Quantum Physics Of Atoms And Materials
9Quantum Physics of Atomsand MaterialsThe first postulate enunciates the existence of stationary states of an atomic system. The second postulate states that the transition of the system from one stationary state to another is accompanied by the emission of one quantum of radiation.Niels Bohr(1913)Niels Bohr, Danish physicistwho in 1913 discovered thequantum model of the atom andthe relation of an atom’s changein energy to the light emitted orabsorbed by it.Physicists Dawn Meekhof and Steve Jefferts with their atomicclock, which would neither gain nor lose 1 sec in 60 million years!Their clock uses the quantum properties of cesium atoms to provide its extreme stability. (Courtesy of the National Institute ofStandards and Technology. Copyright Geoffrey Wheeler, 1999.)299TAF-K10173-08-1107-009.indd 2994/24/09 9:30:05 PM
300The Silicon Web: Physics for the Internet Age9.1 ATOMS, CRYSTALS, AND COMPUTERSModern computers are made with semiconductor-based electronic circuits, which canact as switches, enabling binary data to be stored and logic operations to be performed.Semiconductor-based circuits are made of silicon crystals with small amounts ofother elements added to control their electrical properties. Engineers invented moderncomputers using an understanding of how electrons flow in crystals. This required anunderstanding of the basic properties of atoms and how they combine to form crystals.Gaining a proper understanding of atoms and crystals requires us to learn more aboutthe properties of electrons.In Chapter 5, we discussed the origins of magnetic forces, which, according toAmpère’s law, arise solely from moving electric charges. We learned that magneticdata-recording materials such as iron contain many tiny magnetic regions, calleddomains. If these domains are all aligned and kept in a common direction, the ironbecomes a magnet, allowing us to store a data bit value. A question remained, however:Why is each microscopic domain a permanent magnet itself? The answer lies in atomicphysics, namely in the motion of electrons within atoms.Between 1900 and 1930, there was a rapid and incredibly important advance inscientists’ understanding of the properties and behavior of electrons. They discovered,through careful analysis of experiments, a set of quantum physics principles describingthe behavior of microscopic objects—in particular, electrons in atoms. They found thatthe physics rules for the behavior of microscopic objects are in some ways radicallydifferent from those expounded in the nineteenth century by Newton to explain thebehavior of large objects such as baseballs or the Moon. They found that microscopicobjects obey different laws of motion than do macroscopic ones. This discovery rathershocked them, so much so that some of the founding fathers of the then-new principles—including Einstein—never completely accepted them as correct. Nevertheless, physicists persisted and confirmed that the then-new quantum principles are indeed correct.From the results of many experiments, combined with clever mathematics, physicistsdeveloped a theory that allows us to understand and predict electrons’ properties veryaccurately. Without using mathematics, we can state the main principles in words andillustrate them using pictures. This will allow us to build up a set of rules and guidelines that provide a mental picture of how electrons behave in atoms.We will learn how to read the Periodic Table of the Elements, which summarizesthe structure and properties of atoms—the building blocks of matter. We will see howatoms combine to form crystals, and how the atomic structure of each type of crystaldetermines its electrical properties, that is, whether it is a good electric-current conductor, insulator, or in between. In the following chapters, we will develop models forthe operation of semiconductor devices and see how semiconductor computer logicworks.To start at the beginning, let us consider the surprising properties of electrons, andhow their behavior leads to understanding the structure of atoms.9.2THE QUANTUM NATURE OFELECTRONS AND ATOMSBefore we discuss the structure and behavior of atoms, we need to review a few descriptive facts. A proton is a tiny object having a small mass and positive ( ) electric charge. A neutron is a tiny neutral object (zero electric charge) having approximatelythe same mass as a proton.TAF-K10173-08-1107-009.indd 3004/24/09 9:30:07 PM
Quantum Physics of Atoms and Materials301 An electron is an even tinier object having negative (–) charge and mass about1/(2000) that of a proton or neutron. A nucleus is made of protons and neutrons bound tightly together by so-callednuclear forces, which we will not discuss in this text. An atom consists of a nucleus and one or more electrons moving around it. An element is a substance made of a single kind of atom.As we discussed in Chapter 5, the net charge of an object equals the sum of thecharges of all particles making up the object. This means, for example, that the netcharge on a nucleus equals the number of protons in that nucleus, because the otherparticles in the nucleus—the neutrons—have zero charge. Normally, atoms are neutral:they have zero net charge. This means that the number of electrons surrounding thenucleus equals the number of protons in the nucleus.Many of us have a mental picture of an atom—a small, hard nucleus at the center, with electrons orbiting around the nucleus like wee planets orbiting around a tinysun. We can think crudely of the electrons as particles moving in an orbit around thenucleus, as shown in Figure 9.1 for the case of a helium atom. This picture of the atomis somewhat naive and is not entirely correct. In fact, no one has a truly satisfactorymental picture of exactly how electrons behave, although we do have a good mathematical theory describing their behavior.By studying how atoms absorb and emit light of different colors, and how electronstravel through electric and magnetic fields, scientists discovered after 1900 that atomsand electrons do not obey the classical principles of mechanics that were put forth byNewton and described in Chapter 3. Scientists of the time could not understand how thebasic “laws of motion,” which were so successful in describing the motions of typicallarge objects, could fail when applied to atoms.Perhaps, with hindsight, it is not so surprising that electrons don’t follow Newton’slaws. The objects that we can see directly—those at the human scale—do obey Newton’slaws. By human scale we mean the scale of baseballs, racing cars, and space shuttles.Newton’s theory is extremely accurate for large, slow-moving objects, but fails whenthe object is on the scale of electrons and atoms; that is, roughly a billion–billion timesless massive than a baseball. Newton did not take into account the behavior of such tinyobjects when he formulated his laws, because at that time nothing was known aboutsuch objects. Any successful theory of atoms must take such behaviors into account.One of the limitations of the naive Newtonian view of the atom was the faulty assumption that an electron is actually a particle, as illustrated by the dots in Figure 9.1. Whatis meant here by “particle”? A particle is an entity or thing with mass, a definite location in space, and a definite speed. Surprisingly, this description does not apply to electrons. It is not simply that we lack information about where an electron is at a particularmoment. Rather, the very concepts of location and speed are not strictly appropriate toelectrons. It is as if the electron is spread or smeared throughout some region in space,rather than being at a specific place. This is one of the mysterious properties referredElectronElectronNucleusFIGURE 9.1 Naïve picture of a helium atom, showing two electrons orbiting around a nucleus, comprised of two protons (black) and two neutrons (gray). The drawing of the atom is not drawn to scale.TAF-K10173-08-1107-009.indd 3014/24/09 9:30:07 PM
302The Silicon Web: Physics for the Internet Ageto as the “quantum nature of electrons,” which distinguishes them from the classicalconcept of particles used by Newton and his followers. We can crudely represent thespread-out nature of electrons by drawing a fuzzy region as in Figure 9.2.Although we need to keep this spread-out picture in mind, it is cumbersome to drawit in this way, especially when there are many fuzzy orbits that need to be drawn. Sowe will use the simpler style of drawing shown in Figure 9.2a to symbolically representthe more accurate picture in Figure 9.2b.We should wonder what the spread-out picture of an electron really represents. Themathematics of quantum theory, which we will not study here, shows that a spread-outelectron behaves in some ways like a wave. A wave—such as waves in the ocean—isnot located at a particular position. A water wave is made of many separate water (H2O)molecules, moving in an organized pattern. In contrast, the electron wave is associatedwith only one electron. We believe in the validity of this wavelike description becausethe mathematics that goes along with it is in excellent agreement with all of the experimental observations on electrons.This description of an electron might seem strange, but physicists have an interpretation of the meaning of the electron’s wave. The wave’s amplitude in a region of spacetells us the likelihood that the electron will be found in that region. In Figure 9.2b,the darker shaded regions are the places with higher likelihood for the electron to belocated. Before we make the measurement, the electron is not at a definite location, butthe very act of measuring causes the electron to appear at a definite location.To further develop the water-wave analogy for the electron in an atom, consider thesurface of water in a drinking cup. The wave is confined within the cup. The patternillustrated in Figure 9.3 is a circular wave, rotating counterclockwise as time goeson. In the example shown, there are eight wave peaks around the edge of the circular(a)(b)FIGURE 9.2 (a) Naïve classical picture of an electron orbit as a localized particle traveling around alocalized path. (b) Quantum picture of an electron orbit as a spread out region in space. The darker theshading, the more likely it is to find the electron at that location.FIGURE 9.3 Frames (left to right) showing a rotating circular wave, in top view and side view. Thediamond labels a particular spot on the wave, showing how it rotates in time. In this simple model of anelectron’s wave, the likelihood that the electron is in some region is highest at the edges of the circularregion, where the amplitude is greatest.TAF-K10173-08-1107-009.indd 3024/24/09 9:30:08 PM
Quantum Physics of Atoms and Materials303pattern. This means that the wavelength along the edge equals one-eighth of the circumference of the circular edge of the pattern.THINK AGAINWhen you think of a wave, such as a water wave, you usually think of manyparticles (H2O molecules) moving in an organized pattern. However, thewave describing an electron corresponds to only a single electron. This isvery different from the idea of a wave in classical physics.An analogy that is simpler to visualize is that of a water wave traveling around a circular canal, as in Figure 9.4. In this example, the wave travels around the canal in theclockwise direction and has 16 wavelengths fitting precisely around the circular lengthof the canal. This leads to constructive interference of the wave when it goes aroundonce and meets up with its “tail.” This reinforces and makes a stable wave.The condition for stability of an electron wave is shown in Figure 9.5. For a wavemoving in a circular path to be stable, there must be an integer number of wavelengths exactly fitting around the edge circumference. If instead the wavelengthequaled, for example, 1/(8.5) of the edge circumference, as shown in the middle ofFigure 9.5, the wave would not constructively reinforce itself; rather it would tendto cancel, leading to an unstable wave. This means that only certain discrete wavelengths are allowed for stable circular waves of a given circumference. (Discretemeans distinct or unconnected.) The figure also shows a wave with 20 wavelengthsfitting around a somewhat larger circumference; this is also a stable wave.WavelengthFIGURE 9.4 A water wave traveling around a circular canal. The circumference of the canal must equalan integer number of wavelengths (in this example, 16), otherwise the wave cannot be continuous and stable.WavelengthWavelengthWavelengthFIGURE 9.5 Constructive interference of electron waves. The circumference of the edge of an orbitmust be an integer number of electron wavelengths, otherwise the wave cannot be continuous and stable.TAF-K10173-08-1107-009.indd 3034/24/09 9:30:09 PM
304The Silicon Web: Physics for the Internet AgeNiels Bohr, in the chapter-opening quote, called these stable conditions stationarystates. We refer to the discreteness of wavelengths by saying that the values of the wavelengths are “quantized.” This is the origin of the term quantum physics. The quantizednature of a wave’s wavelength is a result of its being confined to a small region—in thewater case, the region of the cup. In the case of atoms, the electron is confined to thesmall region around the nucleus.The electron’s wave has a frequency as well as a wavelength. For the simple modelin Figure 9.5, the wave’s frequency could be observed by sitting at a fixed point onthe outer edge and counting the number of oscillations of the passing wave’s displacement during a certain time interval. As for any wave, a decreased wavelengthmeans an increased frequency, although the precise relation depends on the typeof wave and the shape of the small volume to which it is confined. Because theelectron’s wavelength in an atom is quantized, its possible frequency values are alsoquantized.According to quantum theory, an electron’s wavelike motion determines its energy.When an electron is confined to the volume of an atom, this motion is quantized, andtherefore the electron’s energy is quantized. This means that when an electron is confined to the volume of an atom, its energy can take on only certain discrete values.This behavior is quite unlike a moon orbiting around a planet. Such a moon can haveany energy as it flies, depending on how fast it moves; that is, the energy of an orbitingmoon is not quantized. The mathematics of quantum theory makes it possible to createaccurate pictures of the electron’s wave within a hydrogen atom. A few examples ofelectron waves of different energies are shown in Figure 9.6.The quantization of an electron’s energy, first proposed by Niels Bohr in 1913, isa remarkable property, totally outside the realm of classical, Newtonian physics. Itsdiscovery led to a revolution in our understanding of the nature of atoms, molecules,and crystals and paved the way for developing computer technology. It was arrivedat—not by purely intellectual reasoning—but by thinking hard about how to understand the results of experiments carried out around 1900. Next we review some of thoseexperiments.THINK AGAINWhen we say that the electron behaves in a discrete manner, we do not meanthat its position is discrete (that would be more like a particle than a wave).We mean that the electron’s energy is discrete, or quantized.FIGURE 9.6 Realistic computer-generated images of electron waves in a hydrogen atom. (Createdusing Atom in a Box, http://daugerresearch.com/orbitals. With permission of Dauger Research, Inc.)TAF-K10173-08-1107-009.indd 3044/24/09 9:30:11 PM
Quantum Physics of Atoms and Materials3059.3 THE EXPERIMENTS BEHIND QUANTUM THEORYHow do we know that these claims about electron behavior are true? Three crucialexperiments that were carried out around 1900 paved the way for the discovery of thequantum nature of electrons and other particles.9.3.1The Spectrum of Light Emitted by a Hot ObjectWhen a piece of any material, such as a metal light-bulb filament, is heated to a hightemperature, it glows and emits light. When it is not very hot, it emits mostly red andinfrared light. The hotter it is, the more yellow and blue is the light. When heated tovery high temperatures, it emits light of all colors, so it looks white. Light can be analyzed for the different colors it contains, using a prism or other device to spread out thecolors into a spectrum. As shown in Figure 9.7, light from an incandescent light bulb ispassed through a narrow slit in an opaque screen to make a narrow beam of light, andthen is spread out by a prism. A smooth, continuous spectrum is observed.In 1900 Max Planck, a German scientist, found that the prediction for this spectrumbased on the theories of Newton and Maxwell did not agree with experiments. It did notcorrectly predict the relative amounts of light at different colors in the spectrum illustrated in Figure 9.7. Those theories predicted blue light that was far too intense relativeto the intensity or brightness of the red light. Resolving this discrepancy led to a revolution in our understanding of the physics of the universe. Max Planck found that hecould alter the theory of Newton and Maxwell by making a radical assumption abouthow light behaves. He hypothesized that light cannot exist in continuous amounts ofenergy, but rather comes only in indivisible, discrete bundles of electromagnetic (EM)energy. We call these energy bundles photons, as will be described in more detail inChapter 12. Planck found that by making this alteration to the old (“classical”) theories,he could derive a formula that accurately predicts the relative intensity of each color inthe spectrum of light emitted by a hot object.Planck was on to something, but he did not know precisely what. In fact, he spent thenext couple of decades trying to wriggle out of his 1900 hypothesis, thinking that it wassimply an accident of the mathematics. That is, he looked hard for an alternative, lessradical explanation for the spectrum of colors that did not use the idea of light energybundles. His hard work failed. Since then, thousands of physicists have looked for convincing alternative theories, but they have also failed. Their failure greatly strengthensROYWavelengthGBVFIGURE 9.7 The spectrum of light from a hot metal filament in an incandescent light bulb is madeup of smooth, continuous bands of colors: red, orange, yellow, green, blue, and violet.TAF-K10173-08-1107-009.indd 3054/24/09 9:30:11 PM
306The Silicon Web: Physics for the Internet Agephysicists’ confidence that no other satisfactory explanation exists. It seems that weare stuck with the idea of photons, that is, the idea that the energy in light comes inlittle indivisible amounts, lumps, or bundles. This means that light cannot simply bedescribed as a wave of EM fields, as we implied in earlier chapters. The classical wavepicture of light is not completely wrong, but it did need to be refined with the quantumtheory.9.3.2Sharp-Line Atomic Lamp SpectraIn contrast to the case of an incandescent light bulb, if we analyze light emitted by anatomic-vapor lamp, such as those in a neon display light or a yellow sodium streetlamp,we observe sharp, discrete lines of color. We discussed atomic-vapor lamps in In-DepthLook 5.2. The experiment showing the discrete lines of color is shown in Figure 9.8.Linelike spectra of this type were observed as early as 1885 by Jakob Balmer, a scienceteacher in a Swiss girls’ school. He found a mathematical formula that matched the pattern of the line positions in the spectrum, but he had no explanation for why this patternarose. This effect was a mystery to scientists at the time. The theories of the time, basedon Newton’s theory and Maxwell’s theory, predicted a smooth, continuous spectrum,such as that seen in Figure 9.7.In 1913 Niels Bohr proposed that these sharp lines of color were associated withlight given off by
Quantum Physics of Atoms and Materials 301 An electron is an even tinier object having negative (–) charge and mass about 1/(2000) that of a proton or neutron. A nucleus is made of protons and neutrons bound tightly together by so-called nuclear forces, which we will not discuss in this text. An atom consists of a nucleus and one or more electrons moving around it.
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