2Quick Review over the Last Lecture Wave / particle duality of an electron :Particle natureWave natureKinetic energyMomentumBrillouin zone (1st & 2nd) :Fermi-Dirac distribution (T-dependence) :kxkyf(E)ET11T21/2T3T1 < T2 < T3
3Contents of Introductory Nanotechnology First half of the course :Basic condensed matter physics1. 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)
4How Does an Electron Travel in a Material ? Group / phase velocityEffective massHall effectHarmonic oscillatorLongitudinal / transverse wavesAcoustic / optical modesPhoton / phonon
5How 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 isAccordingly,Therefore, electron wave velocity depends ongradient of energy curve E(k).
6Equation 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 byAt the same time E is defined to beFrom these equations, Equation of motion for an electron with k
7Effective Mass By substituting into By comparing with acceleration for a free electron : Effective mass
8Hall Effect - - - - - + + + + + + + + + - - - - Under an applications of both a electrical current i an magnetic field B :xyz----i -++++lEBxyzBil++++hole +----EHall coefficient
9Harmonic Oscillator Lattice vibration in a crystal : spring constant : kumass : MHooke’s law :Here, we define 1D harmonic oscillation
10Strain Displacement per unit length : Young’s law (stress = Young’s modulus strain) :S : areaxx + dxHere,u (x)u (x + dx)For density of , Wave eqution in an elastomerTherefore, velocity of a strain wave (acoustic velocity) :
11Longitudinal / Transverse Waves Longitudinal wave : vibrations along their direction of travelTransverse wave : vibrations perpendicular to their direction of travel
12Transverse Wave Amplitude Propagation direction *
13Longitudinal Wave Propagation direction Amplitude sparse dense *
14Acoustic / Optical Modes For a crystal consisting of 2 elements (k : spring constant between atoms) :un-1vn-1unvnun+1vn+1MmaxBy assuming,
15Acoustic / 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 vibrationAcoustic vibrationFor qa = ,
16Why 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 : 1All the atoms move in parallel.k =0small klarge kOpticalAcoustic* M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).
17Photon / PhononQuantum hypothesis by M. Planck (black-body radiation) :Here, h : energy quantum (photon)mass : 0, spin : 1Similarly, for an elastic wave, quasi-particle (phonon) has been introducedby P. J. W. Debye.Oscillation amplitude : larger number of phonons : larger
18What is a conductor ?Number of electron states (including spins) in the 1st Brillouin zone :Ek1stHere, N : Number of atoms for a monovalent metalAs there are N electrons, they fill half of the states.By applying an electrical field E, the occupied statesbecome asymmetric.E increases asymmetry.Elastic scattering with phonon / non-elasticscattering decreases asymmetry.Ek1stForbiddenAllowed Stable asymmetry Constant current flow Conductor :Only bottom of the band is filled by electronswith unoccupied upper band.