ELECTRON AND PHONON TRANSPORT

• The Hall Effect

• General Classification of Solids

• Crystal Structures

• Electron band Structures

• Phonon Dispersion and Scattering

• Electron Emission and Tunneling

• Electrical Transport in Semiconductor Devices

The Hall Effectz

y

x+-

Id

V

electron

Electron: a fundamental subatomic particle that carries a negative electric charge

Hole: An electric charge carrier with a positive charge, equal in magnitude but opposite in polarity to the charge on the electron

magnetic fieldV

-

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

+ + + + + + + + + + +

l

dI

zy

xHV

B

d F qu B lI B

q: charge of each carrierud: drift velocity of the carrier l: length of the conductor

FF

F E

I

-

B

d F qE qu B

d- 0 0

0 0

x y z

y y

e e e

qE e q u

B

.- 0y d x z yF q E u B e

d. , y x zE u B E V

H d, y x zV E t u B t

d d, I I

I nqu A unqA nqd

H IB

Vnqd

n: number density of free electron

zy

x

d

l

Klitzing constant

H IB

Vnqd

Hall coefficient HH

1

V d

IB nq

Hall resistance H HH

V BR

I d

Hall resistivity HH H

y

x

EV dr B B

IB J Jx: current density

Hall coefficientp – type semiconductor (+)n – type semiconductor (-)

Magneto resistance - The change in resistance of a material - The Giant Magneto resistance (GMR) → magnetic hard disk

2K / 25,812.807449 0.000086 R h e

General Classification of Solids

periodic table of elements

by the number of proton

100

s

1

2

K

2

30, 1, 2

0, ± 1, ±2

s, p, d

1, 3, 5

2, 6, 10

M

18

…..n….(n-1)…..± l

g, h….

(2l+1)

2(2l+1)

O, P…..

n l

m

Subshelldesignation

Orbital in subshell

Subshellcapacity

Principal shellcapacity

40, 1, 2, 3

0, ±1, ±2, ±3

s, p, d, f

1, 3, 5, 7

2, 6, 10, 14

N

32

20, 1

0, ±1

s, p

1, 3

2, 6

L

8 22n

Electrons in Atoms

The number of quantum states

incorrect

incorrect7

2N

1s

Pauli’s exclusion principle

Each quantum state can have no more than one electron.At most only two electrons can share the same orbit. (one with +1/2, the other with -1/2 spin)

Aufbau principle

Electron fills the lowest energy states first.

2s 2 p

incorrect

N

Hund’s rule

Every orbital in a subshell is singly occupied with one electron before any one orbital is doubly occupied and all electrons in singly occupied orbitals have the same spin.

72N

1s 2s 2 p

03d63 p 24s23s62 p22s21s

Ca

20Ca

4s orbits are filled before the 3d orbits - The associated energy level of a 3d orbit is higher than that of 4s orbit

93d63 p 24s23s62 p22s21s

63 p 14s23s62 p22s21s 103d

29Cu

A half-filled or filled ‘d’ subshell is more stable than the ‘s’ shell of the next level.

2 2 6 2 6 2 10 6 2 10 6 2 14 10 6 2 14 101 2 2 3 3 4 3 4 5 4 5 6 4 5 6 7 5 6s s p s p s d p s d p s f d p s f d

2 He 10 Ne 18 Ar 36 Kr 54 Xe 86 Rn

11 Na

Ionization energy: the energy required to separate an electron from the atomic nucleus

easily lose

He: 24.6 eV, Li: 5.4 eV

Classification of solids

arrangement of atoms in the solid

crystalline polycrystalline amorphous

electrical conductivities - insulators, semiconductors, conductorchemical bonds - ionic, covalent, molecular, hydrogen bond metallic bond

Insulators, Conductors, and Semiconductors

Band structure

free electron gasElectron energies are limited to a nearly continuous band from the zero energy level up to the Fermi energyelectrons in single atomElectron energies are in various discrete energy levels

Allowable band

Forbidden band

Allowable band

FE

Insulator (Dielectric)

conduction band

valance band

gEFE

valance bandband made up of the occupied molecular orbitals

conduction bandband free to move about the crystal

Eg = 5 ~ 15 eV

SiO2: 8 eV

Transparent to visible light Valence electrons can not be excited.

Metal & Semimetal

conduction band

valance band

conduction band

valance bandFE FE

metal semimetal

Ex) Bi, Sn …

Metals interact with electromagnetic radiation [→ free electrons ]strong absorption by thin metallic films, high reflection from polished metal

Semiconductor

conduction band

valance band

FEgE

Eg ~ 1 eVdiamond: 5.5 eV, Si: 1 eV

Pure or intrinsic semi-conductor Low Temperature → insulator High Temperature → conductorBand gap absorption interaction semi-conductor with optical radiation

Doping the semi-conductor with impurities→ electrons ≠ holes

n-type vs p- type Semi-conductor

n – type semi-conductor

p – type semi-conductor

- increase the number of free (negative) charge carries - group V element- produce an abundance of mobile or "carrier" electrons in the material - phosphorous (donor )

cEFE

iE

VE

cE

FEiE

VE

- increase the number of free (positive) charge carries- group III element- create an abundance of holes - boron (acceptor)

Atomic Binding in Solid

Cl _

Na+Na+

Cl _

Cl _

Cl _

Cl _

Na+

Na+

Ion crystals(ionic)

C

C

CC

C

Covalent crystals(covalent)

AA

A AA

AA

Crystals of inert gases

(van der Waals)

F _

H+ F _

Hydrogen bond

Na+

Na+Na+

Na+Na+

Metal(metalic)

Ionic bond

Alkali & alkaline earth metal [ H, Li, Na, k, Be, Mg, Ca ] - Valance electron are loosely bonded

NaNa

The elements in group VIa & VIIa [ F, Cl, Br..] - Gain additional electrons

Cl Cl

Cl _

Na+Na+

Cl _

Cl _

Cl _

Cl _

Na+

Na+

Ionic crystals (NaCl, CsCl..) – hard, high melting pointTransparent InsulatorSome these crystal is soluble → the solution become conductiveThe interaction energy between ion i and j

Cl _

Na+

a0 1 22

q qF

r

electrostatic force

contribution of the Coulomb attraction

1-

r

contribution of the repulsive force1mr

NaCl m = 6 ~10

Covalent bonds

Nonmetallic elementThe attractive force, very strong bindingHigh melting point & thermal conductivity

C C

linear Planar Tetrahedral

C

C

C CC

Diamond

tetrahedronmelting point 3820 Kk = 2300 W/mKNo absorb radiation at frequencies lower than that of the corresponding Eg

III-V semiconductor (GaN, GaAS, InSb) II-IV semiconductor (ZnO, CdS) → covalent bond characteristics (30%)SiC (dipole form) → some ionic bond characteristics

Molecular bonds

Inert gases can be solidified at low temperatureElectron distribution is very close to that of the free atomsThe atoms induce dipole moments in each other (Bound together by Van der Waals forces)

12 6

0 00( ) 4ij

ij ij

r ru r

r r

Lennard-Jones Potential

Transparent insulatorWeakly boundLow melting temperatureCrystal structures are FCC except He3 and He4

Hydrogen bonds

An atom of hydrogen is attracted by strong forces to two atoms.Largely ionic in characterFormed only between the most electronegative atoms (F, O, N)Important part of the interaction between water molecules and ferroelectric crystals

H2F F

Metallic bonds

Some valance electrons leave the ion cores andare shared by all the crystal.Supplemented by covalent and molecular bondMetallic crystals are more flexible.Weaker binding than othersBond formed by conduction electron is not so strong.

Na+

Na+Na+

Na+Na+

electron sea