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

Quick Review over the Last Lecture 3 states of matters : solid liquid gas density ( large ) ( large ) ( small ) ordering range ( long ) ( short ) rigid time scale ( long ) ( short ) 4 major crystals : soft solid ( van der Waals crystal ) ( 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 Degeneracy Electron Potential Brillouin Zone Fermi Distribution

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

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

Electron Transport in a Metal 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. Effective energy for electrons Inside a metal decrease in E free electron 　　-

Free Electron Model - + - - - - - - - - - Equilibrium state : For each free electron : thermal velocity (after collision) : v0i acceleration by E for i 　　- + v0i 　　- 　　- i 　　- + - 　　- 　　- 　　- 　　- 　　- 　　- 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

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

Relaxation 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 /  Number of non-collided Electrons N time  For the initial condition : v = 0 at t = 0 N0 For a steady state (t >> ),  : relaxation time  : collision time

Mobility 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, Here, = 0, as v0 is random.

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 Distance from the atomic nucleus Electron potential energy : V = -A / r 3s 2p 2s 1s Na 11+

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

Free Electrons in a Solid Free electrons in a crystal : Total electron energy Wave number k 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. electron beam Ni crystal Particle nature Wave nature Kinetic energy Momentum (h : Planck’s constant)

Brillouin Zone Bragg’s law :  In general, forbidden bands are a Total electron energy k  reflection Therefore, no travelling wave for n = 1, 2, 3, ... Allowed band  Forbidden band Allowed band : Forbidden band  1st Brillouin zone Allowed band Forbidden band Allowed band 2nd 1st 2nd

Periodic Potential in a Crystal Allowed band Forbidden band Allowed band Forbidden band Allowed band k 2nd 1st 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  ± kx ± ky = 2/a a kx ky nx, ny = 0, ± 1, ± 2, ... Reciprocal lattice : kx ky 1st Brillouin zone is defined by nx = 0, ny = ± 1  kx = ± /a nx = ± 1, ny = 0  ky = ± /a Fourier transformation = Wigner-Seitz cell

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) E T = 0 1  : chemical potential (= Fermi energy EF at T = 0) kB : Boltzmann constant T1 ≠ 0 1/2 T2 > T1 

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).

Fermi velocity and Mean Free Path 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 g(E) E Density of states : Number of quantum states at a certain energy in a unit volume

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