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Nanoelectronics 02 Atsufumi Hirohata Department of Electronics 10:00 Tuesday, 13/January/2015 (B/B 103) Quick Review over the Last Lecture Nano-scale miniaturisation : reduction of ( effective electron paths ) reduction of ( ( faster electron scattering ) ) operation nano-fabrication ; ( complicated ( higher ( larger ) processes ) cost ) distributions in device properties ( leakage ) current ( Joule ) heating electron ( confinement ) Electron transport : •( •( diffusive ( ) transport electron scattering ) ballistic ) transport ( negligible electron scattering ) Contents of Nanoelectronics I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ? II. Electromagnetism (02 & 03) 02 Maxwell equations 03 Scalar and vector potentials III. Basics of quantum mechanics (04 ~ 06) IV. Applications of quantum mechanics (07, 10, 11, 13 & 14) V. Nanodevices (08, 09, 12, 15 ~ 18) Lecture notes and files can be found at http://www-users.york.ac.uk/~ah566/ 02 Maxwell Equations • • • Electromagnetic field Origins of an electromagnetic field Boundary conditions of an electromagnetic field Maxwell Equations Maxwell equations are proposed in 1864 : ì ¶D rot H = J + ï ¶t ï ï ¶B í rot E = ¶t ï ï div D = r ïî div B = 0 E : electric field, B : magnetic flux density, H : magnetic field, D : electric flux density, J : current density and : charge density Supplemental equations for materials : ì D = eE ï í B = mH ï J = sE î Definition of an electric flux density Definition of a magnetic flux density Ohm’s law * http://www.wikipedia.org/ Maxwell Equations - Origins of an electromagnetic field Maxwell equations : ì D = eE ï í B = mH ï J = sE î ì ¶D rot H = J + ï ¶t ï ï ¶B í rot E = ¶t ï ï div D = r ïî div B = 0 For a time-independent case, rot H = J Ampère’s law Biot-Savart law i dH H i Gauss law : An electrical charge induces an electric field. E Maxwell Equations - Boundary conditions of an electromagnetic field Maxwell equations : ì ¶D rot H = J + ï ¶t ï ï ¶B í rot E = ¶t ï ï div D = r ïî div B = 0 ì D = eE ï í B = mH ï J = sE î Faraday’s law of induction : N magnetic field force magnetic field N current current S force S Gauss law for magnetism : Conservation of magnetic flux * http://www.wikipedia.org/ Maxwell Equations in Free Space Maxwell equations : ì ¶D rot H = J + ï ¶t ï ï ¶B í rot E = ¶t ï ï div D = r ïî div B = 0 ì D = eE ï í B = mH ï J = sE î In free space (no electron charge, and , and : constant), ì ¶E rot H = s E + e ï ¶t ï ï ¶H rot E = m í ¶t ï ï div E = r e ïî div H = 0 By differentiating the first equation with t and substituting the second equation, ¶ ¶ ¶ æ ¶E ö ( rot H ) = (sE) + çèe ÷ø ¶t ¶t ¶t ¶t ¶E ¶ 2E \- rot rot E = sm + em 2 ¶t ¶t Maxwell Equations in Free Space (Cont'd) Here, the left term can be rewritten as Similarly, For an ideal insulating matrix, Electric field Magnetic field Electromagnetic wave propagation speed : v = in a vacuum, v= 1 e0m0 1 em =c * http://www.molphys.leidenuniv.nl/monos/smo/index.html Electromagnetic Wave * http://www.wikipedia.org/ Essence of the Maxwell Equations Maxwell equations unified electronics and magnetism : Electronics Magnetism Electron charge 1 Q1Q2 r 4 pe r 2 r 1 Q r E= 4 pe r 2 r 1 Q V= 4 pe r F= Source Force (Coulomb’s law) D = eE Field Potential Flux (Gauss’ law) Further unification with the other forces Einstein’s theory of relativity Magnetic dipole moment m1m2 r 4 pm r 2 r 1 m r H= 4 pm r 2 r 1 m f= 4 pm r F= 1 B = mH Michelson-Moley Experiment In 1881, Albert A. Michelson and Edward W. Morley precisely designed experiment to prove the presence of Ether : Ether was believed exist as a matrix to transfer an electromagnetic wave. No interference between parallel / perpendicular to Ether flow No sign of Ether No relative speed ! * http://www.wikipedia.org/ Einstein's Theory of Relativity In 1905, Albert Einstein proposed the theory of special relativity : Lorentz invariance for Maxwell’s equations (1900) Poincaré proved the Lorentz invariance for dynamics. Lorentz invariance in any inertial coordinates Speed of light (electromagnetic wave) is constant. c= 1 e0m 0 * http://www.wikipedia.org/ 0 10-43 s 10-35 s 10-12 s Unified Theory beyond the Maxwell Equations Big bang and Grand Unification Theory Gravity Weak nuclear force Big bang Weinberg-Salam Theory Maxwell Equation -decay Electromagnetic force Strong nuclear force nucleus * http://map.gsfc.nasa.gov