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Department of Electronics

Nanoelectronics 10. Atsufumi Hirohata. Department of Electronics. 10:00 17/February/2014 Monday (G 001). Quick Review over the Last Lecture. Major surface analysis methods :. Contents of Nanoelectonics. I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ?.

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Department of Electronics

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  1. Nanoelectronics 10 Atsufumi Hirohata Department of Electronics 10:00 17/February/2014 Monday (G 001)

  2. Quick Review over the Last Lecture Major surface analysis methods :

  3. Contents of Nanoelectonics I. Introduction to Nanoelectronics (01) 01 Micro- or nano-electronics ? II. Electromagnetism (02 & 03) 02 Maxwell equations 03 Scholar and vector potentials III. Basics of quantum mechanics (04 ~ 06) 04 History of quantum mechanics 1 05 History of quantum mechanics 2 06 Schrödinger equation IV. Applications of quantum mechanics (07, 10, 11, 13 & 14) 07 Quantum well 10 Harmonic oscillator V. Nanodevices (08, 09, 12, 15 ~ 18) 08 Tunneling nanodevices 09 Nanomeasurements

  4. 10 Harmonic Oscillator • 1D harmonic oscillator • 1D periodic potential • Brillouin zone

  5. Harmonic Oscillator spring constant : k mass : M u Lattice vibration in a crystal : Hooke’s law : Here, we define  1D harmonic oscillation

  6. 1D Harmonic Oscillator spring constant : k mass : M x For a 1D harmonic oscillator, Hamiltonian can be described as : Here, k=m2. By substituting this to the Schrödinger equation, Here, for x ,   0. By substituting xwith  ( : a dimension of length and  : dimensionless) By dividing both sides by in order to make dimensionless, Simplify this equation by defining

  7. 1D Harmonic Oscillator (Cont'd) For || , (zero-point energy) (lowest eigen energy) In general, By substituting this result into the above original equation, = Hermite equation by classical dynamics

  8. 1D Periodic Potential In a periodic potential energy V(x) at ma (m=1,2,3,…), V (x) K : constant (phase shift : Ka) Here, a periodic condition is x 4a 0 a 2a 3a A potential can be defined as Now, assuming the following result (0x<a), (1) For a x<2a, (2) Therefore, for ma x<(m+1)a, by using

  9. 1D Periodic Potential (Cont'd) By taking x a for Eqs. (1) and (2), continuity conditions are In order to obtain A and B( 0), the determinant should be 0.

  10. 1D Periodic Potential (Cont'd) Now, the answers can be plotted as In the yellow regions,  cannot be satisfied.  forbidden band (bandgap) * http://homepage3.nifty.com/iromono/kougi/index.html

  11. Brillouin Zone Total electron energy k 0 Bragg’s law :  In general, forbidden bands are a n=1, 2, 3, ... For  ~ 90° ( / 2),  reflection Therefore, no travelling wave for n=1, 2, 3,... Allowed band  Forbidden band Allowed band : Forbidden band Allowed band  1st Brillouin zone Forbidden band Allowed band 1st 2nd 2nd

  12. Periodic Potential in a Crystal E Allowed band Forbidden band Allowed band Forbidden band Allowed band k 0 1st 2nd 2nd Energy band diagram (reduced zone)  extended zone

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