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3.23 Fall 2007 – Lecture 15 ANHARMONICITYLast timeStudyCarrier concentration in a p-n junctionWhat is the built-in voltage Vbi?Qualitative Effect of BiasRectificationSemiconductor solar cellsBipolar Junction TransistorField-effect TransistorBloch oscillationsConductivity in semiconductorsOhmic to ballistic conductanceElectron transport at the nanoscaleElectron transport at the nanoscaleBallistic TransportConductance from transmissionQuantum transport in CNTsPhonons 1Phonon dispersions in diamondPhonon dispersions in graphitePhonons 2Phonons 3Phonons 4Phonons 5Phonon 6Thermal Contraction in 2-d and 1-d CarbonGrüneisen parameters for SWNTMIT OpenCourseWare http://ocw.mit.edu 3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)3.23 Fall 2007 – Lecture 15ANHARMONICITYGalloping Gertie (Tacoma Narrows Bridge, the old one…)Image from Wikimedia Commons, http://commons.wikimedia.org/wiki/Main_Page.3.23 Electronic, Optical and Magnetic Properties of Materials ‐ Nicola Marzari (MIT, Fall 2007)Last time1. Chemical potential as a function of T: intrinsic and extrinsic case2. Population of impurity levels3. Equilibrium carrier densities in impure semiconductors, and simplified expressions4. p-n junction: depletion layer/space charge, built-in voltage, operation under bias and rectificationStudy• Singleton, most appropriately, scattered around.3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Carrier concentration in a p-n junctionPlease see http://commons.wikimedia.org/wiki/Image:Pn-junction-equilibrium.svg.Image removed due to copyright restrictions.Image removed due to copyright restrictions. Please see http://commons.wikimedia.org/wiki/Image:Pn-junction-equilibrium-graphs.png. .pngWhat is the built-in voltage Vbi?EcµEvpqVbi=µn‐ µpnlnDnibiNkTnμμ⎛⎞=+⎜⎟⎝⎠lnapibiNkTnμμ⎛⎞=−⎜⎟⎝⎠2lnbadbiikT N NVqn⎛⎞∴=⎜⎟⎝⎠nnnppnppqVbiqVbi• Forward bias (+ to p, - to n) decreases depletion region, increases diffusion current exponentially• Reverse bias (- to p, + to n) increases depletion region, and no current flows ideallyEcEvnnnppnppqVbi‐q|Va|Qualitative Effect of BiasEcµEvnnnppnppqVbi+q|Va|qVbi+q|Va|qVbi‐q|Va|Forward BiasReverse Bias+Va‐⎟⎟⎠⎞⎜⎜⎝⎛−=⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛+= 1122TkqVoTkqVdihhaieebabaeJeNnLDNnLDqJqTkDbii=μiiiDLτ=VILinear,OhmicRectification,Non‐linear, Non‐OhmicV=IRV=f(I)Solve minority carrier diffusion equations on each side and determine J at depletion edgeRectification3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Image removed due to copyright restrictions.Please see: Fig. 29.5 in Ashcroft, Neil W., and Mermin, N. David. Solid State Physics.Belmont, CA: Brooks/Cole, 197.Semiconductor solar cells3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Image removed due to copyright restrictions. Please see http://commons.wikimedia.org/wiki/Image:Pn-junction-equilibrium.svg.Bipolar Junction Transistor3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)++--forward backwardField-effect Transistor3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Image removed due to copyright restrictions. Please see http://commons.wikimedia.org/wiki/Image:Lateral_mosfet.svg.Bloch oscillations3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)Image removed due to copyright restrictions. Please see: Fig. 9.1 in Singleton, John. Band Theory and Electronic Properties of Solids.Oxford, England: Oxford University Press, 2001.Conductivity in semiconductors3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)eheheheeehheeene nemmememττστμτμ=+==2jneveEvmnejEmττ=−=−=Text removed due to copyright restrictions.Please see: Table 3 in Kittel, Charles. "Introduction to Solid State Physics." Chapter 8 in Semiconductor Crystals. New York, NY: John Wiley & Sons, 2004.Ohmic to ballistic conductanceWhat happens when electric field is applied?• If we reduce the length conductance grows indefinitely! • Experiment shows limiting value Gc. • This resistance comes from contactsImage removed due to copyright restrictions. Please see Fig. 1.7.2 in Datta, Supriyo. Electronic Transport in Mesoscopic Systems.New York, NY: Cambridge University Press, 1995.Electron transport at the nanoscale3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)–Short length > Few scattering events > Phase coherency–Wave character becomes importantS. Franks et al., Science 280, 1744 (1998)–~μm, room temperature– 50 % of the theoretical value – Very high current density > non-dissipative transportMulti-walled carbon nanotubesImages removed due to copyright restrictions.Please see: Fig. 1 and 2 in Frank, Stefan, et al. "Carbon Nanotube Quantum Resistors." Science 280 (June 1998): 1744-1746.Electron transport at the nanoscale3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)W. Liang et al., Nature 411, 665 (2001)Images removed due to copyright restrictions.Please see: Fig. 1 and 3a in Liang, Wenjie, et al. "Fabry-Perot Interference in a Nanotube Electron Waveguide." Nature 411 (June 2001): 665-669.Ballistic Transport3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Quantum conductance of an ideal ballistic conductor∫∑∑+∞∞−++++=∂∂== (E)dEfhe(E)fkELe(E)νfLeIkk21=VheeμμhedEEfEfheIII22122)(2 )](-)( [2-=−==−=∫∞+∞−+−+heVdIdG22==conductance quantumNo scattering, length-independent ! NchNch=3µ1µ1µ2µ2I+I-kEFigure by MIT OpenCourseWare.Conductance from transmission• Predominant “wave” character ¨ Solve the Schrödinger equation ( = t*t)Quantum transport in CNTs3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007) Temperature / Length / Phonons …– Very short CNT >conductance independent of length and temperature– Longer CNT >conductance decreases as temperature increasesdue to the


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