The Hall Effect & General Classification of Solids

(Chap 6.1 ~ 6.2)

Yoon kichulDepartment of Mechanical EngineeringSeoul National University

Multi-scale Heat Conduction

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Contents

1. Overview of Chap. 6

2. 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

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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 Chap-ter,

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

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2. The Hall Effect

1) What is the Hall Effect?

B

HV

l

dt

z yx

I- - - - - - - - - - - -

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

-

∙ By Lorentz force electrons move towards –y axis

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

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2. The Hall Effect

2) Derivation of the Hall voltage & Hall coefficient

B

HV

l

dt

z yx

I- - - - -+ + + + +

d du A uA

II nqnq

V E =

d H d HBE u B E u B IV t t V

nqd = = = VH : Hall voltage

0= =( )q dF E u B When forces are balanced∙

∙ HH

1B

V dI nq

: Hall coefficientH

∙ H HH

BBVRI nqd d

RH : Hall resistance

∙ dH

d

u BB

uy

Hx

Er

J nq rH : Hall resistivity du , A

Ax

xIJ nq td

A t d

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2. The Hall Effect

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

HH

1B

V dI nq

: Hall coefficientH

∙ By measuring VH with known values I, B, and d

Hall coefficient( ) can be calculatedH

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

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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

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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 > amorphous k follows the same order

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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)

04 (EA)

0 / 0, 19 (EA)

0 / 0, 1 / 0, 1, 2(EA)

0 / ∙∙∙ / 0, ∙∙∙, (n-1)

QS in shell 2 (EA) 8 (EA) 18 (EA) 2 (EA)

2n

2nⅹ2

1 1, spin2 2

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

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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)

1 1, spin2 2

∙ Aufbau principal : Electrons fill the lowest energy state first21s2 62 2s p2 6 103 3 3s p d2 6 10 144 4 4 4s p d f2 6 10 145 5 5 5s p d f2 6 106 6 6s p d27s

2 2 6 2 6 2 10 6 2 10 6 21 | 2 2 | 3 3 | 4 3 4 | 5 4 5 | 6s s p s p s d p s d p s He2 Ne10 Ar18 Kr36 Xe54

2 2 6 2 6 1 101 | 2 2 | 3 3 | 4 3s s p s p s dCu :

4s orbits are filled before 3d orbits b/o energy level

∙ Ionization energy : required energy to separate an electron from the atom

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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)

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

∙ Electronic band states near the Fermi surface

- Fermi surface : thermal, electrical, magnetic, and optical properties deter-mined

FE

Conduction Band

Valance Band

gE

Insulators Metals Semimetals Semiconductors

EmptyFilled with electrons

(such as Bi and Sn)

Conduction Band

Valance Band

Conduction BandConduction Band

Valance Band Valance Band

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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

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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

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∙ 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 holesacceptor p-type semiconductor

- Impurities and defects by doping increase phonon scattering reduce k

C

FEI

V

C

FEIV

IE

Ionization energy- Arsenic is substituted with germanium extra valence electrons excited

donor n-type semiconductorIE

IE

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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

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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 QS1 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

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

m=6~10 for alkali halides

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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

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4) Hydrogen bond

4.3 Atomic Binding in Solids

- 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 bond-ing

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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