Introduction to nanotechnology

introduction to micro and nanotechnology and introduction to nanoelectronics science nanotechnology engineering and applications
AbbieBenson Profile Pic
AbbieBenson,United States,Professional
Published Date:13-07-2017
Your Website URL(Optional)
Introduction to nanotechnology Henrik Bruus MIC – Department of Micro and Nanotechnology Technical University of Denmark Lyngby, spring 2004iiPreface In the spring 2002 MIC launched a new fourth semester course at the Technical University of Denmark (course no. 33320, 10 ECTS) to provide a general and broad introduction to the multi-disciplinary field of nanotechnology. The number of students attending the course has grown steadily from 24 in 2002, to 35 the following year and now more than 50 in 2004. Based on the feed-back from the students I have changed part of the course and expanded the lecture notes. The aim of the course remains the same. It is intended for students who have completed three semesters in any engineering or science study programme at college level. During the course the students will be introduced to many fascinating phenomena on the nanometer scale, and they will hopefully acquire basic knowledge of the theoretical concepts and experimental techniques behind the recent vastly improved ability to observe, fabricate and manipulate individual structures on the nanometer scale. The first part of the course, which is covered by these lecture notes, is an introduction to the top-down approach of microelectronics and micromechanics. Here selectred topics like the AFM and quantum transport are studied in some detail. The second part has a much broader focus. Here the aim is to give the students an overview of the on-going merge of the top-down approach with the bottom-up approach of chemistry/biochemistry; a development that is creating new and exciting cross-disciplinary research fields and tech- nologies. Much of the material used in this part of the course is provided by guest lecturers Henrik Bruus MIC – Department of Micro and Nanotechnology Technical University of Denmark 26 January 2004 iiiiv PREFACEContents 1 Top-down micro and nanotechnology 1 1.1 MicrofabricationandMoore’slaw. ... .... ... .... ... ... ... 3 1.2 Clean room facilities . .... ... ... .... ... .... ... ... ... 4 1.3 Photolithography ... .... ... ... .... ... .... ... ... ... 5 1.4 Electronbeamlithography.. ... ... .... ... .... ... ... ... 6 1.5 Nanoimprintlithography... ... ... .... ... .... ... ... ... 8 2 A brief intro to quantum physics 11 2.1 Theparticle-waveduality . . ... ... .... ... .... ... ... ... 11 2.2 deBrogliewaves ... .... ... ... .... ... .... ... ... ... 14 2.3 Thequantumpressure .... ... ... .... ... .... ... ... ... 15 2.4 The Schr¨odingerequationinonedimension . . ... .... ... ... ... 16 2.5 The Schr¨odingerequationinthreedimensions . ... .... ... ... ... 17 2.6 Superpositionandinterferenceofquantumwaves . . .... ... ... ... 18 2.7 Energyeigenstates .. .... ... ... .... ... .... ... ... ... 19 2.8 The interpretation of the wavefunction ψ ... ... .... ... ... ... 19 2.8.1 Theintensityargument . . ... .... ... .... ... ... ... 20 2.8.2 Thecontinuityequationargument ... ... .... ... ... ... 20 2.8.3 Quantum operators and their expectation values . . . . . . . . . . . 21 2.9 Many-particlequantumstates ... ... .... ... .... ... ... ... 22 2.9.1 The N-particlewavefunction .. .... ... .... ... ... ... 22 2.9.2 Permutation symmetry and indistinguishability . . . . . . . . . . . . 22 2.9.3 Fermions: wavefunctions and occupation number . . . . . . . . . . . 23 2.9.4 Bosons: wavefunctions and occupation number . . . . . . . . . . . . 24 2.9.5 Operatorsactingonmany-particlestates . . .... ... ... ... 25 3 Metals and conduction electrons 27 3.1 The single-electron states: travelling waves . . . . . .... ... ... ... 28 3.2 Thegroundstatefornon-interactingelectrons. ... .... ... ... ... 29 3.3 Theenergyofthenon-interactingelectrongas. ... .... ... ... ... 31 3.4 Theenergyoftheinteractingelectrongas ... ... .... ... ... ... 32 3.5 Thedensityofstates . .... ... ... .... ... .... ... ... ... 34 3.6 Theelectrongasatfinitetemperature . .... ... .... ... ... ... 35 vvi CONTENTS 4 Atomic orbitals and carbon nanotubes 37 4.1 The Schr¨odingerequationforhydrogen-likeatoms . .... ... .... .. 37 4.1.1 The azimuthal functions Φ (φ) . .... ... .... ... .... .. 38 m 4.1.2 The polar functions Θ (θ) ... .... ... .... ... .... .. 39 lm m 4.1.3 The spherical harmonics Y (θ,φ)=Θ (θ)Φ (φ) . ... .... .. 39 m l lm 4.1.4 The radial functions R (r) ... .... ... .... ... .... .. 40 nl 4.2 Theenergiesandsizesoftheatomicorbitals. . ... .... ... .... .. 42 4.3 Atomicorbitals: shapeandnomenclature ... ... .... ... .... .. 43 4.4 Angular momentum: interpretation of l and m ... .... ... .... .. 44 2 4.5 The carbon atom and sp hybridization .... ... .... ... .... .. 45 4.6 Graphene,sigmaandpibonds .. ... .... ... .... ... .... .. 48 4.7 Carbonnanotubes .. .... ... ... .... ... .... ... .... .. 50 5 Atomic force microscopy (AFM) 53 5.1 ThebasicprinciplesoftheAFM . ... .... ... .... ... .... .. 53 5.2 Thecantilever: springconstantandresonancefrequency.. ... .... .. 54 5.3 Contactmode.. ... .... ... ... .... ... .... ... .... .. 57 5.4 Non-contactmode .. .... ... ... .... ... .... ... .... .. 57 5.4.1 Atomicpolarization . ... ... .... ... .... ... .... .. 58 5.4.2 vanderWaalsforces . ... ... .... ... .... ... .... .. 59 5.5 Tappingmode.. ... .... ... ... .... ... .... ... .... .. 60 6 Transport in nanostructures 61 6.1 Nanostructuresconnectedtoelectronreservoirs ... .... ... .... .. 61 6.2 Currentdensityandtransmissionofelectronwaves . .... ... .... .. 62 6.2.1 Electronwavesinconstantpotentialsin1D . .... ... .... .. 62 6.2.2 The current density J ... ... .... ... .... ... .... .. 63 6.2.3 The transmission and reflection coefficients T andRin1D ... .. 64 6.3 Electronwavesandthesimplepotentialstep . ... .... ... .... .. 65 6.4 Tunneling through a potential barrier . .... ... .... ... .... .. 67 6.4.1 Transmissionbelowthebarrier . .... ... .... ... .... .. 68 6.4.2 Transmissionabovethebarrier . .... ... .... ... .... .. 70 6.4.3 The complete transmission functionT (ε) . . .... ... .... .. 71 6.5 Transferandscatteringmatrices . ... .... ... .... ... .... .. 71 6.6 Conductance and scattering matrix formalism . . . . .... ... .... .. 72 6.6.1 Electronchannels... ... ... .... ... .... ... .... .. 73 6.6.2 Current,reservoirs,andelectronchannels .. .... ... .... .. 73 6.6.3 The conductance formula for nanostructures . .... ... .... .. 74 6.7 Quantized conductance.... ... ... .... ... .... ... .... .. 75CONTENTS vii 7 Scanning Tunneling Microscopy (STM) 79 7.1 ThebasicprincipleoftheSTM .. ... .... ... .... ... ... ... 79 7.2 Thepiezo-electricelementandspectroscopy . . ... .... ... ... ... 80 7.3 Thelocalelectronicdensityofstates .. .... ... .... ... ... ... 81 7.4 AnexampleofaSTM .... ... ... .... ... .... ... ... ... 82 AExercises 85 Exercis... es for... Chap.... . 1 ... .... ... ... .... ... .... ... ... ... 85 Exercis... es for... Chap.... . 2 ... .... ... ... .... ... .... ... ... ... 85 Exercis... es for... Chap.... . 3 ... .... ... ... .... ... .... ... ... ... 87 Exercis... es for... Chap.... . 4 ... .... ... ... .... ... .... ... ... ... 89 Exercis... es for... Chap.... . 5 ... .... ... ... .... ... .... ... ... ... 91 Exercis... es for... Chap.... . 6 ... .... ... ... .... ... .... ... ... ... 94 Exercis... es for... Chap.... . 7 ... .... ... ... .... ... .... ... ... ... 96viii CONTENTSChapter 1 Top-down micro and nanotechnology Nanotechnology deals with natural and artificial structures on the nanometer scale, i.e. −9 ˚ in the range from 1 µmdown to10 A. One nanometer, 1 nm = 10 m, is roughly the distance from one end to the other of a line of five neighboring atoms in an ordinary solid. The nanometer scale can also be illustrated as in Fig. 1.1: if the size of a soccer ball −1 (∼ 30 cm = 3× 10 m) is reduced 10.000 times we reach the width of a thin human hair −5 (∼ 30 µm=3× 10 m). If we reduce the size of the hair with the same factor, we reach −9 the width of a carbon nanotube (∼3nm=3× 10 m). It is quite remarkable, and very exciting indeed, that we today have a technology that involves manipulation of the ultimate building blocks of ordinary matter: single atoms and molecules. Nanotechnology owes it existence to the astonishing development within the field of micro electronics. Since the invention of the integrated circuit nearly half a century ago in 1958, there has been an exponential growth in the number of transistors per micro chip and an associated decrease in the smallest width of the wires in the electronic circuits. As −1 Figure 1.1: (a) A soccer ball with a diameter ∼ 30 cm = 3× 10 m. (b) The width of 4 a human hair (here placed on a microchip at the white arrow) is roughly 10 times, i.e. −5 ∼ 30 µm= 3× 10 m. (c) The diameter of a carbon nanotube (here placed on top of 4 −9 some metal electrodes) is yet another 10 times smaller, i.e.∼3nm=3× 10 m. 12 CHAPTER 1. TOP-DOWN MICRO AND NANOTECHNOLOGY (a) (b) Figure 1.2: (a) Moore’s law in the form of the original graph from 1965 suggesting a doubling of the number of components per microchip each year. (b) For the past 30 years Moore’s law has been obeyed by the number of transistors in Intel processors and DRAM chips, however only with a doubling time of 18 months. a result extremely powerful computers and efficient communication systems have emerged with a subsequent profound change in the daily lives of all of us. A modern computer chip contains more than 10 million transistors, and the smallest wire width are incredibly small, now entering the sub 100 nm range. Just as the American microprocessor manufacturer, Intel, at the end of 2003 shipped its first high-volume 90 nm line width production to the market, the company announced that it expects to ramp its 1 new 65 nm process in 2005 in the production of static RAM chips. Nanotechnology with active components is now part of ordinary consumer products. Conventional microtechnology is a top-down technology. This means that the mi- crostructures are fabricated by manipulating a large piece of material, typically a silicon crystal, using processes like lithography, etching, and metallization. However, such an approach is not the only possibility. There is another remarkable consequence of the development of micro and nanotechnology. Since the mid-1980’ies a number of very advanced instruments for observation and manipulation of individual atoms and molecules have been invented. Most notable are the atomic force microscope (AFM) and the scanning tunnel microscope (STM) that will be treated later in the lecture notes. These instruments have had en enormous impact on fundamental science as the key elements in numerous discoveries. The instruments have also boosted a new approach to technology denoted bottom-up, where instead of making small structure out over large structures, the small structures are made directly by assembling of molecules and atoms. In the rest of this chapter we shall focus on the top-down approach, and describe some 1 Learn more about the 65 nm SRAM at MICROFABRICATION AND MOORE’S LAW 3 Figure 1.3: Moore’s law applied to the shrinking of the length of the gate electrode in CMOS transistors. The length has deminished from about 100 nm in year 2000 to a projected length of 10 nm in 2015 from the International Technology Roadmap for Semi- conductors, 2003 Edition ( of its main features. 1.1 Microfabrication and Moore’s law The top-down approach to microelectronics seems to be governed by an exponential time dependence. I 1965, when the most advanced integrated circuit contained only 64 tran- sistors, Gordon E. Moore, Director of Fairchild Semiconductor Division, was the first to note this exponential behavior in his famous paper Cramming more components onto in- tegrated circuits Electronics, 38, No. 8, April 19 (1965): ”When unit cost is falling as the number of components per circuit rises, by 1975 economics may dictate squeezing as many as 65,000 components on a single silicon chip”. He observed a doubling of the number of transistors per circuit every year, a law that has become known as Moore’s law. It is illustrated in Fig. 1.2. Today there exist many other versions of Moore’s law. One of them is shown in Fig. 1.3. It concerns the exponential decrease in the length of the gate electrode in standard CMOS transistors, and relates to the previous quoted values of 90 nm in 2003 and 65 nm in 2005. Naturally, there will be physical limitations to the exponential behavior expressed in Moore’s law, see Exercise 1.1. However, also economic barriers play a major if not the decisive role in ending Moore’s law developments. The price for constructing microproces- sor fabrication units also rises exponentially for each generation of microchips. Soon the4 CHAPTER 1. TOP-DOWN MICRO AND NANOTECHNOLOGY Figure 1.4: The clean room facilities DANCHIP, situated next to MIC at the Technical University of Denmark. The large building in background to the left is the original MIC clean room from 1992. The building in the front is under construction until the summer of 2004. level is comparable to the gross national product of a mid-size country, and that might very well slow down the rate of progress. 1.2 Clean room facilities The small geometrical features on a microchip necessitates the use of clean room facilities during the critical fabrication steps. Each cubic meter of air in ordinary laboratories may 7 contain more than 10 particles with diameters larger than 500 nm. To avoid a huge flux of these ”large” particles down on the chips containing micro and nanostructures, micro and nanofabrication laboratories are placed in so-called clean rooms equipped with high-efficiency particulate air (HEPA) filtering system. Such systems can retain nearly all particles with diameters down to 300 nm. Clean rooms are classified according to the maximum number of particles per cubic foot larger than 500 nm. Usually a class-1000 or class-100 clean room is sufficient for microfabrication. The low particle concentration is ensured by keeping the air pressure inside the clean room slightly higher than the surroundings, and by combining the HEPA filter system with a laminar air flow system in the critical areas of the clean room. The latter system let the clean air enter from the perforated ceiling in a laminar flow and leave through the perforated floor. Moreover, all personnel in the clean room must be wearing a special suit covering the whole body to minimize the surprisingly huge emission of small particles from1.3. PHOTOLITHOGRAPHY 5 Figure 1.5: The basic principles of photolithography. The left-most figures illustrates the use of a negative resist to do lift-off. The right-most figures illustrates the use of a positive resist as an etch mask. See the text for more details. each person. 5 3 −1 The air flow inside the DANCHIP clean room is about 1.3×10 m h ,mostofwhich is recirculated particle-free air from the clean room itself. However, since the exhaust air from equipment and fume hoods is not recirculated, there is in intake of fresh air of 5 3 −1 0.3× 10 m h . 1.3 Photolithography Almost all top-down manufacturing involves one or more photolithography fabrication steps, so we give a brief outline of this technique here. A generic photolithography process is sketched in Fig. 1.5. From a lightsource light is directed through a mask carrying the circuit design down onto the substrate wafer covered with a photo-sensitive film, denoted the photoresist. Depending on the local photo-exposure defined by the photolitographic mask the photoresist can be partly removed by a chemical developer leaving well-defined parts of the substrate wafer exposed to etching or metal deposition. The substrate wafer is typically a very pure silicon disk with a thickness around 500 µm and diameter of 100 mm (for historic reasons denoted a 4 inch wafer). Wafers of different purities are purchased at various manufacturers. The photolithographic mask contains (part of) the design of the microsystem that is to be fabricated. This design is created using computer-aided design (CAD) software. Once6 CHAPTER 1. TOP-DOWN MICRO AND NANOTECHNOLOGY completed the computer file containing the design is sent to a company producing the mask. At the company the design is transferred to a glass plate covered with a thin but non-transparent layer of chromium. The transfer process is normally based on either the relatively cheap and fast laser writing with a resolution of approximately 1.5 µmand a delivery time of around two weeks, or the expensive and rather slow electron beam writing with a resolution of 0.2 µm and a delivery time of several months. The photo exposure is typically performed using the 356 nm UV line from a mercury lamp, but to achieve the line widths of sub 100 nm mentioned in Sec. 1.1 an extreme UV source or even an X-ray source is needed. To achieve the best resolution must minimize note only the wavelength λ of the exposure light, but also the distance d between the photolithographic mask and the photoresist-covered substrate wafer, and the thickness t of the photoresist layer. The minimum line width w is given by the approximate min expression  3 w = λ(d + t). (1.1) min 2 If d = 0 nm the mask is touching the photo-resist. This situation, denoted contact printing, improves the resolution but wears down the mask. If d 0 nm, a case denoted proximity printing, the resolution is pourer but the mask may last longer. It is difficult to obtain w 2 µm using standard UV photolithography. min The photoresist is a typically a melted and thus fluid polymer that is put on the substrate wafer, which then is rotated at more than 1000 rounds per minute to ensure an even and thin layer of resist spreading on the wafer. The photoresists carry exotic names like SU-8, PMMA, AZ4562 and Kodak 747. The solubility of the resists is proportional to the square of the molecular weight of the polymer. The photo-processes in a polymer photoresist will either cut the polymer chains in small pieces (chain scission) and thus lower the molecular weight, or they will induces cross-linking between the polymer chains and thus increase the molecular weight. The first type of resists is denoted the positive tone photoresists, they will be removed where they have been exposed to light. The second type is denoted the negative tone photoresists, they will remain where they have been exposed to light. 1.4 Electron beam lithography To obtain resolutions better than the few µm of photolithography it is necessary to use either X-ray lithography or electron beam lithography. Here we give a brief overview of the latter technique. After development of the resist one can choose to etch the exposed part of the wafer. Acid will typically not etch the polymer photoresist but only the substrate, so etching will carve out the design defined by the mask. The shape of the etching depends on the acid and the substrate. It can be isotropic and have the same etch rate in all spatial directions, or it can by anisotropic with a very large etch rate in some specific directions. One can choose the etching process that is most suitable for the design. Metal deposition followed by lift-off is another core technique. Here a thin layer of metal (less than 500 nm) is deposited by evaporation technique on the substrate after de-1.4. ELECTRON BEAM LITHOGRAPHY 7 (a) (b) Figure 1.6: (a) The various components of an scanning electron microscope (SEM) from the electron gun in the top to the sample moved by a stepper motor in the bottom. (b) A computer simulation of the back scattering of electrons in the substrate. Note how the electrons enters the resist from the back in a much wider beam (500 - 1500 nm) than the incoming beam. The higher the energy (e.g., 20 kV) the higher is the spread of electrons (e.g., 1.5 µm), and the lower is the resolution. veloping the resist. At the exposed places the metal is deposited directly on the substrate, and elsewhere the metal is residing on top of the remaining photoresist. After the metal deposition the substrate is rinsed in a chemical that dissolves the photoresist and thereby lift-off the metal residing on it. As a result a thin layer of metal is left on the surface of the wafer in the pattern defined by the photography mask. The above mentioned process steps can be repeated many times with different masks and very complicated devices may be fabricated that way. Electron beam lithography is based on a electron beam microscope, see Fig. 1.6, in which a focused beam of fast electrons are directed towards a resist-covered substrate. No mask is involved since the position of the electron beam can be controlled directly from a computer through electromagnetic lenses and deflectors. The electrons are produced with an electron gun, either by thermal emission from hot tungsten filament or by cold field emission. The emitted electrons are then accelerated by electrodes with a potential U ≈ 10 kV and the beam is focused by magnetic lenses and steered by electromagnetic deflectors. As we shall discuss in great detail in Sec. 2.1 the electron is both a particle and a wave. The wavelength λ of an electron is given in terms by the momentum p of the electron and Planck’s constant h by the de Broglie relation Eq. (2.3) λ = h/p. In the electron beam microscope the electron acquires a kinetic energy given by the acceleration voltage U as 1 2 mv = eU,where m and−e is the mass and charge of the electron, respectively. Since 2 p = mv the expression for the wavelength λ becomes h λ = √ , (1.2) 2meU8 CHAPTER 1. TOP-DOWN MICRO AND NANOTECHNOLOGY which for a standard potential of 10 kV yields λ=0.012 nm. However, the resolution of an electron beam microscope is not given by λ. First of all, one can not focus the electron beam on such a small length scale. A typical beam spot size is around 0.1 nm. But more importantly are the scattering processes of the electrons inside the resist and the substrate. As illustrated by the computer simulation shown in Fig. 1.6(b) the backscattering of the electrons implies that an area much broader area is exposed to electrons than the area of the incoming electrons. This results in an increase of the resolution. It turns out that in practice it is difficult to get below a minimum linewidth of 10 nm. Electron beam lithography is still the technique with the best resolution for lithography. A major drawback of the method is the long expose time required to cover an entire wafer with patterns. The exposure time t is inversely proportional to the current I in the exp electron beam and proportional to the clearing dose D (required charge per area) and the exposed area A, DA t = . (1.3) exp I This formula is discussed further in Exercise 1.3. In photolithography the entire wafer is exposed in one flash, like parallel processing, whereas in electron beam lithography it is necessary to write one pattern after the other in serial processing. For mass production electron beam lithography is therefore mainly used to fabricate masks for photolithography discussed in Sec. 1.3 and nanoimprint lithography discussed in Sec. 1.5. 1.5 Nanoimprint lithography Nanoimprint lithography (NIL) is a relatively young technique compared to photolithog- raphy and electron beam lithography. The first results based on NIL was published in 1995. The technique is in principle very simple. It consists of pressing machine that presses a stamp or mold containing the desired design down into a thin polymer film spun on top of a wafer and heated above its glass transition temperature. The basic principle of nanoimprint lithography is sketched in Fig. 1.7. Naturally, a stamp is needed, and often its is produced by making a nanostructured surface in some wafer by use of electron beam lithography as described in Sec. 1.4 and subsequent etching techniques. Different materials have successfully been used as stamps among them silicon, silicon dioxide, metals, and polymers. Often it is necessary to coat the stamp with some anti-stiction coating to be able to release the stamp form the target material after pressing. The target material is a polymer, which is useful for two reasons. First, above the glass transition temperature polymers are soft enough to make imprinting possible. Sec- ond, polymers can be functionalized to become sensitive various electric, magnetic, ther- mal, optical and biochemical input. Thus the resulting nanostructure can become very sophisticated indeed. The pressing machine needs be able to deliver the necessary pressure. Moreover, it must contain an efficient temperature control in the form of heater plates and a thermostat,1.5. NANOIMPRINT LITHOGRAPHY 9 (a) (b) (c) Figure 1.7: (a) The principle of nanoimprint lithography: after having made nanostruc- tures in the surface of the stamp wafer using electron beam lithography, it is pressed into a thin layer of polymer film spun on top of a gold-plated silicon wafer. (b) A sketch of the setup before imprinting, after imprinting, and after etching. (c) A sketch of the pressing machine at MIC to used to perform nanoimprint lithography. since it is crucial to operate at the correct temperature somewhat above the glass transition temperature of the polymer. To avoid impurities it must also operate under a sufficiently low vacuum, and finally it should allow for correct alignment of the sample before pressing. Nanoimprint lithography is one of the few nanotechnologies that seems to be capable of mass production. Once the stamp is delivered, and the pressing machine is correctly set up, it should be possible to mass fabricate nanostructured wafer. The cycle time of a typical nanoimprint machine is of the order of minutes. This time scale is determined by the actual time it takes to press the stamp down and the various thermal time scales for heating and cooling of the sample.10 CHAPTER 1. TOP-DOWN MICRO AND NANOTECHNOLOGYChapter 2 A brief intro to quantum physics It is crucial to realize that the physics on the nanometer scale tends to become dominated by quantum physics. In the nanoworld one must always be prepared to take seemingly strange quantum phenomena into account and hence give up on an entirely classical de- scription. Although this is not a course in quantum physics it is nevertheless imperative to get a grasp of the basic ideas and concepts of quantum physics. Without this it is not possible to reach a full understanding of the potentials of nanotechnology. Serious students of nanotechnology are hereby encouraged to study at least a minimum of quantum theory. 2.1 The particle-wave duality The first laws of quantum physics dealt with energy quantization. They were discovered in studies of the electromagnetic radiation field by Planck and Einstein in 1900 and 1905, respectively. A new universal constant, Planck’s constant h, was introduced in physics in addition to other constants like the speed of light, c, the gravitation constant, G,andthe 1 charge quantum, e. The 1998 CODATA values for h and  = h/2π are −34 h=6.62606876(52) × 10 Js, (2.1a) −34 =1.054571596(82)× 10 Js. (2.1b) The energy E of light quanta (photons) in light of frequncy f or angular frequency ω =2πf is given by E = hf = ω. (2.2) Already in 1906 Einstein applied this energy quantization on other objects than light, namely on oscillating atoms in his work on the heat capacity of solids. But it was first with Bohr’s analysis of the stationary states in the hydrogen atom that quantum physics really proved to be essential for the understanding of not only the radiation field, but also of matter. With Heisenberg and Schrod ¨ inger’s seminal papers from 1925 and 1926, respectively, modern quantum theory was born, a theory that eversince has been the foundation of our understanding of the physical world in which we live. 1 See the NIST reference on constants, units, and uncertainty 1112 CHAPTER 2. A BRIEF INTRO TO QUANTUM PHYSICS A central concept in quantum physics is the particle-wave duality, the fact that fun- damental objects in the physical world, electrons, protons, neutrons, photons and other leptons, hadrons, and field quanta, all have the same dual nature: they are at the same time both particles and waves. In some situations the particle aspect may be the dominant feature, in other vice versa; but the behavior of any given object can never be understood fully by ascribing only one of these aspects to it. Historically, as indicated, the particle-wave duality was first realized for the electro- magnetic field. It is interesting to note that right from the beginning when the first theories of the nature of light was proposed in the 17th century, it was debated whether light were particles (corpuscles), as claimend by Newton, or waves, as claimed by Huygens. The de- bate appeared to end in the beginning of the 19th century with Young’s famous double-slit interference experiments that demonstrated that light were waves. With Maxwell’s theory (1873) and Hertz’s experiments (1888) the light waves were shown to be of electromag- netic nature. But after nearly one hundred years of wave dominance Planck’s formula for the energy distribution in black-body radiation demonstrated that light possesses some element of particle nature. This particle aspect became more evident with Einstein’s explanation of the photoelectric effect: small energy parcels of light, the so-called light quanta, are able to knock out electrons from metals just like one billiard ball hitting an- other. For some years theorists tried to give alternative explanations of the photo-electric effect using maxwellian waves, but it proved impossible to account for the concentration of energy in a small point needed to explain the photoelectric effect without postulating the existence of light quanta – today called photons. In 1913 Niels Bohr published his theory of the hydrogen atom, explaining its stability in terms of stationary states. Bohr did not explain why stationary states exist. He boldly postulated their existence and from that assumption he could explain the frequencies of the experimentally observed spectral lines, and in particular he could derive Balmer’s empirical expression for the position of the spectral lines. He also derived a formula for the Rydberg constant appearing in that expression. In the following years it was postulated still without an explanation that a particle of momentum p = mv,where m and v is its mass and velocity, respectively, moving in a closed orbit must obey the Bohr-Sommerfeld  quantization rule, p· dr = nh, where the integral is over one revolution and n is an integer. Figure 2.1: Resonance modes or eigenmodes in one and two dimensions: (a) a vibrating string described by sin(kx), and (b) a vibrating membrane described in polar coordinates in terms of a Bessel function by J (kr)cos(φ). 1