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Department of Electronics

Introductory Nanotechnology ~ Basic Condensed Matter Physics ~. Atsufumi Hirohata. Department of Electronics. Quick Review over the Last Lecture. k y. f(E ). 0. k x. 0. E. 0. Wave / particle duality of an electron :. Fermi-Dirac distribution ( T -dependence) :.

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Department of Electronics

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  1. Introductory Nanotechnology ~ Basic Condensed Matter Physics ~ Atsufumi Hirohata Department of Electronics

  2. Quick Review over the Last Lecture ky f(E) 0 kx 0 E 0 Wave / particle duality of an electron : Fermi-Dirac distribution (T-dependence) : Brillouin zone (1st & 2nd) : T1 1 T2 1/2 T3  T1 < T2 < T3

  3. Contents of Introductory Nanotechnology First half of the course : Basic condensed matter physics 1. Why solids are solid ? 2. What is the most common atom on the earth ? 3. How does an electron travel in a material ? 4. How does lattices vibrate thermally ? 5. What is a semi-conductor ? 6. How does an electron tunnel through a barrier ? 7. Why does a magnet attract / retract ? 8. What happens at interfaces ? Second half of the course : Introduction to nanotechnology (nano-fabrication / application)

  4. How Does an Electron Travel in a Material ? • Group / phase velocity • Effective mass • Hall effect • Harmonic oscillator • Longitudinal / transverse waves • Acoustic / optical modes • Photon / phonon

  5. How Fast a Free Electron Can Travel ? - - - - - Electron wave under a uniform E : Phase-travel speed in an electron wave : Wave packet Phase velocity : Group velocity : Here, energy of an electron wave is Accordingly, Therefore, electron wave velocity depends on gradient of energy curve E(k).

  6. Equation of Motion for an Electron with k For an electron wave travelling along E : Under E, an electron is accelerated by a force of -qE. In t, an electron travels vgt, and hence E applies work of (-qE)(vgt). Therefore, energy increase E is written by At the same time E is defined to be From these equations,  Equation of motion for an electron with k

  7. Effective Mass By substituting into By comparing with acceleration for a free electron :  Effective mass

  8. Hall Effect z z x x y y i l B Under an applications of both a electrical current i an magnetic field B : - - - - i - + + + + l E B + + + + hole + - - - - E Hall coefficient

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

  10. Strain x + dx x u (x + dx) u (x) Displacement per unit length : Young’s law (stress = Young’s modulus  strain) : S : area Here, For density of , Wave eqution in an elastomer Therefore, velocity of a strain wave (acoustic velocity) :

  11. Longitudinal / Transverse Waves Longitudinal wave : vibrations along their direction of travel Transverse wave : vibrations perpendicular to their direction of travel

  12. Transverse Wave Amplitude Propagation direction * http://www12.plala.or.jp/ksp/wave/waves/

  13. Longitudinal Wave Amplitude Propagation direction sparse dense sparse sparse dense sparse * http://www12.plala.or.jp/ksp/wave/waves/

  14. Acoustic / Optical Modes For a crystal consisting of 2 elements (k : spring constant between atoms) : un-1 vn-1 un vn un+1 vn+1 M m x a By assuming,

  15. Acoustic / Optical Modes - Cont'd Optical vibration Acoustic vibration * N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, London, 1976). Therefore, For qa = 0, For qa ~ 0, For qa = ,

  16. Why Acoustic / Optical Modes ? Oscillation amplitude ratio between M and m (A / B): Optical mode : Neighbouring atoms changes their position in opposite directions, of which amplitude is larger for m and smaller for M, however, the cetre of gravity stays in the same position. Acoustic mode : 1 All the atoms move in parallel. k =0 small k large k Optical Acoustic * M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

  17. Photon / Phonon Quantum hypothesis by M. Planck (black-body radiation) : Here, h : energy quantum (photon) mass : 0, spin : 1 Similarly, for an elastic wave, quasi-particle (phonon) has been introduced by P. J. W. Debye. Oscillation amplitude : larger  number of phonons : larger

  18. What is a conductor ? E E k k 0 0 1st 1st Forbidden E Allowed Number of electron states (including spins) in the 1st Brillouin zone : Here, N : Number of atoms for a monovalent metal As there are N electrons, they fill half of the states. By applying an electrical field E, the occupied states become asymmetric. • E increases asymmetry. • Elastic scattering with phonon / non-elastic scattering decreases asymmetry.  Stable asymmetry  Constant current flow  Conductor : Only bottom of the band is filled by electrons with unoccupied upper band.

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