Report

Introductory Nanotechnology ~ Basic Condensed Matter Physics ~ Atsufumi Hirohata Department of Electronics Quick Review over the Last Lecture 3 states of matters : solid gas liquid density ( large ) ( large ) ordering range ( long ) ( short ) rigid time scale ( long ) ( short ) ( small ) 4 major crystals : soft ( van der Waals crystal ) solid ( metallic crystal ( ) ( covalent crystal ionic crystal ) ) 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) What Is the Most Common Atom on the earth? Phase diagram • Free electron model • Electron transport • • Electron Potential • • • Degeneracy Brillouin Zone Fermi Distribution Abundance of Elements in the Earth Ca Al Ni S Na Mg (12.70 %) Fe (34.63 %) Si (15.20 %) O (29.53 %) Mason (1966) Only surface 10 miles (Clarke number) Mantle : Fe, Si, Mg, ... Can We Find So Much Fe around Us ? Iron sand (Fe3O4) Iron ore (Fe2O3) Desert (Si + Fe2O3) "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. Major Phases of Fe Fe changes the crystalline structures with temperature / pressure : 56 T [K] Liquid-Fe Fe : Most stable atoms in the universe. 1808 -Fe bcc 1665 Phase change -Fe (austenite) fcc 1184 Martensite Transformation : (-Fe) ’-Fe (martensite) 1043 -Fe -Fe (ferrite) bcc hcp 1 p [hPa] Electron Transport in a Metal Free electrons : - r = 0.95 Å + Na + 3.71 Å r~2Å Na bcc : 3s electrons move freely a = 4.28 Å among Na+ atoms. Uncertainty principle : xp (x : position and p : momentum) When a 3s electron is confined in one Na atom (x r) : p p r E p 2m r 2m 2 2 2 2mr2 (m : electron mass) For a free electron, r large and hence E small. Effective energy for electrons Inside a metal - free electron decrease in E Free Electron Model Equilibrium state : - + For each free electron : + - i + + - - + v0i + - + + - i + thermal velocity (after collision) : v0i - acceleration by E for i v i v 0i - Average over free electrons : collision time : Free electrons : vd q E m mass m, charge -q and velocity vi Using a number density of electrons n, Equation of motion along E : current density J : m dvi qE dt Average over free electrons : q E i m drift velocity : vd J v d qn q 2 n E m Free Electron Model and Ohm’s Law Ohm’s law : V iR i S For a small area : V i V i E J S S where : electric resistivity (electric conductivity : = 1 / ) By comparing with the free electron model : n E m 1 n q2 m J q2 Relaxation Time Resistive force by collision : + + + v + If E is removed in the equation of motion : - + + + m + For the initial condition : Equation of motion : with resistive force mv / m dv m qE v dt For the initial condition : v = 0 at t = 0 v q E1 exp t m For a steady state (t >> ), v q E vd m : collision time dv m v dt v = vd at t = 0 v v d expt Number of N 0 non-collided Electrons N t N 0 expt / N : relaxation time t time Mobility Equation of motion under E : m dv qE dt Also, t For an electron at r, collision at t = 0 and r = r0 with v0 1 qE 2 r r0 v0 t t 2 m therefore, by taking an average over non-collided short period, qE 2 r r0 v0 t t 2m Since t and v0 are independent, v0 t v0 t Here, v 0 = 0, as v0 is random. t exp d t 2 2 t2 0 Accordingly, Here, ergodic assumption : temporal mean = ensemble mean 2 qE 2 m q E vd m r r0 r r0 Finally, vd = -E is obtained. = q / m : mobility Degeneracy For H - H atoms : Total electron energy Unstable molecule 1s state energy isolated H atom 2-fold degeneracy Stable molecule r0 H - H distance 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 r0 Distance between atoms Electron Potential Energy Potential energy of an isolated atom (e.g., Na) : Na For an electron is released from the atom : vacuum level 0 Distance from the atomic nucleus 3s 2p 2s 1s Na 11+ Electron potential energy : V = -A / r Periodic Potential in a Crystal Potential energy in a crystal (e.g., N Na atoms) : Vacuum level Distance 3s 2p 3s 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 Free Electrons in a Solid Free electrons in a crystal : Total electron energy Total electron energy Energy band Wave number k Wave / particle duality of an electron : Wave nature of electrons was predicted by de Broglie, and proved by Davisson and Germer. Ni crystal Kinetic energy electron beam h h p mv Momentum (h : Planck’s constant) Particle nature Wave nature mv 2 2 mv h h k h 2 , 2 , k 2 Brillouin Zone Bragg’s law : n 2d sin In general, forbidden bands are a k For ~ 90° ( / 2), n 2a k 2 n a n kn n = 1, 2, 3, ... d sin Total electron energy reflection Therefore, no travelling wave for 2 n = 1, 2, 3, ... Allowed band Forbidden band Allowed band : a k Forbidden band a Allowed band 1st Brillouin zone Forbidden band Allowed band 2 a 2nd a 0 1st a 2 a 2nd k Periodic Potential in a Crystal E Allowed band Forbidden band Allowed band 2 a a 1st 2nd a 0 2 a Forbidden band Allowed band k 2nd Energy band diagram (reduced zone) extended zone Brillouin Zone - Exercise 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 ky ± kx ± ky = 2/a a kx kx n x ky n y a nx 2 ny 2 nx, ny = 0, ± 1, ± 2, ... Reciprocal lattice : 1st Brillouin zone is defined by 2 a nx = 0, ny = ± 1 kx = ± /a nx = ± 1, ny = 0 ky = ± /a ky a 0 Fourier transformation = Wigner-Seitz cell kx a 2 a 2 a a 0 a 2 a 3D Brillouin Zone * C. Kittel, Introduction to Solid State Physics (John Wiley & Sons, New York, 1986). Fermi Energy Fermi-Dirac distribution : E T≠0 T=0 EF Pauli exclusion principle At temperature T, probability that one energy state E is occupied by an electron : f E 1 expE kBT 1 : chemical potential (= Fermi energy EF at T = 0) kB : Boltzmann constant f(E) 1 T=0 T1 ≠ 0 1/2 T2 > T1 0 E 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 Maxwell-Boltzmann distribution quantum mechanical Fermi-Dirac distribution (small electron number density) (large electron number density) * M. Sakata, Solid State Physics (Baifukan, Tokyo, 1989). Fermi velocity and Mean Free Path Fermi wave number kF represents EF : Fermi velocity : v F kF m 2EF m 2mEF vF 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 v F Mean free path g(E) Density of states : Number of quantum states at a certain energy in a unit volume gE 2 1 2 3 32 4 2m EdE 2 2 0 E Density of States (DOS) and Fermi Distribution Carrier number density n is defined as : n f EgEdE T=0 g(E) f(E) EF 0 E T≠0 g(E) f(E) n(E) 0 EF E