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Introduction to Nanoscience and Nanotechnology: A Workbook M. Kuno May 14, 20042Contents 1 Introduction 1 2 Structure 11 3 Length scales 29 4 Con¯nement 35 5 Density of states 57 6 More density of states 65 7 Even more density of states 73 8 Joint density of states 85 9 Emission 97 10 Bands 115 11 Tunneling 137 12 The WKB approximation 155 13 Synthesis 183 14 Tools 201 15 Applications 209 Acknowledgments 231 iii CONTENTSList of Figures 1.1 CdSe quantum dot . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Quantum con¯nement . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 Size dependent absorption and emission of CdSe . . . . . . . 8 1.5 Arti¯cial solid . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1 14 3D Bravais lattices . . . . . . . . . . . . . . . . . . . . . . 12 2.2 Atoms per unit cell . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Atom sharing . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 FCC unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.5 BCC unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 Primitive hexagonal unit cell . . . . . . . . . . . . . . . . . . 17 2.7 Diamond structure unit cell . . . . . . . . . . . . . . . . . . . 18 2.8 Zincblende or ZnS structure unit cell . . . . . . . . . . . . . . 19 2.9 NaCl structure unit cell . . . . . . . . . . . . . . . . . . . . . 19 2.10 CsCl structure unit cell . . . . . . . . . . . . . . . . . . . . . 20 2.11 Primitive wurtzite unit cell . . . . . . . . . . . . . . . . . . . 20 2.12 Miller index examples . . . . . . . . . . . . . . . . . . . . . . 22 2.13 More Miller index examples . . . . . . . . . . . . . . . . . . . 23 4.1 Particle in an in¯nite box . . . . . . . . . . . . . . . . . . . . 36 4.2 Half a harmonic oscillator . . . . . . . . . . . . . . . . . . . . 38 4.3 Particle in a ¯nite box . . . . . . . . . . . . . . . . . . . . . . 39 4.4 Particle in a ¯nite box: solutions . . . . . . . . . . . . . . . . 40 4.5 Particle in a ¯nite well: Mathcad solutions. . . . . . . . . . . 44 4.6 Particle in an in¯nite circle . . . . . . . . . . . . . . . . . . . 45 4.7 Particle in a sphere . . . . . . . . . . . . . . . . . . . . . . . . 49 5.1 3D density of states . . . . . . . . . . . . . . . . . . . . . . . 59 5.2 2D density of states . . . . . . . . . . . . . . . . . . . . . . . 61 iiiiv LIST OF FIGURES 5.3 1D density of states . . . . . . . . . . . . . . . . . . . . . . . 62 5.4 0D density of states . . . . . . . . . . . . . . . . . . . . . . . 63 6.1 3D density of CB and VB states . . . . . . . . . . . . . . . . 70 6.2 3D Fermi level . . . . . . . . . . . . . . . . . . . . . . . . . . 72 7.1 2D density of CB states . . . . . . . . . . . . . . . . . . . . . 74 7.2 2D density of VB states . . . . . . . . . . . . . . . . . . . . . 76 7.3 1D density of CB states . . . . . . . . . . . . . . . . . . . . . 78 7.4 1D density of VB states . . . . . . . . . . . . . . . . . . . . . 81 8.1 Vertical transitions . . . . . . . . . . . . . . . . . . . . . . . . 87 8.2 3D joint density of states . . . . . . . . . . . . . . . . . . . . 89 8.3 2D joint density of states . . . . . . . . . . . . . . . . . . . . 92 8.4 1D joint density of states . . . . . . . . . . . . . . . . . . . . 95 8.5 Summary, joint density of states . . . . . . . . . . . . . . . . 96 9.1 Einstein A and B coe±cients . . . . . . . . . . . . . . . . . . 98 9.2 Derived emission spectrum: Einstein A and B coe±cients . . 110 9.3 Pulsed experiment and lifetime . . . . . . . . . . . . . . . . . 110 9.4 Radiative decay of excited state . . . . . . . . . . . . . . . . . 112 9.5 Multiple pathway decay of excited state . . . . . . . . . . . . 112 10.1 Kronig-Penney rectangular potential . . . . . . . . . . . . . . 115 10.2 Kronig-Penney delta function potential . . . . . . . . . . . . . 126 10.3 General Kronig Penney model: Mathcad solutions . . . . . . 129 10.4 General Kronig Penney model continued: Mathcad solutions 130 10.5 Kronig Penney model revisited: Mathcad solutions . . . . . . 131 10.6 Kronig Penney model, delta functions: Mathcad solutions . . 132 10.7 Kronig Penney model, delta functions continued: Mathcad solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 10.8 From metals to insulators . . . . . . . . . . . . . . . . . . . . 134 11.1 Potential step . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 11.2 Potential step ("V) . . . . . . . . . . . . . . . . . . . . . . 138 11.3 Potential step ("V) . . . . . . . . . . . . . . . . . . . . . . 139 11.4 Potential barrier . . . . . . . . . . . . . . . . . . . . . . . . . 144 11.5 Potential barrier ("V). . . . . . . . . . . . . . . . . . . . . 145 11.6 Potential barrier ("V). . . . . . . . . . . . . . . . . . . . . 150 11.7 Semiconductor junction . . . . . . . . . . . . . . . . . . . . . 153LIST OF FIGURES v 12.1 Arbitrary potential step . . . . . . . . . . . . . . . . . . . . . 157 12.2 Arbitrary potential drop . . . . . . . . . . . . . . . . . . . . . 162 12.3 Arbitrary potential barrier . . . . . . . . . . . . . . . . . . . . 166 12.4 Field emission . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 12.5 Shottky barrier . . . . . . . . . . . . . . . . . . . . . . . . . . 176 12.6 Parabolic barrier . . . . . . . . . . . . . . . . . . . . . . . . . 177 12.7 Linear barrier . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 12.8 Parabolic barrier . . . . . . . . . . . . . . . . . . . . . . . . . 180 13.1 Cartoon of a MBE apparatus . . . . . . . . . . . . . . . . . . 184 13.2 Cartoon of a MOCVD apparatus . . . . . . . . . . . . . . . . 185 13.3 Colloidal synthesis apparatus . . . . . . . . . . . . . . . . . . 187 13.4 LaMer model . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 13.5 LaMer model: size distribution . . . . . . . . . . . . . . . . . 195 14.1 Transmission electron microscopy . . . . . . . . . . . . . . . . 202 14.2 Secondary electron microscopy . . . . . . . . . . . . . . . . . 203 14.3 Atomic force microscopy . . . . . . . . . . . . . . . . . . . . . 204 14.4 Scanning tunneling microscopy . . . . . . . . . . . . . . . . . 205 14.5 Dip pen nanolithography. . . . . . . . . . . . . . . . . . . . . 206 14.6 Microcontact printing . . . . . . . . . . . . . . . . . . . . . . 207 15.1 Nanowire device . . . . . . . . . . . . . . . . . . . . . . . . . 210 15.2 Nanowire sensor . . . . . . . . . . . . . . . . . . . . . . . . . 211 15.3 Quantum dot/dye photobleaching. . . . . . . . . . . . . . . . 213 15.4 Quantum dot/dye absorption/emission spectra . . . . . . . . 214 15.5 Density of states for lasing . . . . . . . . . . . . . . . . . . . . 216 15.6 Solar spectrum and QD absorption/emission spectra . . . . . 218 15.7 Quantum dot LED schematic . . . . . . . . . . . . . . . . . . 220 15.8 Orthodox model of single electron tunneling . . . . . . . . . . 222 15.9 Coulomb Staircase . . . . . . . . . . . . . . . . . . . . . . . . 226vi LIST OF FIGURESList of Tables 2.1 Common metals . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Group IV semiconductors . . . . . . . . . . . . . . . . . . . . 24 2.3 Group III-V semiconductors . . . . . . . . . . . . . . . . . . . 24 2.4 Group II-VI semiconductors . . . . . . . . . . . . . . . . . . . 25 2.5 Group IV-VI semiconductors . . . . . . . . . . . . . . . . . . 25 viiviii LIST OF TABLESPreface Thissetoflecturenotesaboutnanoscienceandnanotechnologywasinitially written over the spring and summer of 2003. After my initial appointment as an assistant professor in chemistry, I agreed to teach an introductory class on nanoscience and nanotechnology for incoming graduate students at the University of Notre Dame. However after accepting this task, it quickly became apparent to me that there were few resources available for teach- ing such a class, let alone any textbook. So while waiting for equipment to arrive, I undertook it upon myself to compile a series of lecture notes that would explain to the student some of the underlying concepts behind \nano". The motivation for this was to describe to the student the physics behind each concept or assumption commonly encountered in the nano lit- erature rather than just providing a qualitative overview of developements in the ¯eld. I have also tried to illustrate and motivate these concepts with instances in the current literature where such concepts are applied or have been assumed. In this manner, the goal is to provide the student with a foundation by which they can critically analyze, and possibly in the future, contributetotheemergingnano¯eld. Itisalsomyhopethatoneday, these lecture notes can be converted into an introductory text so that others may bene¯t as well. Masaru Kenneth Kuno Notre Dame, IN May 14, 2004 ixx LIST OF TABLESChapter 1 Introduction Preliminaries Whatis\nano"? Well, without providingade¯niteanswerto thisquestion, nano is a popular (emerging) area of science and technology today. It has attracted the attention of researchers from all walks of life, from physics to chemistry to biology and engineering. Further impetus for this movement comes from the temendous increase in public and private funding for nano over the last ten years. A prime example of this is the new National Nan- otechnology Initiative (NNI) created by former President Bill Clinton. The NNI increases funding in national nanoscience and nanotechnology research by hundreds of millions of dollars yearly. In addition, private sector con- tributions have jumped dramatically as evidenced by the plethora of small startup ¯rms lining the tech corridors of the east and west. Nano has even entered popular culture. It's used as a buzzword in con- temporary books, movies and television commercials. For example, in the recentblockbuster,Spiderman,theWillemDafoecharacter(theGreenGob- lin) is a famous (and wildly wealthy) nanotechnologist whose papers the Tobey McGuire character (Spiderman) has read and followed (see the scene outside of Columbia university). Likewise, in the movie \Minority Report" TomCruise'scharacterundergoeseyesurgerytoavoidbiometric¯ngerprint- ing. This scene involves a retinal eye transplant aided by so called \nano reconstructors". A scene in the DC metro shows him reading a newspaper with the headline \nanotechnology breakthrough". In television, a current GE commercial for washers and dryers features the storyline of: geeky nan- otechnologist bumps into a supermodel at the laundromat resulting in love at ¯rst sight. We're therefore, implicitly, told to believe that their mix of 12 CHAPTER 1. INTRODUCTION brains and beauty embody GE's new washer/dryer line. In books, the New York Times bestseller \Prey" by Michael Crighton features nanotechnology run amok with spawns of tiny nano bots escaping from the laboratory and hunting down people for food. (Sounds like the \Andromeda Strain" except recast with nano as opposed to an alien virus.). The mantle of nano has also been adopted by various scienti¯c visionar- ies. Perhaps the most prominent is Eric Drexler who has founded an insti- tute,calledtheForesightInstitute, devotedtoexploringhisideas. Concepts beingdiscussedincludedevelopingtinynanorobotsthatwill\live"insideus andrepairour bloodvesselswhendamaged, preventingheartattacks. They will also kill cancer, cure us when we are sick, mend bones when broken, make us smarter and even give us immortality. These nano robots will also serve as tiny factories, manufacturing anything and everything from food to antibiotics to energy. In turn, nanotechnology will provide a solution to all of mankind's problems whether it be hunger in developing countries or pollution in developed ones. Drexler therefore envisions a vast industrial revolution of unprecendented size and scale. At the same time, concurrent with his visions of a utopian future is a darker side, involving themes where such nano robots escape from the laboratory and evolve into sentient beings completely out of mankind's control. Such beings could then sow the seeds to mankind's own destruction in the spirit of recent movies and books such as The Terminator, The Matrix and Prey. Now, whether such predictions and visions of the future will ever become reality remains to be seen. How- ever, any such developments will ultimately rely on the scienti¯c research of today, which is, on a daily basis, laying down the foundation for tomorrow's nanoscience and nanotechnology. Intoday'sscienti¯crealm, thewordnanodescribesphysicallengthscales that are on the order of a billionth of a meter long. Nanoscale materials therefore lie in a physical size regime between bulk, macroscale, materi- als (the realm of condensed matter physics) and molecular compounds (the realm of traditional chemistry). This mesoscopic size regime has previously been unexplored and beckons the researcher with images of a scienti¯c wild wild west with opportunites abound for those willing to pack their wagons and head into the scienti¯c and technological hinterland. In this respect, nanoscale physics, chemistry, biology and engineering asks basic, yet unan- swered,questionssuchashowtheopticalandelectricalpropertiesofagiven material evolve from those of individual atoms or molecules to those of the parent bulk. Other questions that nanoscience asks include: ² How does one make a nanometer sized object?3 ² How do you make many (identical) nanometer sized objects? ² How do the optical and electrical properties of this nanoscale object change with size? ² How does its optical and electrical properties change with its \dimen- sionality"? ² How do charges behave in nanoscale objects? ² How does charge transport occur in these materials? ² Do these nanoscale materials posess new and previously undiscovered properties? ² Are they useful? Thetransitiontonanosciencebeginsatthislastpointwhenweaskhowthese nanoscalematerialsmightbeexploitedtoimproveourlives. Venturecapital ¯rms and mainstream industry have therefore taken up this challenge with many small startups trying to apply nanoscale materials in products rang- ing from better sunscreen lotions to °uorescent labels for biological imaging applications to next generation transistors that will one day store the entire content of the Library of Congress on the head of a pin. More established companies, such as GE, HP, Lucent and IBM, have also started their own in house nano programs to revolutionalize consumer lighting, personal com- puting, data storage and so forth. So whether it be for household lighting or consumer electronics, a nano solution exists and there is very likely a company or person pursuing this vision of a nano future. So what is nano? This series of lecture notes tries to answer this ques- tion by explaining the physical concepts behind why such small, nanoscale, materials are so interesting and potentially useful. Overview The idea behind these lecture notes is as follows: First in Chapter 2, the composition of solids is discussed to introduce common crystal structures found in nanomaterials. Solids come in a number of forms, from amor- phous(glass-like)topolycrystalline(multipledomains)tocrystalline. Much of nanoscience and nanotechnology focuses on nanometer sized crystalline solids, hence the emphasis on crystal structure. In addition, the structure4 CHAPTER 1. INTRODUCTION section also illustrates the increase in surface to volume ratio for nanoma- terials over bulk. This is because in nanometer sized systems up to 50% of the atoms lie at the surface of a nanostructure, in direct contrast to macro- scopic solids where such numbers are typically much smaller. The surface is therefore potentially important in dictating a material's optical and electri- cal properties when nanometer sized. Furthermore, the increase in surface area is important to applications where the surface to volume ratio plays a critical role such as in catalysis as well as in photovoltaics. Developments in this area, using nanostructures, have led to increasingly e±cient solar cells such as the Gratzel cell. Finally, the concept of crystal structure and the periodic potential due to the ordered arrangement of atoms is central to the concept of electronic bands, which we will discuss later on. Figure 1.1: Transmission electron micrograph of individual CdSe quantum dots Chapter 3 introducs the concept of length scales to put into perspective the actual physical lengths relevant to nano. Although being nanometer sized if often considered the essence of \nano", the relevant physical length scalesareactuallyrelativetothenaturalelectronorholelengthscalesinthe parent bulk material. These natural length scales can either be referred to by their deBroglie wavelength or by the exciton Bohr radius. Thus, while a givennanometersizedobjectofonematerialmayqualifyfornano, asimilar sized object of another material may not. Next the concept of quantum con¯nement is introduced in Chapter 4 through the simple quantum mechanical analogy of a particle in a 1 di-5 mensional, 2 dimensional and 3 dimensional box. Quantum con¯nement is most commonly associated with nano in the sense that bulk materials gen- erally exhibit continuous absorption and electronic spectra. However, upon reaching a physical length scale equivalent to or less than either the exciton Bohr radius or deBroglie wavelength both the optical and electronic spec- tra become discrete and more atomic-like. In the extreme case of quantum dots, con¯nement occurs along all three physical dimensions, x,y,and z such that the optical and electrical spectra become truly atomic-like. This is one reason why quantum dots or nanocrystals are often called arti¯cial atoms. Analogiescomparingtheparticleinaonedimensionalboxtoaquantum well,theparticleinatwodimensionalboxtoaquantumwireandtheparticle inathreedimensionalboxtoaquantumdotprovideonlyhalfthesolution. If oneconsidersthatinaquantumwellonlyonedimensioniscon¯nedandthat two others are \free", there are electronic states associated with these extra two degrees of freedom. Likewise in the case of a quantum wire, with two degrees of con¯nement, there exists one degree of freedom. So solving the particleinatwodimensionalboxproblemmodelstheelectronicstatesalong the two con¯ned directions but does not address states associated with this remaining degree of freedom. To gain better insight into these additional states we introduce the concept of density of states (DOS) in Chapters 5,6,and 7. The density of states argument is subsequently applied to both the valence band and conduction band of a material. Putting together both valence and conduction band density of states introduces the concept of the joint density of states (JDOS) in Chapter 8 which, in turn, is related to the absorption coe±cient of a material. Afterdescribingtheabsorptionspectraof3D(bulk),2D(quantumwell), 1D (quantum wire), and 0D (quantum dot) systems we turn to the concept of photoluminescence. Generally speaking, in addition to absorbing light, systems will also emit light of certain frequencies. To describe this process, the Einstein A and B coe±cients and their relationships are introduced and derived in Chapter 9. Finally, the emission spectrum of a bulk 3D material is calculated using the derived Einstein A and B coe±cients. The concept of quantum yields and lifetimes, which describe the e±ciency and timescale of the emission, completes this section. Bands are introduced in Chapter 10. This topic is important because metals, semiconductors, and semi-metals all have bands due to the periodic potential experienced by the electron in a crystal. As mentioned earlier in the section on structure, this periodic potential occurs due to the ordered and repeating arrangement of atoms in a crystal. Furthermore, metals, semiconductors,insulators,andsemi-metalscanallbedistinguishedthrough6 CHAPTER 1. INTRODUCTION Figure 1.2: Photograph of the size dependent emission spectra of both HgS (top) and CdSe (bottom) quantum dots. Small quantum dots absorb and emit blue/green light, larger dots absorb and emit red light.7 Figure 1.3: Cartoon of con¯nement along 1, 2 and 3 dimensions. Analogous to a quantum well, quantum wire and quantum dot. the occupation of these bands by electrons. Metals have \full" conduction bandswhilesemiconductorsandinsulatorshave\empty"conductionbands. At the same time, in the case of semiconductors and insulators there is a rangeofenergiesthatcannotbepopulatedbycarriersseparatingthevalence band from the conduction band. This forbidden range of energies (a no man's land for electrons) is referred to as the band gap. The band gap is extremely important for optoelectronic applications of semiconductors. For example, the band gap will generally determine what colors of light a given semiconductormaterialwillabsorboremitand,inturn,willdeterminetheir usefulness in applications such as solar energy conversion, photodetectors, or lasing applications. An exploration of the band gap concept ultimately touchesonthee®ectsquantumcon¯nementhasontheopticalandelectrical properties of a material, which leads to the realization of a size dependent band gap. Introducing the concept of bands is also important for another reason since researchers have envisioned that ordered arrays of quantum wells, or wires, or dots, much like the arrangement of atoms in a crystal, can ulti- matelyleadtonew\arti¯cial"solidswitharti¯cialbandsandcorresponding band gaps. These bands and associated gaps are formed from the delocal- ization of carriers in this new periodic potential de¯ned by the ordered ar- rangement of quantum dots, quantum wires, or quantum wells. Talk about designermaterials. Imaginearti¯cialelementsorevenarti¯cialmetals,semi- metals, andsemiconductors. NowondervisionariessuchasDrexlerenvision8 CHAPTER 1. INTRODUCTION Figure1.4: SizedependentabsorptionandemissionspectraofcolloidalCdSe quantum dots.