Introductory Nanotechnology ~ Basic Condensed Matter Physics ~

Slides:



Advertisements
Similar presentations
APPLIED PHYSICS CODE : 07A1BS05 I B.TECH CSE, IT, ECE & EEE UNIT-3
Advertisements

Department of Electronics Introductory Nanotechnology ~ Basic Condensed Matter Physics ~ Atsufumi Hirohata.
Free Electron Fermi Gas
Lecture #5 OUTLINE Intrinsic Fermi level Determination of E F Degenerately doped semiconductor Carrier properties Carrier drift Read: Sections 2.5, 3.1.
Department of Electronics Nanoelectronics 13 Atsufumi Hirohata 12:00 Wednesday, 25/February/2015 (P/L 006)
Anandh Subramaniam & Kantesh Balani
LECTURE 2 CONTENTS MAXWELL BOLTZMANN STATISTICS
Happyphysics.com Physics Lecture Resources Prof. Mineesh Gulati Head-Physics Wing Happy Model Hr. Sec. School, Udhampur, J&K Website: happyphysics.com.
Non-Continuum Energy Transfer: Phonons
Introductory Nanotechnology ~ Basic Condensed Matter Physics ~
Computational Solid State Physics 計算物性学特論 第2回 2.Interaction between atoms and the lattice properties of crystals.
Energy Bands in Solids: Part II
Department of Electronics Introductory Nanotechnology ~ Basic Condensed Matter Physics ~ Atsufumi Hirohata.
Lecture Notes # 3 Understanding Density of States
CHAPTER 3 Introduction to the Quantum Theory of Solids
The mathematics of classical mechanics Newton’s Laws of motion: 1) inertia 2) F = ma 3) action:reaction The motion of particle is represented as a differential.
Solid state Phys. Chapter 2 Thermal and electrical properties 1.
Molecules and Solids (Cont.)
CHAPTER 5 FREE ELECTRON THEORY
Exam Study Practice Do all the reading assignments. Be able to solve all the homework problems without your notes. Re-do the derivations we did in class.
Statistical Mechanics
The Photoelectric Effect
Last Time Free electron model Density of states in 3D Fermi Surface Fermi-Dirac Distribution Function Debye Approximation.
Read: Chapter 2 (Section 2.4)
Metals: Drude Model and Conductivity (Covering Pages 2-11) Objectives
PHYS3004 Crystalline Solids
AME Int. Heat Trans. D. B. GoSlide 1 Non-Continuum Energy Transfer: Electrons.
Department of Electronics Nanoelectronics 10 Atsufumi Hirohata 10:00 Tuesday, 17/February/2015 (B/B 103)
Department of Electronics Nanoelectronics 07 Atsufumi Hirohata 10:00 Tuesday, 27/January/2015 (B/B 103)
Chapter 6: Free Electron Fermi Gas
Solid State Physics Bands & Bonds. PROBABILITY DENSITY The probability density P(x,t) is information that tells us something about the likelihood of.
Usually a diluted salt solution chemical decomposition
Quantum Distributions
Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 5 Lecture 5: 2 nd Order Circuits in the Time Domain Physics of Conduction.
Chapter 19: Fermi-Dirac gases The Fermi energy Fermi-Dirac statistics governs the behavior of indistinguishable particles (fermions). Fermions have.
6. Free Electron Fermi Gas Energy Levels in One Dimension Effect of Temperature on the Fermi-Dirac Distribution Free Electron Gas in Three Dimensions Heat.
Specific Heat of Solids Quantum Size Effect on the Specific Heat Electrical and Thermal Conductivities of Solids Thermoelectricity Classical Size Effect.
Chapter 21: Molecules in motion
Lecture 4 OUTLINE Semiconductor Fundamentals (cont’d)
UNIT 1 FREE ELECTRON THEORY.
1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2013 DEE4521 Semiconductor Device Physics Lecture.
Today’s Objectives Finish energy level example Derive Fermi Dirac distribution Define/apply density of states Review heat capacity with improved model.
EEE 3394 Electronic Materials
ELECTRON AND PHONON TRANSPORT The Hall Effect General Classification of Solids Crystal Structures Electron band Structures Phonon Dispersion and Scattering.
Statistical mechanics How the overall behavior of a system of many particles is related to the Properties of the particles themselves. It deals with the.
Copyright©2000 by Houghton Mifflin Company. All rights reserved. 1 Electromagnetic Radiation Radiant energy that exhibits wavelength-like behavior and.
Metals I: Free Electron Model
EEE 3394 Electronic Materials Chris Ferekides SPRING 2014 WEEK 2.
ELECTRON THEORY OF METALS 1.Introduction: The electron theory has been developed in three stages: Stage 1.:- The Classical Free Electron Theory : Drude.
Free electron theory of Metals
Electron & Hole Statistics in Semiconductors A “Short Course”. BW, Ch
BASICS OF SEMICONDUCTOR
1 Prof. Ming-Jer Chen Department of Electronics Engineering National Chiao-Tung University October 1, 2012 DEE4521 Semiconductor Device Physics Lecture.
Lecture 2.0 Bonds Between Atoms Famous Physicists’ Lecture.
Chapter 7 in the textbook Introduction and Survey Current density:
제 4 장 Metals I: The Free Electron Model Outline 4.1 Introduction 4.2 Conduction electrons 4.3 the free-electron gas 4.4 Electrical conductivity 4.5 Electrical.
Nanoelectronics Part II Many Electron Phenomena Chapter 10 Nanowires, Ballistic Transport, and Spin Transport
제 4 장 Metals I: The Free Electron Model Outline 4.1 Introduction 4.2 Conduction electrons 4.3 the free-electron gas 4.4 Electrical conductivity 4.5 Electrical.
Department of Electronics
Electrical Engineering Materials
Review of solid state physics
Electrical Engineering Materials
Department of Electronics
Prof. Jang-Ung Park (박장웅)
Electrical Properties of Materials
Lecture #5 OUTLINE Intrinsic Fermi level Determination of EF
Thermal Properties of Matter
FERMI-DIRAC DISTRIBUTION.
Lecture 2.0 Bonds Between Atoms
Department of Electronics
Presentation transcript:

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