Chapter 5 Semiconductor Equations of State (Silicon Earth) PDF

Semiconductors: Lite! 143 And we can do this trivially by simply controllably introducing impurities intothe semi- conductor! Read: POWERFUL tool. There is one final concept we need to address in doping, and that is the notion...

Semiconductors: Lite! 143 And we can do this trivially by simply controllably introducing impurities intothe semi- conductor! Read: POWERFUL tool. There is one final concept we need to address in doping, and that is the notion of "charge neutrality." For most folks, when Isay "charge," you would say, "oh, there is only one type of chargeresponsible for current-electrons." Not so with semiconduc tors, In principle, for any given extrinsic semiconductor, there are actually four types of charges that may be present: (1) electrons (1), (2) holes (p), (3) ionized donor impu rities (Np), and (4) ionized acceptor impurities (Na). Electrons and holes are clearly mobile (they can produce current), whereas the donor and acceptor impurities are sta tionary (they cannot produce current). However, as we will see, under certain condi tions, those fixed impurities, although they cannot contribute directly to current flow, cancreate their own internalelectric field! The net charge density (Psemi) is the sum of the four, or Psemi q(p-n+ND-Nã). (5.14) For our preceding1-type silicon example, Nb =1x10" cm, NA =0, n=1x 1017 cm,and p= l,000 cm,and thus psemi 0, since 1,000 is tiny compared to 100,000,000,000,000,000 and can be trivially neglected. When Psemi = 0, the sample is said to be "charge neutral," and "charge neutrality" is satisfied. For a uniformly doped semiconductor sitting on your desk, the net charge density is thus zero, even though there is a vast amount of charge present in it (100,000 trillion electrons/em³ to be precise). This is why if Itouch my finger to the sampie of silicon sitting on my desk Idon't get instantly electrocuted! There is a tremendous amount of charge present, but it isbalanced positive to negative, effectively nulling-out the net charge. The sample is "charge-neutral"-so go ahead, it's safe to pick it up! Let me torque the thumbscrew slightly. Brace yourself! Poisson's equation, one of Maxwell's equations, is nature's way of mathematically relating the net charge density and the electric field (or electrostatic potential Y, aka the voltage), and is given in three dimensions by V.8= Psemi (5.15) KsemiEo where ksemi iS the dielectric constant of the semiconductor relative to vacuum (11.9 for sili con) and e, is the permittivity of free space (8.854 x 10-4 F/cm). Don't worry about the math; the point is this: If the net charge density is zero, Poisson's equation guarantees that the electric field is zero, and thus the electrostaticpotential is constant (8 = -dY/dx). This is equivalent to saying that the energy band diagram is constant (lat), because E=-. Said another way, if E and Ey are drawn flat in the energy band diagram tor a partic ular situation, there cannot be an electric field present. Conversely, if the energy bands are not flat, then an electric field is present, and this condition is known affectionately as "band-bending" (Figure 5.20). The magnitude of that induced electric field in one dimen sion can be calculated from the slope (mathematically, the derivative) of the band, namely, E=-1/qdE/dx), As we will see,creative band-bending is at the very heart of transistor operation.  Introduction to. Microelectronicsand 144 Silicon Earth: Nanotechnolo Electron movement Electron scattering events Electricfield Semiconductor E, Ketix Ec=ÈNotental Electrc icd= l/lq dE l£t E Hole scattering events Hole movement A net movement of ht semiconductor, resulting in band-bending FIGURE 5.20 electric field to a processes, and Application of an holes in response to that field is produced, subject to the operative scattering electrons and right to left generating a net current flow from Drift andDiffusion Transport Carrier Drift Wonderful. We now have energy bands, electron and hole densities, and ways to easily dynamic range, by means of doping manipulate those carrier populations over a wide in partially filled bands, enabling them for instance. Now that we have carriers present simple questions: What magnitude or in principle to move, we need to ask a couple of And what does that current depend on current actually flows in response to a stimulus? device that we will employ Appreciate that it is the current flowing in a semiconductor aradio for asmartphone. do useful work, say builda microprocessor for a computer or in semiconduc- Interestingly enough, there are in fact two ways for carriers to move Drift is more drift" and "carrier diffusion." tors; twO mechanisms for transport: "carrier theapplied in drift transport is intuitive, so let's start there. The fundamental driving force sili doped of onthe electric field(i.e, put a voltage across asample of length ). Imagine a sample (dependentsample. con at 30O K, which has a finite density of electrons and holes presentVacrooss the doping density, which is known). Now imagine applying a voltage response The question of the day: If I put an ammeter in the circuit, what Current flows in question! to that applied voltage, and what system variables does it depend on? Tough= V/), and the filled If we apply a voltage across the sample, an electric field is induced (6 partially single electrons and holes will move (drift) in response to that "drift field" in themotion ofa a the follow conduction and valence bands, respectively. Imagine zooming in on microscopically and hole in the valence band as it moves (Figure 5.21). The hole will start verv circuitous route through the material in response to the drift field. It will Semiconductors: Lite! 145 Electric field Microscopicmotion Si Free hole (Si Si Drift field Average motion Darit FIGURE 5.21 Representation of drift transport inside a semiconductor, with the electrons and holes moving in the applied electric field. A microscopic view of the hole drift response to motion tional to its velocity) is also shown, together with a simplified picture (the vector length in each case is propor that assumes that all holes move with an average drift velocity. stop, speed up and slow down, and even change directions abruptly as it moves along in the field, depending on what it confronts in itspath. Anything that impedes the of carriers is called a "scattering event" (this is illustrated within the energy bandtransport in Figure 5.20). Think of scattering as a "collision" process. A carrier diagram gains kinetic energy (velocity increases) from the applied field,and then scatters, reducing its velocity (energy). There are many types of scattering in semiconductors, but the two most important are "lattice scattering" the interaction of the carriers with the thermally induced vibrations of the crystal lattice, and "impurity scattering," the interaction of the carriers with the charged dopant impurities. If we now imagine a sea of carriers moving in this drift field (recall: with a boron doping level of 1 x 106 cm- there are 1 x 10 cm- holesa bunch!), We can invent an "average drift velocity" (7anit) for the moving sea of holes (in statistical physics, this is called an "ensemble average"). The net result is important. In response to an applied drift field, and in the presence of the various scattering processes: (1) the average drift velocity of the carriers depends on the magnitude of the applied drift field (hence the force driving their motion) and (2) the maximum average velocity of the carriers is finite, and is called the "saturation veloc ity" (u). In lightly doped silicon at 300 K, for instance, v, is about 1x 10 cm/s, 223,694 mi/h, about 1/3,000 the speed of light in vacuum (read: mighty darn tast!}). The so-called "velocity-field" characteristics of Si, Ge, and GaAsare shown in Figure 5.22. You will note that,for electrons in silicon, it takes an applied electric field of about 20,000 V/em toreach Saturation velocity. This feels like a pretty large field. It is! Here's the example Ilike to use to illustrate big versus smallfields. If you walked into a room, and saw a cable with a ongnt yellow sign saying "DANGER-20,000 VI" pasted on it, would you walk up, stand  to Microelectronics and Silicon Earth: Introduction 146 l08 Nanotechnology Electrons Holes (cm/s) GaAs velocity 107 drift Carrier Ge' 106 l05 10 10 106 10 103 Electric field (V/cm) FIGURE 5.22 for Si, Ge, and GaAs at 300 K. Electron and hole drift velocities as functions of the applied electric field is in fact equivalent in apuddle of water, and bring your finger to within 1cm of it (this it one! to a 20,000 V/cm electric field)? As I like to say, you might, but youd only get toisdotrivial it is afield that to Clearly, 20,000 V/cm is avery, very large field, but amazingly induce in asemiconductor. Consider: Let's apply 2 Vacross a transistor only 1 um long Congratulations, you have just applied a field of 20,000 V/cm! (Psssttt... Check the math.) Moral: In the micro/nano world, smallvoltages across tiny distances generate huge het and hence VERY fast carriers. This is an IMPORTANT result! You willalso note from Figure 5.22 that at "'low" electric fields (sav below 1000 V/cm tor electrons in silicon), the average driftvelocity depends linearly on the drift feld. this functional relationship, for convenience, we define a new parameter knownVs),asuch "low-field carrier mobility," or just"mobility"(, or Hp measured in units of cm?/V that Darit = u. pieceof The mobility is very useful because: (1) it collapses an enormously complex mass)and(2)it carrier-scattering physics into a single parameter (much like the effective at300K can fairly easily be measured. Figure 5.23 shows electron and hole mobility data else holes, all as functions of doping, As can be seen, for silicon, electrons are faster than being equal, and this will have important circuit design implications, as we will see. current Okay. With those bare-bones ideas in place, we can rather easily derive the drift in density that flows in response to the applied field. Consider the toy model illustrated intui- Figure 5.24(we'll choose Iwill use aan holes in ap-type tive step-by-step "dimensional" argument semiconductor for simplicity). to get the current expression |1| " Step (1) Varitt (cm): All holes this distance back from A intimet will cross the plane " Step (2) VarifitA (cm'): All holes in this volume will cross A in time t Step (3) poarintA (# of holes): Number of holes crossing Ain time t Semiconductors: Lite! 147 10 Silicon (cm/V.s) mobility H, Majority H, Majority field H, Minority. Low H, MinorityMinority 102 Majority l015 10l6 1017 1018 l019 1020 Doping concentration Na or Np (cm-3) FIGURE 5.23 Low-field electron and hole mobilities as functions of doping concentration at 300 K. Drift field larift ldrift FIGURE 5.24 Model for determnining the drift current that flows in response to the applied electric field. Step (4) qpvanntA (C):Charge crossing Ain time t Step (5) qpvarifA (C/s): Charge per unit time crossing A (i.e., current) Step (6) qpvarift (A/cm'): Drift current density crossing A Step (7) qH,pE (A/em): Drift current density written in terms of mobility Shazam! Thus, the electron and hole drift current densities, in three dimensions (they are vector quantities), are given by (5.16) (5.17)  Silicon Earth: Introduction to 148 Microelectronics and Nanotechnol gy of total We come full circle now. Armed with this definition drift current Connect our seniconductor resistivity with the electron and density, We can hole densities, hence constructed (doped). Because how itis J= q(u,n+H,p)8 (519 and j-, 620) we thus find that 1 (5.21 q(u,n +4,p) Observe: Give me the doping and I will calculatethe resistivity. For example, most of the world's integrated circuits are built on 8-10 S2crn p-type silicon, which we now can easij see corresponds to about 1-2 x 1015 cm-³ boron doping (go ahead, check it!). Carrier Diffusion If you asked most people who know a little about electricity what actually produes cur rent in materials, they willinevitably say, "Well, electrons move in response to a according to V=IR." Fine in most cases, but NOT for semiconductors. This is Only the truth! We already know that we have both electron drift current andand hoiethe valennk rent (twO sources), because carrier motion in both the conduction band band contribute to current flow. But that is only 1/2of the truth! It is an amazts Sutd flou. Sound in semiconductors, even with zero field applied (no voltage), current can still "difusiont. aka magical?! It isn't. This new" type of current flows by diffusive transport, entrenchedin Although perhaps less familiar to most people, diffusion is actually firmly youwalkn your normal life experience. Consider: Iam in the front of theclassroom, and momenN Within a few carrying a biscuit(morning class!), and sit down in the back row. the smell of your biscuit has my mouth watering. How? molecules in the nth Well, imagine zooming in to a microscopic view of the airnolecules, hopetiully the (you know, a little oxygen, alot of nitrogen, and a few trace of kT),all dnother ing toxic or illegal!). Because of the thermal energy present (vep, same off one injetd molecules are whizzing back and forth at very high speeds, bouncingtheroom that massive numbers of collisionsit's a chaotic mess. Now, at One end of Well,allottie few "great-biscuit-smell" molecules into the mix, and see what happens. intothetaus molecules Mom chaotic motion willtend to spread out the"great-biscuit-smell" yOurroom di volume of the room. TwO minutes later my mouth is watering (hopefully presentin the »stirrins no yousomething about sharing!), Note, it is NOT the air currents with this spreading; the same result will occur in acontrolled environment Semiconductors: Lite! 149 of the air. In short, entropy is at work here, the great levelerof the universe. Said another way, for very deep reasons, nature much prefers the lowest-energy Svstem, the so-called "equilibrium state" (see the "Equilibrium" configuration sidebar of ay for areminder). In essence, nature seesa large concentration of "great-biscuit-smell" molecules at one end of the room, and says, "hey, let's equilibrate the system of the great-biscuit-smell' molecules uniformly over(lower its energy) by spreading out all smell" molecules "diffuse" from point A to point B. volume V" Ergo, the "great-biscuit EQUILIBRIUM The term "equilibrium" is used to describe the unperturbed state of any system in nature. Under equilibrium conditions, there are no external forces stresses, etc.), and all "observables" (things we can measure)acting on the system (fields, are time invariant (don't depend on time). Knowledge of any given system's simplest "frame of reference" for the system.equilibrium state is vital because it forms the It is also the lowest-energy system, and hence nature will strive to return the system to configuration of the it by any means. Being able to theoretically (and equilibrium when we perturb experimentally) understand any (semiconductor or otherwise) in equilibrium thus gives us a powerful physical basissystem ascertaining the condition of the system once a perturbation is applied. for When we declare that a semiconductor is in external electricor magnetic fields are applied, equilibrium, for instance, we mean that no no is shining on the sample. More formally, equilibriumtemperature mandates gradient exists, and no light that ota =0. That is, drift and diffusion transport processes are balanced, such that no net charge transport occurs. As can be formally proven, this further guarantees that the Fermi diagram; refer to Figure 5.17) is position independent (constantlevel (E, in our energy band in x, spatially "flat"). Here is the million-dollar question: What is the fundamental sion? Give up? It is the position-dependent variation in particle driving force in diffu Ato point B. More precisely (yep, think calculus), in concentration from point semiconductors it is the carrier con centration gradient, Vn or Vp, (dn/dx and dp/dx in 1D) that drives the diffusion Kemember, if charge moves, by whatever means, current will flow. Said anothercurrents. way. if 1can induce avariation in nor p such that it is not constant in position in the crystal, its gradient function is nonzero, and current will flow, even with n0 applied drift field Think about thisit is rather remarkable. present. Sadly, deriving an expression for diffusion current density is abit more for drift current, and slightly beyond us here, but let me point you in theinvolved than right direc tion. Consider Figure 5.25, which shows a p-type sample with no field applied, but has nduced hole gradient present. ByFick's first law of diffusion for particles (be it electronsan Toles, or even "great-biscuit-smell molecules), we can write (in 3D) 0=-DVn (5.22) where Fis the particle "flux" (particles/cm' s) nis the particle concentration or density (number of particles/cm") Dis the s0-called "diffusion coefficient," simply a proportionality constant dependent on the physical context (dimensionally, (D| =cm²/s)  Silicon Earth: Introduction to Microelectronics and 150 Drift field = 0 Nanotechnolozy laitf P gradient flowsin response to the carrier concentration FIGURE 5.25 Model for determining the diffusion current that have Applying Fick's law to semiconductors, we Jp,difrusion=-qD,Vp (5.23) for the holes, and (5.24 Jn,diffusion =qD,Vn the negative charge of the elecu for the electrons (the difference in sign is just due to mechanisms, transport for both electrons anu Finally, combining our drift and diffusion holes, we have (523) n, total =Jn,drit +Jn,difusion = qHnn8 +qD,Vn, Jp,total Jpdrit +Jp,diftusion qHpp8-qD,Vp, with atotal current density of transport "drift-diffusion semicondu Not surprisingly, these two equations are known as the flow in t oreal equations." Clearly, we have four potential sources for current relevant typicalh tors-pretty neat, huh?! Are all four of these transport mechanisms re transport tras semiconductor devices? You bet! In a MOSFET, for instance, dritt ditfusion dominates the current, whereas for pn junctions and bipolar transistors, port dominates. Semiconductors: Lite! 151 Oneother interesting tidbit. Given this final result for current flow, a logical question nresents itself (for you deep thinkers!). Given that drift anddiffusion represent twofunda mentally different means nature has for moving charge in asemiconductor, is there some ekace between the two transport mechanisms? There is! As first proved by Einstein, one can show at avery deep level that D, kT (5.28) for electronsand D, kT (5.29) for holes. That is, drift and diffusion are linked by the respective mobility and diffusion coefficients and connected by our most fundamental parameter in the universe-kT. Slick, huh?! These are the so-called "Einstein relations" (refer to the "Geek Trivia: The Einstein Relation'" sidebar for some Geek trivia). Appealingly simple. GEEK TRIVIA: THE EINSTEIN RELATION lf you are a student of history, you might be curious as to how Einstein arrived at this pivotal result. Hint: lt had nothing to do with semiconductors! In1905, the so-called annus mirabilis of physics, Einstein published "On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat" (5], the result of his study of Brownian motion as a means for examining the basic underpinnings of Boltzmann's kinetic theory of heat. The oh-so-simple Einstein relation elegantly falls out of that analysis and applies generally to any "dilute gas" (in our case, electrons and holes in semiconductors). You can also derive it directly from Equation 5.25 by assuming the total current is zero (ie., equilibrium). Dare ya! Generation and Recombination h quick review. Semiconductors contain energy bands, and the electron and hole densities In those bands can be varied controllably by means of, say, doping. We can also move those electrons and holes around at high speed by means of either drift or diffusion (or both) to Ceate current flow. Fine. What else? Last topic (I promise!). If Ttake a semiconductor out Of its equilibrium state, by whatever means, what tools does nature have at its disposal to reestablish equilibrium? Goodquestion! In general, whenasemiconductor is perturbed from equilibrium, either an excess or a MeiCit of localcharge density (electron or hole, or both) is inevitably produced relative to their equilibrium concentrations. Nature does not take kindly to this, given that it repre- DEnts ahigher-energy configuration, and will promptly attempt to rectify the situation (G/R for short) td restore equilibrium. How? Well, carrier generation-recombination  Microelectronics and N Silicon Earth: Introduction to 152 is nature's order-restoring mechanism in semiconductors. In essence, G/R is the means Nanotechnology stabilized (if the perturbation carrier density excess or deficit is remains) by which or eliminated either alocal(if the perturbation is subsequerntly removed). Formally, "genera- created, whereas electrons and holes are tion" is any process any process which "recombination" by which electrons and holes are annihilated. Because nonequilibrium Condi-is by tions prevail in all semiconductor devices while in use, clearly G/R processes play a piv- how they are designed, built, and utilized. otalrole in shaping how transistors are used to spaWn a we can describe in a nutshell Intuitively, Communications Revolution. (Ms.?) Transistor. " Step 1: Build Mr. equilibrium twiddling its thumbe transistor sits comfortably in Step 2: Said screaming) momentarily o.. said transistor (kicking and " Step 3: We take equilibrium. Mother Nature (yep, definitely female!) doesn't take kindly to Step 3 (or iuch Step 4: out of her pocket to restore es:! in and pulls G/R " Step 5: Mother Nature steps transistor to equilibrium. the current Nature or said transistor, we farm Step 6: Unbeknownst to Mother useful work! processes to do somne generated by the G/R restoration Revolution! Step 7: Presto-Communications fooled Mother Nature?* Perhaps. After all, we humans are pretty darn clever. Have we on. Still, nothing comes totally for free. Read work? Let's revisit our energy band model. The two So how do G/R processes actually "band-to-band illustrated in Figures 5.26 and 5.27: most important G/R processes are what energy bands are, but what about "traps"? All G/R" and "trap-center G/R." We know impurities. If contain some finite, albeit tiny, density of defects or imperfections oraffectionately crystals with the electrons or holes, they are those imperfections can communicate or referred to as "trap" states (meaning: they for can capture [trap] and hold onto an electron i). some amount of time tr and then release hole from the conduction or valence band we need to create an electron to The band-to-band G/R processes are straightforward. If conductton directly from the valence band to the remove adeficit, an electron is excited the e event effectively annihilates involved, band. Conversely, the band-to-band recombination between Ec and Ey is tron. In trap-center G/R, a trap state located at energy E, create an electron, the electron and the G/R process is now atwo-step process. Thatand is, to band then released by the trap is first captured by the trap from the valence involved Conduction band. is nature is for free) equal Clearly energy exchange (read: nothing that happens in 5.27, AE to first order is the in G/R processes, and, as can be seen in Figures 5.26 and annihilated, where ddoes go to the bandgap energy E,. If, for instance, an electron is -it has to conservation of energy Caffeine energy go? (Please don't feel tempted to violate materials (refer to energy-release somewhere!) In recombination events, direct bandgap dominant phonons(lat- Alert #1, sorry!) favor the production of photons (light) as the of behind which mechanism, whereas indirect bandgap materials favor the production tice vibrations-heat). Not surprisingly, the fundamental differences the197s. commercialsfrom * Older readers may recall the "It's not nice to fool Mother Nature!" margarine Semiconductors: Lite! 153 Band-to-band Trap-center generation generation Thermal energy (k7) E AE -Eç-Ef AE= E, Ef ----0 AE = E,-Ey Step 1 Step 2 FIGURE 5.26 Illustration of band-to-band and trap-center thermalgeneration of an electron and a hole. Band-to-band Trap-center recombination Tecombination Eç AE =Ec-Et AE =Eg E, -------- AE =Er-Ey Ey Xo Xo Thermal energy (kT) Step 1 Step 2 FIGURE 5.27 IIlustration of band-to-bandand trap-center thermal recombination of an electron and a hole. Semiconductors mnake the best electronicversus photonic devices originate here. Silicon, for instance,cannot be used to create a good diode laser for a DVD player. Makes a great transistor, though! This is why. For analyzing semiconductor devices, knowing the netrate of recombination U(number of carrier annihilations per second per unit volume) is important. Although the solution to this fundamental semiconductor problem is beyond us here (solved first by Shockley and cOworkers), with sufficient approximation a very simple answer results: T, (5.30)  Silicon Earth:Introduction to Microelectronics and N 154 Nanotechnolozy 10-4 (s) Electrons lifetime 10-5 carrier -Holes l0-6 Minority 10-/ 10-8 10-9 10-10| lo19 1020 1015 1016 1017 1018 Doping concentration N or Np (cm) FIGURE 5.28 Electron and hole minority carrier lifetimes in silicon as furnctions of doping concentration at 300 K. That is, the net rate of electron recombination in a p-type material, say, is approximarey equal to the "excess" minority electron density (An =1, - n0, Where the 0subscript repre sents the equilibrium value), divided by a newly defined parameter called the "minorty carrier G/R lifetime" or just "lifetime" (t). The carrier lifetime is an intuitive concepl baru It measures the average amount of time it takes an excess electron in the conduction variables(the recombine with a hole and thus be annihilated. Clearly, T, depends on many level,the the doping energy of the G/R traps, their density, the trap-capture probabilities, measured. Figure? bandgap, temperature, etc.), but importantly, it can be fairly easily functions of doping shows minority electron and hole lifetime data for silicon at 300 K, as microseconds of level. Good rule of thumbin silicon at 300 K, lifetimes are in the range increasing dop with (10) for reasonable doping levels. The observed decrease in lifetime timetorecombie ing should not surprise you. If Icreate an extra electron, it willl take lessdoesn't havetowait if lots of holes are present (high p-type doping). That is, Mr. Electron long to find Ms. Hole (or viceversa!) and consummate their suicide pact! The Semiconductor Equations of State ot rlket sigh toughest Finally, we come to the end of this nmodestly weighty chapter (okay, breathea the and repeat after me: "Not so bad. Igot this!"). Cheer up; you've just conquered ditfusiontrats semion chapter in the book! Hooray! Energy bands, electrons and holes, drift and any i a port, and now G/R processes. This is actually all one needs to: solve virtuallyequations ductor device problem. I close this chapter by bringing together all of our  Semiconductors:Lite! 155 And God Said ßn ßt - G, - U, : -G, -U, - p9ppE -q D, Vp J, =q H, n 8 + q D, Vn p= g(p -n+N, - N,) And The Transistor Was Born. FIGURE 5.29 The semiconductor equations of state one place. Shown in Figure 5.29 are the so-called Maxwell's equations if you will, which embody all semiconductor "equations of state," our of the physics we need to tackle any electronic or photonic problenm. You will recognize the drift-diffusion equation, and charge neutrality. The first two equations are known as equations, Poisson's tinuity equations," and are simply charge-accounting equations that couplethe "current-con dynamics (time dependence) and G/R processes to the currents. If youdo not the carrier speak vec tor calculus, don't be dismayed. If you happen to, appreciate the beauty. With equations, the transistor was indeed conceived and born! these seven Iime Out: Gut Check on Your Comprehension (Read'um and Weep!) 1. What is the difference between asemiconductor, a metal, and an insulator? 2. How many different forms does silicon come in? Name them. 3. What fundamentally makes semiconductors so useful for building electron and photonic devices?. How does resistivity differ from resistance? What does that matter? O. In general terms, what is a scientific model? 156 Silicon Earth: Introduction to Microelectronics and 6. What is the energy bandgap in a semiconductor? 7. What is a hole? Nanotechnol gy 8. What is the law of mass action? Why is it useful? 9. What is the driving force behind drift transport? Diffusion transport> 10. What defines equilibrium in a semiconductor? 11. What is generation/recombination? Why do wecare? References 1. R.F. Pierret, Semiconductor Device Fundamentals, Addison-Wesley, Reading, MA, 1996. 2. B.L. Anderson and R.L. Anderson, Fundamentals of Semiconductor Devices, McGraw:ltil New York, 2005. 3. R.M. Warner Jr. and B.L. Grung, Transistors: Fundamentals for the Integrated-Circuit Enginer Wiley, New York, 1983. 4. R. de L. Kronig and W.G. Penney, Quantum mechanics of electrons in crystal lattices, in Proceedings of the Royal Society, A130, 499, 1931. 5. A. Einstein, Investigations on the theory of Brownian Movement, Annalen der Physik, 17, 549-560, 1905. 6. L. Pauling and R. Hayward, The Architecture of Molecules, Freeman, San Francisco, CA, 1964.