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Department of Electronics Nanoelectronics 03 Atsufumi Hirohata 17:00 17/January/2014 Friday (P/L 001)

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Quick Review over the Last Lecture Maxwell equations : Time-independent case : ( Ampère’s / Biot-Savart ) law ( Gauss ) law ( Gauss ) law for magnetism ( Faraday’s ) law of induction ( Initial conditions ) ( Assumptions ) ( Time-dependent equations ) Electromagnetic wave : ( Electric ) field ( Magnetic ) field Propagation direction propagation speed : in a vacuum,

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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)

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03 Scalar and Vector Potentials Scalar potential Vector potential A Lorentz transformation

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Maxwell Equations Maxwell equations : E : electric field, B : magnetic flux density, H : magnetic field, D : electric flux density, J : current density and : charge density Supplemental equations for materials : Definition of an electric flux density Definition of an magnetic flux density Ohm’s law

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Electromagnetic Potentials Scalar potential and vector potential A are defined as By substituting these equations into Eq. (2), (1) (2) (3) (4) Similarly, y - and z -components become 0. Here, Also, Satisfies Eq. (2).

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Electromagnetic Potentials (Cont'd) (1) (2) (3) (4) Satisfies Eq. (4). By substituting these equations into Eq. (4),

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Electromagnetic Potentials (Cont'd) Gauge transformation Here, A ’ and ’ provide E and B as the same as A and . By assuming x as a differentiable function, we define Therefore, electromagnetic potentials A and contains uncertainty of x. In particular, when A and satisfies the following condition : Laurentz gauge (1) (2) (3) (4) Under this condition, Eqs. (1) and (3) are expressed as Equation solving at home

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Einstein's Theory of Relativity In 1905, Albert Einstein proposed the theory of special relativity : Speed of light (electromagnetic wave) is constant. * Lorentz invariance for Maxwell’s equations (1900) Poincaré proved the Lorentz invariance for dynamics. Lorentz invariance in any inertial coordinates

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Scalar Potential Scholar potential holds the following relationship with a force : * The concept was first introduced by Joseph-Louis Lagrange in 1773, and named as scalar potential by George Green in **

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Scalar Potential and Vector Potential A Scalar potential holds the following relationship with a force : * Voltage potential induced by a positive charge Electrical current Vector potential around a current rot A Magnetic field Vector potential A (decrease with increasing the distance from the current) Electric field E (induced along the direction to decrease the voltage potential) Magnetic field H (induced along the axis of the rotational flux of the vector potential, rot A )

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Vector Potential A Faraday found electromagnetic induction in 1831 : * Faraday considered that an “electronic state” of the coil can be modified by moving a magnet. induces current flow. In 1856, Maxwell proposed a theory with using a vector potential instead of “electronic state.” Rotational spatial distribution of A generates a magnetic flux B. Time evolution of A generates an electric field E.

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Maxwell's Vector Potential * magnetic line of forcevector potential electrical current electrical current magnetic field By the rotation of the vector potentials in the opposite directions, rollers between the vector potentials move towards one direction. Ampère’s law After the observation of a electromagnetic wave, E and B : physical quantities A : mathematical variable

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Aharonov-Bohm Effect ** Yakir Aharonov and David Bohm theoretically predicted in 1959 : * Electron can modify its phase without any electrical / magnetic fields. electron source coil shield phase shift interference magnetic field

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Observation of the Vector Potential ** org/jpn/books/kaishikiji/200012/ html In 1982, Akira Tonomura proved the Aharonov-Bohm effect : * Nb NiFe No phase shift Phase shift = wavelength / 2

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