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**Introductory Nanotechnology ~ Basic Condensed Matter Physics ~**

Atsufumi Hirohata Department of Electronics

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**Quick Review over the Last Lecture**

Wave / particle duality of an electron : Particle nature Wave nature Kinetic energy Momentum Brillouin zone (1st & 2nd) : Fermi-Dirac distribution (T-dependence) : kx ky f(E) E T1 1 T2 1/2 T3 T1 < T2 < T3

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**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)

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**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

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**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).

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**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 vgt, and hence E applies work of (-qE)(vgt). 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

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**Effective Mass By substituting into**

By comparing with acceleration for a free electron : Effective mass

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**Hall Effect - - - - - + + + + + + + + + - - - -**

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

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**Harmonic Oscillator Lattice vibration in a crystal :**

spring constant : k u mass : M Hooke’s law : Here, we define 1D harmonic oscillation

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**Strain Displacement per unit length :**

Young’s law (stress = Young’s modulus strain) : S : area x x + dx Here, u (x) u (x + dx) For density of , Wave eqution in an elastomer Therefore, velocity of a strain wave (acoustic velocity) :

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**Longitudinal / Transverse Waves**

Longitudinal wave : vibrations along their direction of travel Transverse wave : vibrations perpendicular to their direction of travel

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**Transverse Wave Amplitude Propagation direction**

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**Longitudinal Wave Propagation direction Amplitude sparse dense**

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**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 a x By assuming,

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**Acoustic / Optical Modes - Cont'd**

Therefore, For qa = 0, For qa ~ 0, * N. W. Ashcroft and N. D. Mermin, Solid State Physics (Thomson Learning, London, 1976). Optical vibration Acoustic vibration For qa = ,

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**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).

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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

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What is a conductor ? Number of electron states (including spins) in the 1st Brillouin zone : E k 1st 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. E k 1st Forbidden Allowed Stable asymmetry Constant current flow Conductor : Only bottom of the band is filled by electrons with unoccupied upper band.

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Lattice Vibrations, Part I

Lattice Vibrations, Part I

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