and of the relation between the wave and particle natures of light, is also key to understanding deviations from...

## Quantum Technologies Based on Magnetism Electron and Nuclear Spin and Superconductivity 173

8.1 The Stern-Gerlach Experiment Observation of Spin V2 Angular Momentum of the Electron 176 8.2 Two Nuclear Spin Effects MRI (Magnetic Resonance Imaging) and the 21.1 cm Line 177 8.3 Electron Spin V2 as a Qubit for a Quantum Computer Quantum Superposition, Coherence 180 8.4 Hard and Soft Ferromagnets 183 8.5 The Origins of GMR (Giant Magnetoresistance) Spin-dependent Scattering of Electrons 184 8.6 The GMR Spin Valve, a Nanophysical Magnetoresistance Sensor 186 8.7 The Tunnel Valve, a Better...

## Linear Engines on Tracks

The spring and ratchet systems, structurally ill-defined but clearly electrostatic in mechanism, contrast with well defined linear and rotary engines, which move in stepwise fashions, however, with mechanisms that remain unclear. In this discussion we mention the linear motors myosin and kinesin 8 and an example of a rotary motor, suitable for driving a flagellum, in this case F1-adenosine triphosphate synthase. In contrast to the spring systems, the energetics of these linear and rotary motors...

## Josephson Effect and the Superconducting Quantum Interference Detector SQUID

The superconducting wavefunction has a reliable phase, 0, which applies to a macroscopic number of electrons. This regularity is enforced by the pairing phenomenon, which leads to a macroscopic number of pairs of electrons in precisely the same quantum state, analogous to the macroscopic number of photons occupying a single electromagnetic mode in a laser. (The phenomenon is also similar to a single ferromagnetic domain, where all electron spins are parallel.) The reliability of the...

## Ion Channels the Nanotransistors of Biology

The smallest forms of life are bacteria, which are single cells of micrometer size. Cells are enclosed by an impermeable lipid bilayer membrane, the cell wall. This hydrophobic layer is akin to a soap bubble. Lipid cell walls are ubiquitous in all forms of life. Communication from the cell to the extracellular environment is accomplished in part by ion channels, which allow specific ion species to enter or leave the cell. Two specific types of transmembrane protein ion channels are the Ca++...

## The Degenerate Semiconductor

Now consider an extreme case, which will lead to an understanding of a heavily doped, metallic semiconductor, as is typical of the important 2DEG electron gas. Consider pure InAs, for which data in Table 5.2 indicate a bandgap of 0.33 eV, electron mass 0.02, hole mass 0.41 and dielectric constant 14.5. First we estimate Ne at 300 K, assuming for simplicity that the Fermi energy is exactly at the center of the energy gap. We find Ne 2(2pme*kBT h2)3 2 exp -(EG-EF) fcBT 7.165 x 1022 x 1.7 x 10-3...

## Piezoelectric Materials

A piezoelectric material can be characterized, e.g., by the fractional change in its x-dimension per unit electric field in the z-direction. As an example, for the useful and commercially available (polycrystalline) material known as PZT-5H, a quoted value of the coefficient D31 S1 E3 strain in 1-direction electric field in 3-direc-tion -2.74 A V. Here the acronym PZTrefers to Strain is a fractional distortion, so that for a bar of length x L, the strain, DL L, is calculated as D31V't, if the...

## Spin Injection the Johnson Silsbee Effect

Johnson and Silsbee 13 demonstrated experimentally that a non-equilibrium density of spin-polarized electrons can be transferred from a ferromagnet into a nonmagnetic metal, like aluminum, under proper conditions. An important corollary, demonstrated experimentally in their pioneering work, is that the same type of fer-romagnet-insulator-normal metal junction responds to the presence of non-equilib- Figure 8.10 Geometry of the spin injection device 14 . (a) SEM image of device with spacing L of...

## What is the Future of Silicon Computer Technology

The ongoing improvement of silicon computer chips continues, with innovative new device designs appearing frequently. In addition to the International Technology Roadmap for Silicon 1 , a further analysis of the trends is given by Meindl et al. 6 . One of the recent trends has been to build the basic field effect transistor, FET, or MOSFET Metal Oxide Silicon Field Effect Transistor , on a silicon chip with a buried oxide layer BOX . The effect of the BOX is intended to reduce the thickness of...

## C60 Buckyball 05 nm

These molecules are empty spherical shells, containing exactly 60 carbon atoms in five- and six-membered rings. It is essentially a graphene sheet closed onto itself, with pentagons added to allow curvature. The pentagonal and hexagonal benzene rings are located in a similar fashion to pentagonal and hexagonal panels in a soccer ball. These molecules are very stable and fully tie up the four valence electrons of each carbon atom. A simple model of this molecule is shown in Figure 6.2 2 . Figure...

## Ethylene C2H4 Benzene C6H6 and Acetylene C2H2

The carbon molecules considered so far have involved only single bonds, based simply on the covalent bonding found in molecular hydrogen that was discussed in Chapter 5, acting on wavefunctions as listed in Table 4.1. Carbon and other atoms also sometimes form double and triple covalent bonds. The linear acetylene molecule C2H2 is formed by carbon atoms which each share three of their four valence electrons with each other, a structure called a triple bond. Thus, six valence electrons are...

## Scaling the xylophone

The familiar xylophone produces musical sounds when its keys a linear array of rectangular bars of dimensions a x b x c, with progressively longer key lengths c producing lower audio frequencies are struck by a mallet and go into transverse vibration perpendicular to the smallest, a, dimension. The traditional middle C in music corresponds to 256 Hz. If the size scale of the xylophone key is reduced to the micrometer scale, as has recently been achieved, using the semiconductor technology, and...