Yoon kichul Department of Mechanical Engineering Seoul National University Multi-scale Heat Conduction

Seoul National University Contents 1. Overview of Chap The Hall Effect 4. General Classifications of Solids 1) What is the Hall Effect? 2) Derivation of Hall Voltage & Hall Coefficient 2) Electrons in Insulators, Conductors, and Semiconductors 1) Electrons in Atoms 3. Magnetoresistance 3) Atomic Binding in Solids 5. Summary

Seoul National University 1. Overview of Chap. 6 ∙ However, in this Chapter, - The hall effect and magnetoresistance - Electronic band theory - Phonon dispersion relations and phonon scattering mechanisms - Electronic emission and tunneling phenomena ∙ In the Previous Chapter, Completely free electrons, spherical and isotropic Fermisurface  Only applicable for good conductors - Drude-Sommerfeld model (free electron model) for solid properties - Assumption :  Well describe electron and phonon transport

Seoul National University 2. The Hall Effect 1) What is the Hall Effect? B z y x ∙ By Lorentz force  electrons move towards –y axis ∙ By the Hall voltage  forces are balanced  electrons move towards –x axis only

Seoul National University 2. The Hall Effect 2) Derivation of the Hall voltage & Hall coefficient B z y x V H : Hall voltage When forces are balanced ∙ ∙ : Hall coefficient ∙ R H : Hall resistance ∙ r H : Hall resistivity

Seoul National University 2. The Hall Effect 2) Derivation of the Hall voltage & Hall coefficient (Continued..) : Hall coefficient ∙ By measuring V H with known values I, B, and d  Hall coefficient( ) can be calculated  Sign of the charge carriers(q) and carrier density(n) can be determined ∙ However, for some metals such as Al, Be, Cd, In, and Zn - Hall coefficient becomes positive (Although it should be negative) - Hall effect cannot be fully accounted by the free electron model - Necessary to understand electronic structures

Seoul National University 3. Magnetoresistance ∙ Magnetoresistance : magnetic field  material’s resistance change ∙ Without the magnetic field  current flows in a radial direction ∙ However, with the magnetic field  current flows in a circular direction as well ※ Resistance b/w inner and outer rims will increase ∙ In free electron theory, resistance is independent of magnetic field strength

Seoul National University 4. General Classifications of Solids ∙ Based on electrical conductivities - Electrical conductivity  determined by free electrons in conduction band - Conductors(metals) > semimetals > semiconductors > insulators(dielectrics) ∙ Based on arrangement’s regularity of the constituents - Crystalline : sharp transition b/w solid and liquid - Amorphous : when heated, softened  melts - Crystalline > polycrystalline > amorphousk follows the same order

Seoul National University 4.1 Electrons in Atoms Determination of quantum states’ number Principal quantum # (n) (Shell) 1 (K shell)2 (L shell)3 (M shell)n (…) Orbital # ( ) (Sub-shell) 1 (EA) 0 (s) 2 (EA) 0 (s), 1 (p) 3 (EA) 0 (s), 1 (p), 2 (d) n (EA) 0, ∙∙∙, n-1 Magnetic quantum # (m) (Orbits) 1 (EA) 0 4 (EA) 0 / 0, 1 9 (EA) 0 / 0, 1 / 0, 1, 2 (EA) 0 / ∙∙∙ / 0, ∙∙∙, (n-1) QS in shell2 (EA)8 (EA)18 (EA)2 (EA) ⅹ2ⅹ2 QS in the th sub-shell = 2 (2 +1) QS of ‘s’ sub-shell ( = 0) = 2, ‘p’ sub-shell ( = 1) = 6, ‘d’ sub-shell ( = 2) = 10

Seoul National University 4.1 Electrons in Atoms (Continued..) ∙ Pauli’s exclusion principal - Each QS can have no more than one electron  at most 2 electrons can share one orbit (one orbit consists of two QS) ∙ Aufbau principal : Electrons fill the lowest energy state first Cu : 4s orbits are filled before 3d orbits b/o energy level ∙ Ionization energy : required energy to separate an electron from the atom

Seoul National University 4.2 Insulators, Conductors, and Semiconductors ∙ Formation of band structures - Electrons occupy atomic orbitals, which form discrete energy levels - When atoms are brought together into a molecule  atomic orbitals split  Produces molecular orbitals (proportional to the number of atoms) - In solids, a large number of atoms are brought together  The number of orbitals becomes exceedingly large  Difference in energy b/w orbitals becomes very small  (Allowable) band - However, some intervals contain no orbitals  Forbidden band (band gaps)

Seoul National University 4.2 Insulators, Conductors, and Semiconductors(Cont..) ∙ Electronic band states near the Fermi surface - Fermi surface : thermal, electrical, magnetic, and optical properties determined Conduction Band Valance Band InsulatorsMetalsSemimetalsSemiconductors Empty Filled with electrons (such as Bi and Sn) Conduction Band Valance Band Conduction Band Valance Band

Seoul National University 4.2 Insulators, Conductors, and Semiconductors(Cont..) ∙ Insulators - Large energy gap (usually b/w 5 and 15 eV) - Valence band is completely filled  electrons are not free to move around - Pure crystalline dielectrics are transparent (because electrons are not excited) ∙ Metals (semimetals) - Uppermost electrons in conduction band  can be excited higher energy level - Semimetals(such as Bi, Sn) : electrical conductivity is quite low - Interaction with electromagnetic radiation is high b/o relatively free electrons - Partially filled conduction band, completely filled valance band - Free electrons  high electrical conductivity

Seoul National University 4.2 Insulators, Conductors, and Semiconductors(Cont..) ∙ Semiconductors - Narrower band gap than insulators (order of 1 eV) - Some have a relatively large band gap : wideband semiconductors - Pure (intrinsic) semiconductors : insulators at low temperature  electrons are excited at high temperature  current flows - Some look dark and opaque (because electrons are excited  absorption) Energy source - Higher energy source  More electrons liberated (or excited)  electrical conductivity increases  negative TCR

Seoul National University ∙ Doped semiconductors (extrinsic) - The number of electrons ≠ the number of holes 4.2 Insulators, Conductors, and Semiconductors(Cont..) - High electron concentration  high electric conductivity - Indium is substituted with germanium  extra holes acceptorp-type semiconductor - Impurities and defects by doping  increase phonon scattering  reduce k Ionization energy - Arsenic is substituted with germanium  extra valence electrons  excited donorn-type semiconductor

Seoul National University 4.3 Atomic Binding in Solids ∙ Two or more atoms combine  molecule (mainly through valence electrons) ∙ Five major chemical bonds - Ionic, covalent, molecular, and hydrogen bonds for insulators - Metallic bond for conductors 1)2)3)4) 5) 1) Ionic bond - Ia, IIa metals tend to loose valence electrons - VIa,VIIa elements tend to gain electrons - One negative, other positive  attract each other Ionic bond formed - Strong bonding  ions cannot move around freely  insulators(Ionic crystals) - Attractive force in a long distance, repulsive force in a short distance  balanced

Seoul National University 4.3 Atomic Binding in Solids 1) Ionic bond (continued..) - As atoms are brought very close to each other  electron orbits overlap  some electrons move to higher QS 1 QS can be occupied by only one electron (Pauli’s exclusion principal)  total energy increase (1/r m )  repulsive force b/w atoms (1/r m+1 ) Energy is integration of force  Atoms bonded at a minimum energy (equilibrium position) 2) Covalent bond - Atoms share electrons  attractive force  Covalent bond - Formed b/w gaseous elements (ex. Cl 2, N 2, and CO 2 ) - Covalent solids : usually very hard, high melting point, high k m=6~10 for alkali halides

Seoul National University 4.3 Atomic Binding in Solids 3) Molecular bond - As temperature decreases, inert gas  liquid  solid by molecular bond - Induced dipole moments  Van der Waal’s force  attraction b/w atoms - Attractive potential ∼ -1/r 6, repulsive potential ∼ 1/r 12 weak interaction - Important for organic molecules

Seoul National University 4) Hydrogen bond 4.3 Atomic Binding in Solids - 2 H + : covalent bond with O 2-  H 2 O molecule - Interaction b/w H 2 O molecules hydrogen bond Essential to organic molecules and polymers 5) Metallic bond - Valence electrons leave ion cores  form electron sea - Free electron gas  high electric and thermal conductivity - Usually supplemented by covalent and molecular bonds - More flexible than nonmetallic crystals b/o not hard bonding

Seoul National University 5. Summary ∙ Current + magnetic field  Lorentz force  electrons move towards –y axis  Stack of electrons  Hall voltage generation  force balance Hall effect ∙ Magnetic field  change of materials resistance Magnetoresistance ∙ Classification of solids by conductivities and arrangement regularity ∙ Determination of quantum state number ∙ Electrons’ occupation by Pauli’s exclusion principal and Aufbau principal ∙ Band structures of insulators, semiconductors, and metals ∙ Five major chemical bonds