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Department of Electronics

Introductory Nanotechnology ~ Basic Condensed Matter Physics ~. Atsufumi Hirohata. Department of Electronics. Quick Review over the Last Lecture. 3 states of matters :. solid. gas. liquid. ( small ). ( large ). ( large ). density. ( long ). ( short ).

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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 3 states of matters : solid gas liquid ( small ) ( large ) ( large ) density ( long ) ( short ) ordering range ( long ) ( short ) rigid time scale 4 major crystals : soft solid ( van der Waals crystal ) ( covalent crystal ) ( metallic crystal ) ( ionic crystal )

  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. What Is the Most Common Atom on the earth? • Phase diagram • Free electron model • Electron transport • Degeneracy • Electron Potential • Brillouin Zone • Fermi Distribution

  5. Abundance of Elements in the Earth Na Al Ca S Ni Mg (12.70 %) Fe (34.63 %) Si (15.20 %) O (29.53 %) Mantle : Fe, Si, Mg, ... Only surface 10 miles (Clarke number)

  6. Can We Find So Much Fe around Us ? Desert (Si + Fe2O3) Iron sand (Fe3O4) Iron ore (Fe2O3)

  7. "Iron Civilization" Today Iron-based products around us : Buildings (reinforced concrete) Qutb Minar : Pure Fe pillar (99.72 %), which has never rusted since AD 415. Bridges * Corresponding pages on the web.

  8. Major Phases of Fe Fe changes the crystalline structures with temperature / pressure : 56 Fe : Most stable atoms in the universe. T [K] Liquid-Fe 1808 -Fe bcc 1665 Phase change -Fe (austenite) fcc 1184 Martensite Transformation : ’-Fe (martensite) (-Fe) 1043 -Fe -Fe (ferrite) bcc hcp 1 p [hPa]

  9. Electron Transport in a Metal + + Effective energy for electrons Free electrons : - - r = 0.95 Å Na 3.71 Å r ~ 2 Å Na bcc : a = 4.28 Å 3s electrons move freely among Na+ atoms. Uncertainty principle : (x : position and p : momentum) When a 3s electron is confined in one Na atom (x  r) : (m : electron mass) For a free electron, r large and hence E  small. Inside a metal decrease in E - free electron

  10. Free Electron Model + + + + - + + + + - i + - Equilibrium state : For each free electron : thermal velocity (after collision) : v0i acceleration by E for i - - v0i - - i - - - - Average over free electrons : collision time :  Free electrons : mass m, charge -q and velocity vi Equation of motion along E : Using a number density of electrons n, current density J : Average over free electrons : drift velocity : vd

  11. Free Electron Model and Ohm’s Law Ohm’s law : For a small area :   where  : electric resistivity (electric conductivity :  = 1 / ) By comparing with the free electron model :

  12. Relaxation Time + + + + + + + + Number of non-collided Electrons N time  Resistive force by collision : If E is removed in the equation of motion : v -  For the initial condition : v = vd at t = 0 Equation of motion : with resistive force mv /  N0 For the initial condition : v = 0 at t = 0 For a steady state (t >> ),  : relaxation time  : collision time

  13. Mobility Here, = 0, as v0 is random. Equation of motion under E : Also, For an electron at r, collision at t = 0 and r = r0 with v0 Accordingly, Here, ergodic assumption : temporal mean = ensemble mean therefore, by taking an average over non-collided short period, Finally, vd = -E is obtained.  = q / m : mobility Since t and v0 are independent,

  14. Degeneracy For H - H atoms : Total electron energy Unstable molecule 1s state energy isolated H atom 2-fold degeneracy Stable molecule H - H distance r0

  15. Energy Bands in a Crystal For N atoms in a crystal : Total electron energy 2p 2p 6N-fold degeneracy 2s 2s 2N-fold degeneracy Forbidden band : Electrons are not allowed Allowed band : Electrons are allowed 1s 1s 2N-fold degeneracy Energy band Distance between atoms r0

  16. Electron Potential Energy Distance from the atomic nucleus Electron potential energy : V = -A / r Potential energy of an isolated atom (e.g., Na) : Na For an electron is released from the atom : vacuum level 0 3s 2p 2s 1s Na 11+

  17. Periodic Potential in a Crystal Potential energy in a crystal (e.g., N Na atoms) : Vacuum level Distance 3s 3s 2p 2p 2s 2s 1s 1s Na 11+ Na 11+ Na 11+ Electron potential energy • Potential energy changes the shape inside a crystal. • 3s state forms N energy levels  Conduction band

  18. Free Electrons in a Solid Total electron energy Wave number k Ni crystal electron beam Free electrons in a crystal : Total electron energy Energy band Wave / particle duality of an electron : Wave nature of electrons was predicted by de Broglie, and proved by Davisson and Germer. (h : Planck’s constant)

  19. 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

  20. 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

  21. Brillouin Zone - Exercise ky a kx ky 0 kx 0 Brillouin zone : In a 3d k-space, area where k ≠ 0. For a 2D square lattice, 2nd Brillouin zone is defined by nx= ± 1 ,ny = ± 1  ± kx±ky= 2/a nx, ny = 0, ± 1, ± 2, ... Reciprocal lattice : 1st Brillouin zone is defined by nx= 0,ny = ± 1  kx= ± /a nx= ± 1,ny = 0  ky= ± /a Fourier transformation = Wigner-Seitz cell

  22. 3D Brillouin Zone * C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986).

  23. Fermi Energy T ≠ 0 f(E) 0 E Fermi-Dirac distribution : E T = 0 EF Pauli exclusion principle At temperature T, probability that one energy state E is occupied by an electron : T = 0 1  : chemical potential (= Fermi energy EF at T = 0) kB : Boltzmann constant T1 ≠ 0 1/2 T2 > T1 

  24. Fermi-Dirac / Maxwell-Boltzmann Distribution Electron number density : Fermi sphere : sphere with the radius kF Fermi surface : surface of the Fermi sphere Decrease number density classical quantum mechanical Maxwell-Boltzmann distribution (small electron number density) Fermi-Dirac distribution (large electron number density) * M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989).

  25. Fermi velocity and Mean Free Path g(E) 0 E Fermi wave number kF represents EF : Fermi velocity : Under an electrical field : Electrons, which can travel, has an energy of ~ EF with velocity of vF For collision time , average length of electrons path without collision is Mean free path Density of states : Number of quantum states at a certain energy in a unit volume

  26. Density of States (DOS) and Fermi Distribution 0 0 E E Carrier number density n is defined as : T = 0 g(E) f(E) EF T ≠ 0 g(E) f(E) n(E) EF

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