VIEWS: 3 PAGES: 31 POSTED ON: 1/5/2011
EE-656: Lecture 1 Introduction: Why Take Another Course? Muhammad A. Alam Electrical and Computer Engineering Purdue University West Lafayette, IN USA Fall 2007 NCN www.nanohub.org Alam / Lundstrom ECE-656 F07 1 outline 1) The reason to take EE656 2) Relation to other courses you have taken 3) Introduction to Boltzmann Transport Equation 4) A few official things Alam / Lundstrom ECE-656 F07 2 EE656 and other courses at Purdue Device-specific EE 695F: EE xx: ?? system design RF Design Opto-system Bio-system Design Application specific EE 654: EE 612: ?? device operation Opto/MW/Magnetics CMOS Bio-Systems Physical Principle of EE 606: EE 604 device Operation Basic Semi Dev. EM, Magnetics EE 656: EE 659: ?? Semi-Transport Quantum Transport Bio-physics Foundation Quantum Statistical Mechanics Mechanics Alam / Lundstrom ECE-656 F07 3 Drift-diffusion and Poisson equations gN n 1 J(x) J(x+dx) J N rN g N The equations are pretty good n t q and have served us well for the last J N qn N E qDN n 50 years… rN p 1 We can analyze complex devices J P rP g P t q like MOSFETs, bipolar transistors, J P qp P E qDP p lasers, etc. with very good agreement with experiments …. E q p n ND N A Then why do we need EE656 ? Alam / Lundstrom ECE-656 F07 4 …. validity of the eqs. no longer guaranteed! Consider a n-type semiconductor in SS without R-G …. n 1 J N rN g N t q J N qn N E qDN n 1 0 JN q p 1 J P rP g P t q J N qn N E J P qp P E qDP p E q p n ND N A Alam / Lundstrom ECE-656 F07 5 Summary of Advanced Semiconductor Fundamentals I J N q n N E V Carrier velocity Density Depends on chemical composition, crystal structure, temperature, doping, etc. Quantum Mechanics + Equilibrium Statistical Mechanics Encapsulated into concepts of effective masses and occupation factors (Ch. 1-4) Transport with scattering, non-equilibrium Statistical Mechanics Encapsulated into drift-diffusion equation with recombination-generation (Ch. 5 & 6) Alam / Lundstrom ECE-656 F07 6 …. n and f computed without E-field! J N qn N E Fn dk g 2 D (k ) f ( x, k ) Ec 1 E(k) m* f ( x, k ) g 2 D (k ) EC 2 2 k 2 1 e 2 m* FN Fn … no distinction between +ve & k0 -ve k; is velocity zero !? Alam / Lundstrom ECE-656 F07 7 … velocity computed without reference to k! J N qn N E x x x x d ( m N * ) m * qE N dt N q N * E N E m N Which k are we talking about ? Average v perhaps, but how should we do the averaging? Alam / Lundstrom ECE-656 F07 8 inconsistent definition of f and v ! J N qn N E E(k) E(k) Fn Fn k0 k0 To resolve this puzzle, need to track of k-specific population … Alam / Lundstrom ECE-656 F07 9 …. even DOS g(E) has problems! J N qn N E dk g 2 D (k ) f ( x, k ) f ( x, k ) 1 q N g 2 D (k ) m* EC 2 2 k FN E N E * 2 1 e 2 m* m N Alam / Lundstrom ECE-656 F07 10 …. even DOS g(E) has problems! 2 2 (r) U C r r E r 2m0 U C r UC r a n,k r un,k r exp ik r 2 2 (r) U C r V r r E r 2m0 ? ? UC r V r UC r a V r a n,k r un,k r exp ik r g ( x, k )? Alam / Lundstrom ECE-656 F07 11 Position variable x …. J N qn N E dk g 2 D ( x, k ) f ( x, k ) 1 q N m* f ( x, k ) ? ( x, k ) E N E ? g 2 D ( x, k ) ? EC 2 2 k FN * 2 mN 1 e 2 m* Homogenous space All possible directions are available for electron scattering Alam / Lundstrom ECE-656 F07 12 Random Dopants and Inhomogenous Space Random Dopants Effective Percolation Media TCAD people do this by Monte Carlo all the time, but a more fundamental analysis is possible … Alam / Lundstrom ECE-656 F07 13 Scattering, effective-mass in nanodevices? What to do when scattering is small? V q N V N * L mN L What happens when L is undefined? What happens when m* is defined? Alam / Lundstrom ECE-656 F07 14 mobility at high fields? q N 0 * E N E 1 E m N E n n 1 EC What causes velocity saturation at high fields? Where does all the mobility formula in Medici come from? E Alam / Lundstrom ECE-656 F07 15 Power-dissipation, energy-flux, etc … Vacuum Bipolar MOSFET Future ?? Tubes Spintronics 1906-1950s 1947-1980s 1960-until now ? Temp Tubes bipolar MOS 1900 1920 1940 1960 1980 2000 2020 Alam / Lundstrom ECE-656 F07 16 Power-dissipation, energy-flux, etc … Power dissipation …. n 1 P (J N J P ) E ??? J N rN g N t q Power dissipation in J N qn N E qDN n forward-biased diode? p 1 J P rP g P t q J P qp P E qDP p Heat flux … E q p n ND N A Q (J N J P ) E … what velocity should I use? Alam / Lundstrom ECE-656 F07 17 Electric field, Magnetic Field, Temperature Many semiconductor devices n 1 J N rN g N work under Temperature t q difference between contacts J N qn N E qDN n (Peltier cooler), or with strong magnetic field (MRAM memories). p 1 J P rP g P t q J P qp P E qDP p www.micropelt.com Google image E q p n ND N A How to modify the equations to account for the additional forces? Alam / Lundstrom ECE-656 F07 18 … so we do need a course! Part 1: q N V High field ballistic transport with N * High field transport with undefined m* mN L Part 2: Scattering dominated transport at low E, T, 0 and B Fields; Balance Equations 1 E E n n 1 Part 3: EC When L is undefined – Percolation theory g(B,T ) .... Part 4: Nonequilibrium high field transport with rigorously computed scattering events and g(E,t, E(k)) bandstructures Alam / Lundstrom ECE-656 F07 19 outline 1) The reason to take EE656 2) Relation to other courses you have taken 3) Introduction to Boltzmann Transport Equation 4) A few official things Alam / Lundstrom ECE-656 F07 20 How is the course related to others at Purdue Device-specific EE 695F: EE xx: ?? system design RF Design Opto-system Bio-system Design Application specific EE 654: EE 612: ?? device operation Opto/MW/Magnetics CMOS Bio-Systems Physical Principle of EE 606: EE 604 device Operation Basic Semi Dev. EM, Magnetics EE 656: EE 659: ?? Semi-Transport Quantum Transport Bio-physics Foundation Quantum Statistical Mechanics Mechanics Alam / Lundstrom ECE-656 F07 21 outline 1) The reason to take EE656 2) Relation to other courses you have taken 3) Introduction to Boltzmann Transport Equation 4) A few official things Alam / Lundstrom ECE-656 F07 22 energy/position resolved transport by f(x,k,t) E(k) E(k0 ) particle E(k0 ) E k k0 E(k) EC (x) ETOT EC (x) E(k) k k0 x Alam / Lundstrom ECE-656 F07 23 f(x,k,t) are defined by the Boltzmann Equation n 1 J N rN g N f N t q ö r f N Fe p f N Cf N t J N qn N E qDN n p 1 J P rP g P f P t q ö r f P Fe p f P Cf P t J P qp P E qDP p E q p n ND N A E q p n ND N A Alam / Lundstrom ECE-656 F07 24 n vs. f(x,k,t): Why BTE is difficult N(x,y,z;t) …………….. 4 independent variables F(x,y,z;kx,ky,kz;t) ………7 independent variables 100x100x100x10 …. 107 million unknowns (100x100x100)x(100x100x100)x10 = 1013 unknowns! A pretty difficult problem ... We will need to make approximations (very large or very small tau, low vs. High-field transport, etc. …… Alam / Lundstrom ECE-656 F07 25 Assumptions of the BTE Independent particle transport -- originally developed for dilute, uncharged gas molecules like H, He, Ar, etc. Application to transport of charge carriers should be always be done carefully and results frequently checked against experiments. Assumes that at each x, an E(k) diagram exists and is governed by local material composition. Modification of DOS due to remote interference requires generalization of BTE to NEGF formulations. Alam / Lundstrom ECE-656 F07 26 consistent n, v, P, Q should follow from f(x,k,t) n( x, t ) dk gC (k ) f ( x, k ) J ( x, t ) dk gC (k ) f ( x, k ) (k ) P( x, t ) dk gC (k ) f ( x, k ) m * (k ) Q( x, t ) dk gC (k ) f ( x, k ) m * (k ) ............ Calculation of f(x,k,t) addresses aforementioned problems … but what equations do we use to compute f(x,k,t) Alam / Lundstrom ECE-656 F07 27 Summary 1) Semiclassical transport equations involving drift-diffusion and Poisson equations have had tremendous success in last 50 years in guiding design of electronic devices. 2) Some of the fundamental assumptions of the theory are falling apart for nanodevices. Also the range of new devices require broadening of the new equations. Moreover, physical origin of mobility models are seldom explored in elementary course. 3) Energy-resolved BTE will help us address many of these challenges. Alam / Lundstrom ECE-656 F07 28 Summary EE656 EE659 Analyze the Simple models assumptions for new devices EE606 The goal is to understand why simple models have worked so well so far and to see if we can develop similar elegant models for modern ultra-scaled semiconductor devices. Alam / Lundstrom ECE-656 F07 29 outline 1) The reason to take EE656 2) Relation to other courses you have taken 3) Introduction to Boltzmann Transport Equation 4) A few official things Alam / Lundstrom ECE-656 F07 30 Course Information Instructors Prof. Lundstrom and Prof. Alam Pre-requisite EE 606 or equivalent elementary Solid State Devices Course Books Advanced Semiconductor Fundamentals (QM, SM, Transport) Fundamentals of Carrier Transport (2nd Ed) by M. Lundstrom. Website http://cobweb.ecn.purdue.edu/~ee656/ -Th Office hours 5:00-6:00 T @ (EE 313D or EE-310) HW/Exams 5-6 HW (25%), Exam 1 (25%), Exam 2 (25%), Final (25%) Discussing HW is okay, copying is not. Will result in F grade Alam / Lundstrom ECE-656 F07 31