交通大學數學建模與科學計算研究中心 Quantum Hydrodynamic Modeling, Numerical Methods, and Applications Semiconductor Transistors Jinn-Liang Liu 劉晉良 National Hsinchu Univ. of Edu. 新竹教育大學 Jan. 25-27, 2010 Classical Computer Bit : Either 0 or 1 Microprocessor Microchips MOSFET Transistor The most important invention of the 20th century? A transistor is an electronic device used as a switch or to amplify an electric current or voltage. 1930 First Transistor Patent Filed by J. E. Lilienfeld in 1926 The First Transistor Invented at Bell Labs in 1947 The original version of the paper was rejected for publication by Physical Review on the referee's unimaginative assertion that it was 'too speculative' and involved 'no new physics.' Received his Ph.D. at University of Tokyo in 1959, Esaki was awarded the Nobel Prize in 1973 for research conducted around 1958 on electron quantum tunneling (Esaki Diode). 假設20歲年輕人之創造力是100%、辨別力是0%,70歲老年人創造力是 0%、辨別力是100%,人生分歧點是45歲。分析諾貝爾獎得主獲獎事由 和年齡關聯性,會發現得獎人年齡大多集中於35歲至39歲時,而我於 44歲發明人造量子結構。 MOSFET (Metal Oxide Semiconductor Field Effect Transistor) Semiconductor A semiconductor is a material that can behave as a conductor or an insulator depending on what is done to it. We can control the amount of current that can pass through a semiconductor. Kingfisher Science Encyclopedia Czochralski Crystal Growth Sand Gold Ingots Ingot Wafer Silicon Ingot Doping IC 12吋矽晶圓 Silicon Crystal Shared electrons Si Si Si Si Si Si Si Si - Si Doping Impurities (n-Type) Si Si Si As Si Si Si Si - Conducting band, Ec Extra Electron Ed ~ 0.05 eV Eg = 1.1 eV Si Valence band, Ev Doping Impurities (p-Type) Si Si Conducting band, Ec Si Hole Si B Eg = 1.1 eV Si Ea ~ 0.05 eV Si Si - Si Electron Valence band, Ev S. Roy and A. Asenov, Science 2005 MOSFET (Metal Oxide Semiconductor Field Effect Transistor) 2003 L = 4 nm Research 2005 L = 45 nm Production 2018 L = 7 nm Production 3D, 30nm x 30nm Gate Length: 90 nm (2005 In Production) (Device Size) 65 nm (2006 In Production) 34 nm (This Talk) Device Sizes Vs. Models Self-Adjoint Energy Transport Model (Chen & Liu, JCP 2003) Quantum Corrected Energy Transport Model (Chen & Liu, JCP 2005) L=IJ=34nm gate contact source contact drain contact I J E B n+ C C’ D’ D n+ B’ E’ interface layer junction layer junction layer pA F bulk contact Doping Concentration Energy Transport Model q S (n p N A N D ), (2.1) J n R, (2.2) J p R, (2.3) n 0 S n J n E n( ), (2.4) n p 0 S p J p E p( ), (2.5) p J n qn n qDnn S n J n n kBTn / q nTn • • • • • • • electrostatic potential n electron density p hole density J current density S energy flux E electric field R generationrecombination rate q(np ni ) R(n, p) 0 n ( p pT ) 0p (n nT ) 2 Auxiliary Relationships Self-Adjoint Formulation n qn qn ni exp u n2 n ni exp VT VT p qp qp ni exp v p2 p ni exp VT VT p n n qn n2 VT ln uni qp p2 VT ln vni New Variables Bohm’s Quantum Potential qn qp n , n 2 p , * 2m p q p 2 2mn* q Planck' s Constant O(10 ) -34 J n q n n( qn ) qDnn J p q p p( qp ) qD pp J n R a fourth order PDE in n Self-Adjoint QCET Model F , (3.1) J n R , (3.2) J p R , (3.3) n Z n , (3.4) p Z p , (3.5) G n Rn , (3.6) G p R p , (3.7) Singularly Perturbed QCET Model (Liu, Lee, & Chen, 2009 Preprint) Dimensionless Scaling • Nano devices extremely singular • Boundary layer • Junction layer • Quantum potential layer Adaptive Algorithm Initial Initialmesh mesh Preprocessing Preprocessing F , Jn R , Gummel Gummelouter outeriteration iteration J p R , (3.3) Solve SolvePoisson PoissonEq. Eq. Solve Solve u , v, n , p Yes Error Error>>TOL TOL Solve Solve g n , g p Error Error>>TOL TOL No Post-Process Post-Process n Z n , (3.4) p Z p , (3.5) G n Rn , (3.6) No Error ErrorEstimation Estimation (3.1) (3.2) Refinement Refinement Yes G p R p , (3.7) Finite Element Method Monotone Iteration Exponential Fitting The Final Adaptive Mesh 100 80 Depth (nm) 60 40 20 0 0 20 40 60 Transverse Distance (nm) 80 100 Electron Temperature Hole Quantum Potential Electron Current Density Drain Current for MOSFET 4.5 ET DG DGET 4 3.5 2.5 2 I DS (mA/ m) 3 1.5 1 0.5 0 0 0.1 0.2 0.3 0.4 0.5 VDS (V) 0.6 0.7 0.8 0.9 1 60% difference predicted by ET and QCET models Alternative Future MOS MOS Scaling Challenges 1. Technology Scaling Parasitic Effects: Leakage, Capacitance, Risistance 2. Power Limits: End of Voltage Scaling 3. Band-Structure Engineering 4. Scattering: e-insulator, e-imp, e-ph, e-e 5. Dopant Atom Fluctuations 6. Non-Equilibrium Electron & Phonon Distrb. 7. Long Range Coulomb Interactions 8. Full-Band Bias-Induced Quantization 9. Phonon Transport Models 10.Automatic Multi-Scale Computing MOS Simulation Challenges Conclusion Self-Adjoint QCET Model: More Advanced Technology Scaling Challenges in Physics and Engineering Muti-Scaling Modeling and Numerical Methods High-Performance Architecture, Algorithms, and Coding