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Yoon kichul Department of Mechanical Engineering Seoul National University. The Hall Effect & General Classification of Solids (Chap 6.1 ~ 6.2). Multi-scale Heat Conduction. Contents. 1. Overview of Chap. 6. 2. The Hall Effect. 1) What is the Hall Effect?.

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Yoon kichul department of mechanical engineering seoul national university

Yoon kichul

Department of Mechanical Engineering

Seoul National University

The Hall Effect & General Classification of Solids(Chap 6.1 ~ 6.2)

Multi-scale Heat Conduction


The hall effect general classification of solids chap 6 1 6 2

Contents

1. Overview of Chap. 6

2. The Hall Effect

1) What is the Hall Effect?

2) Derivation of Hall Voltage & Hall Coefficient

3. Magnetoresistance

4. General Classifications of Solids

1) Electrons in Atoms

2) Electrons in Insulators, Conductors, and Semiconductors

3) Atomic Binding in Solids

5. Summary


The hall effect general classification of solids chap 6 1 6 2

1. Overview of Chap. 6

- Assumption :

∙ However, in this Chapter,

- The hall effect and magnetoresistance

- Electronic band theory

∙ In the Previous Chapter,

- Phonon dispersion relations and phonon scattering mechanisms

- Electronic emission and tunneling phenomena

- Drude-Sommerfeld model (free electron model) for solid properties

 Well describe electron and phonon transport

Completely free electrons, spherical and isotropic Fermisurface

 Only applicable for good conductors


The hall effect general classification of solids chap 6 1 6 2

2. The Hall Effect

1) What is the Hall Effect?

z

y

x

+ + + + + + + + + + + +

- - - - - - - - - - - -

B

-

∙ By Lorentz force  electrons move towards –y axis

∙ By the Hall voltage  forces are balanced  electrons move towards –x axis only


The hall effect general classification of solids chap 6 1 6 2

2. The Hall Effect

2) Derivation of the Hall voltage & Hall coefficient

z

y

When forces are balanced

x

+ + + + +

VH : Hall voltage

- - - - -

B

: Hall coefficient

RH : Hall resistance

rH: Hall resistivity


The hall effect general classification of solids chap 6 1 6 2

2. The Hall Effect

2) Derivation of the Hall voltage & Hall coefficient (Continued..)

∙ By measuring VH 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

: Hall coefficient

∙ 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


The hall effect general classification of solids chap 6 1 6 2

3. Magnetoresistance

∙ Magnetoresistance : magnetic field  material’s resistance change

∙ In free electron theory, resistance is independent of magnetic field strength

∙ 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


The hall effect general classification of solids chap 6 1 6 2

4. General Classifications of Solids

∙ Based on electrical conductivities

- Conductors(metals) > semimetals > semiconductors > insulators(dielectrics)

- Electrical conductivity  determined by free electrons in conduction band

∙ Based on arrangement’s regularity of the constituents

- Crystalline > polycrystalline > amorphous

k follows the same order

- Crystalline : sharp transition b/w solid and liquid

- Amorphous : when heated, softened  melts


The hall effect general classification of solids chap 6 1 6 2

4.1Electrons in Atoms

Determination of quantum states’ number

ⅹ2

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


The hall effect general classification of solids chap 6 1 6 2

4.1Electrons 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


The hall effect general classification of solids chap 6 1 6 2

4.2Insulators, 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)


The hall effect general classification of solids chap 6 1 6 2

4.2Insulators, Conductors, and Semiconductors(Cont..)

∙ Electronic band states near the Fermi surface

- Fermi surface : thermal, electrical, magnetic, and optical properties determined

Empty

Conduction Band

Filled with electrons

Conduction Band

Conduction Band

Conduction Band

Valance Band

Valance Band

Valance Band

Valance Band

Insulators

Metals

Semimetals

Semiconductors

(such as Bi and Sn)


The hall effect general classification of solids chap 6 1 6 2

4.2Insulators, 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)

- Partially filled conduction band, completely filled valance band

- Free electrons  high electrical conductivity

- Semimetals(such as Bi, Sn) : electrical conductivity is quite low

- Uppermost electrons in conduction band  can be excited higher energy level

- Interaction with electromagnetic radiation is high b/o relatively free electrons


The hall effect general classification of solids chap 6 1 6 2

4.2Insulators, 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

Energy source

- Higher energy source  More electrons liberated (or excited)

 electrical conductivity increases  negative TCR

- Some look dark and opaque (because electrons are excited  absorption)


The hall effect general classification of solids chap 6 1 6 2

4.2Insulators, Conductors, and Semiconductors(Cont..)

∙ Doped semiconductors (extrinsic)

- The number of electrons ≠ the number of holes

- Arsenic is substituted with germanium  extra valence electrons  excited

donor

n-type semiconductor

- High electron concentration  high electric conductivity

Ionization energy

- Indium is substituted with germanium  extra holes

acceptor

p-type semiconductor

- Impurities and defects by doping  increase phonon scattering  reduce k


The hall effect general classification of solids chap 6 1 6 2

4.3Atomic Binding in Solids

∙ Two or more atoms combine  molecule (mainly through valence electrons)

∙ Five major chemical bonds

1)

2)

3)

4)

- Ionic, covalent, molecular, and hydrogen bonds for insulators

5)

- Metallic bond for conductors

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

- Attractive force in a long distance, repulsive force in a short distance  balanced

- Strong bonding  ions cannot move around freely  insulators(Ionic crystals)


The hall effect general classification of solids chap 6 1 6 2

4.3Atomic 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/rm)  repulsive force b/w atoms (1/rm+1)

Energy is integration of force

m=6~10 for alkali halides

Atoms bonded at a minimum energy (equilibrium position)

2) Covalent bond

- Atoms share electrons  attractive force  Covalent bond

- Formed b/w gaseous elements (ex. Cl2, N2, and CO2)

- Covalent solids : usually very hard, high melting point, high k


The hall effect general classification of solids chap 6 1 6 2

4.3Atomic 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 12weak interaction

- Important for organic molecules


The hall effect general classification of solids chap 6 1 6 2

4.3Atomic Binding in Solids

4) Hydrogen bond

- 2 H+ : covalent bond with O2- H2O molecule

- Interaction b/w H2O 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


The hall effect general classification of solids chap 6 1 6 2

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