chapter 1. the electron as a particleyu.ac.kr/~hyulee/ssp_lecture/chap1.pdf · the principle of...

Chapter 1.

The Electron as a Particle

Electronic Properties of Materials Hee Young Lee

Introduction

Till now man has been up against Nature;

from now on he will be up against his own nature.

Dennis Gabor Inventing the future

It is a good thing for an uneducated man to read

books of quotations.

W.S. Churchill Roving commission in my early life

(1930)


Introduction: What Are Electrons?- Two stable particles with non-zero mass: electron and proton

- Electrons determine

(1) the physical properties of materials:

ó(electrical conductivity), á(optical absorption coefficient),

ê(thermal conductivity), ì(magnetic permeability), etc.

(2) the behavior of devices and components:

pn junction diode, Schottky diode, LED, laser diode, solar cell, photo-detector,

transistors (BJT and FET), displays (TV, TFT-LCD monitor, PDP, FED), etc.

☞ "The electron is the most important particle in nature."

- Electron is the smallest particle in

mass ( ), and charge ( ).kgm 311011.9 −×= Ce 191060.1 −×=


Discovery of Electrons (1897)

J.J. Thompson, The Cavendish Professor of Cambridge University, postulated that the mysterious cathode rays are streams of charged particles much smaller than atoms, i.e. corpuscles.

Cathode Ray Experiment


The Principle of Wave-Particle Duality

- Electrons can be viewed as particles or waves.

Particles: specified by coordinates; located at (x,y,z) – a point

have kinetic energy given by

Wave: specified by wavelength;

has kinetic energy given by

)221 cv( mvE <<=

)1062.6( 34 sJhmvh ⋅×== −λ

( )

s J.ðh

,ëð k

mk

ëh

mmmv

mvE

⋅×=≡

≡

====

−34

22

2

222

21

100512and

2number) (wavewhere

221

21

h

h

ë↓ ⇒ E↑ or k↑ ⇒ E↑

de Broglie relation


Estimate of Thermal Speed of an Electron- Treat electrons as a gas of particles

Total Energy of a (Free) Electron = Average Thermal Energy of a Gas Molecule

From , we get

≈ average thermal speed of an electron ≈ speed of an electron in Cu or Si at RT

Cf.) Speed of a bullet ≈ 1 km/sec

kTmvth 232

21 =

)103( 10 10119

3001038133 8531

23

m/scm/skg.

KJ/K).(

m

kTvth ×=<<≅

××××

== −

−


- Charged particles experience force(F ) under electric field (E ),

where F // E if q > 0, and F ↔ E if q < 0 .

- Electron has negative charge, i.e. q = - e = -1.60 × 10–19 C

Force è (Net) Motion : Newton’s 2nd Law of Motion

EqF = Electrostatic Force

EaEmq

amaqEF ↔=⇒== where

Electron Current: Drift



mobilitydrift : where me

EEmq

av eeD

τµµ

ττ ≡===

- constant acceleration, if E = constant.

è velocity saturates due to collision with lattice atoms and other electrons

è “drift velocity”, (Note: vD << vth )

where ô : collision time = average time between collisions

∴ If E ≠ 0 è net flow of electrons (in the opposite direction)

= charge flow = electric current !

densityelectron : where, since , eDeDe NtAevNQAevNdtdQ

I ===

τavD =



EEmeN

EeNevNAI

J eeeDe σ

τµ =====

2

- Electric current density (J )

ty conductivi electrical : where, eeeNEJ µσσ ==

or

Ohm’s Law Same as I = V / R

Note: In general, J // E , but if J // E or A ≠ constant ∫ ⋅= AdJI


Georg Simon Ohm

♦ Born: 16 March 1789 in Erlangen, Bavaria (now Germany) Died: 6 July 1854 in Munich, Bavaria, Germany

Georg Ohm gave, between 1825 and 1827, a mathematical description ofconduction in circuits modeled on Fourier's study of heat conduction.

What is now known as Ohm's law appears in Die galvanische Kette,mathematisch bearbeitet (1827). His work strongly influenced theory but it was received with so little enthusiasm that Ohm's feeling were hurt and he resigned his position at the Jesuit College of Cologne (professor of mathematics ).

He accepted a position at Nüremberg in 1833 and his work was eventually recognized by the Royal Society with its award of the Copley Medal in 1841. He became a foreign member of the Royal Society in

1842.


Electrical Resistivity

σρ 1≡- Electrical resistivity (ñ ):

- Resistance vs Resistivity

MATERIAL RESISTIVITY (Ù ·m)

Ag 1.59×10 - 8

Cu 1.673×10 - 8

Au 2.35×10 - 8

Al 2.65×10 - 8

s teel 12 ~ 166×10 - 8

Si (pure) 2 .3×103

Si (extrinsic) 10 - 4 ~102

diamond ~ 106

plast ics 107 ~108

mica ~ 101 2

SiO2 ~ 101 5

Al 2 O3 > 101 2

∫=

=

Adl

R

Al

R

ρ

ρ

or

(if A is not constant)


- The development of transverse electric field normal to both applied electric field and magnetic field.

- due to Lorentz force

- Refer to Hall Effect Measurements web-site.

- Force balance equation under constant dc magnetic field in equilibrium

Hall Effect

( )BvqEqF ×+=

( )

JBRBNqJ

BvE

qEBqvEqBvqFF

HDH

HDHDHB

===

=∴=×=

Thus,

or

H

HH qE

JBN

EBE

== and µ

(RH : Hall coefficient)


Hall Effect

dqVBI

wdw

qVBI

qEJB

N

H

x

H

x

H

=

⋅==

wl

BVV

EBE

x

HHH ⋅== µ

and


♦ Born: 16 July 1853 in Arnhem, Netherlands Died: 4 Feb. 1928 in Haarlem, Netherlands

Hendrik A. Lorentz

Hendrik Lorentz entered the University of Leiden in 1870, and worked for his doctorate while holding the teaching post at Arnhem. Lorentz refined Maxwell’s electromagnetic theory in his doctoral thesis The theory of the reflection and refraction of light presented in 1875. He was appointed professor of mathematical physics at Leiden University in 1878.

Before the existence of electrons was proved, Lorentz proposed that light waves were due to oscillations of an electric charge in the atom. Lorentzdeveloped his mathematical theory of the electron for which he received the Nobel Prize in 1902. The Nobel prize was awarded jointly to Lorentz andPieter Zeeman, a student of Lorentz. Zeeman had verified experimentallyLorentz's theoretical work on atomic structure, demonstrating the effect of a strong magnetic field on the oscillations by measuring the change in the wavelength of the light produced.


Electromagnetic Waves in Solids


James Clerk Maxwell

♦ Born: 13 June 1831 in Edinburgh, ScotlandDied: 5 Nov 1879 in Cambridge, Cambridgeshire, England

• Studied at the University of Edinburgh, and the University of Cambridge. • Postulated the electromagnetic theory of light (ca 1865).

“light consists of transverse waves of electric and magnetic forces”- arrived at this conclusion by his explanation of Michael Faraday’s

discovery of electromagnetic induction in mathematical terms.- calculated that the velocity of these waves to be that of the speed of light. - predicted the existence of other electromagnetic waves, realizing that

there was no set limit on the wave length of these waves. - suggested that one may create electromagnetic waves artificially. • Maxwell's theory was generally disregarded until Heinrich Hertz's

discovery of radio waves in 1887. • Maxwell published his Treatise on Electricity and Magnetism which

contains his famous Maxwell equations (1873). • Became the Cavendish professor of experimental physics at Cambridge

(1871), where he founded the new Cavendish laboratories and a scholarship in physics at Cambridge.


Maxwell’s Equations

.density charge and

/104 vacuum)ofity (permeabil

/10858 vacuum)ofity (permittiv

induction) (magnetic

nt)displaceme (electricwhere

,

0 ,

7

12

=

×=

×=

==

==

∂∂

+=×∇∂∂

−=×∇

=⋅∇=⋅∇

−

−

ρ

πµ

ε

µµµ

εε

ερ

,mH

,mF.

,HHB

,EKED

tD

JHtB

E

HE

o

o

or

o

Current EquationFaraday’s Law:ac induction


tE

BtB

EHE∂∂

=×∇∂∂

−=×∇=⋅∇=⋅∇ µε , , 0 , 0

In charge-free medium such as vacuum or perfect insulators, Maxwell’s equations become ( )0 and 0 == JρQ

Electromagnetic Wave Equation: Insulator

Using the following vector relation, ,

the last two equations may be combined

( ) ( ) ( ) CBABCACBA ⋅−⋅=××

( ) ( ) ( )

tE

xE

tE

E

tE

Btt

BEEE

0

or

2

2

2

2

2

22

2

22

=∂∂

−∂∂

∂∂

=∇

∂∂

−=×∇∂∂

−=

∂∂

−×∇=∇−⋅∇⋅∇=×∇×∇

µεµε

µε

: EM Wave Equation (in 1-dim)


EM Wave Equation: Insulator

tE

xE

0 2

2

2

2

=∂∂

−∂∂

µε xu

vtu

2

22

2

2

∂∂

=∂∂

1-dim. EM Wave Equation 1-dim. Wave Equation

( )

kv

tkxAtxu

ω

ω

=

−=

where

sin),(

solution

( )

k

v

eEtxEE tkxio

ω

µε

ω

≡=

==

== −

velocityphase

1 velocity wave where

),(

Metallurgical and Materials Engineering Electronic and Thin Film Materials Lab

( )

space freein light of speed 1

where

),(

===

== −

ooo

o

txkio

kc

eEtxEE oo

εµω

ω

EM Wave Equation: Insulator(1) Vacuum (or air):

(2) Dielectric: ( )

materials magnetic-nonfor offactor by the

and , , where

),(

Kcv

kkK

c

K

ckv

eEtxEE

oor

tkxio

<⇒

>=≅==

== −

ωωµ

ω

ω

(http://www.glenbrook.k12.il.us/gbssci/phys/mmedia/waves/em.html)

(Source: http://www.phy.ntnu.edu.tw/java/emWave/emWave.html)

All EM waves propagate at the same velocity in free space. è No dispersion

sm

coo

/1000.3

1

8×=

=εµ



(Sources: http://www.purchon.com/physics/electromagnetic.htm,http://webphysics.davidson.edu/Applets/EMWave/EMWave.html)

Electromagnetic Spectrum

(Source: http://imagine.gsfc.nasa.gov/docs/science/know_l1/emspectrum.html)

- Photon energy:length wave

1 frequency Ep ∝∝===λ

νhc

hfh


tEJ

BtB

EHE∂∂

+=×∇∂∂

−=×∇=⋅∇=⋅∇ µεµε

ρ , , 0 ,

In conductors such as metals and semiconductors, Maxwell’s equations become ( )0 and 0 ≠≠ JρQ

Electromagnetic Wave Equation: Conductor

Using the same techniques as before, we get

tE

tE

xE

tE

tE

E

0

0

2

2

2

2

2

22

=∂∂

−∂∂

−∂∂

=∂∂

−∂∂

−∇

µεµσ

µεµσ : EM Wave Equation in 3-dim

: EM Wave Equation in 1-dim


Dispersion Equation

Assuming the same sine-type solution as the case of an insulator, i.e. ,

and substituting into the wave equation, we get (in one dimension)

since

Assume

tE

tE

xE

02

2

2

2

=∂∂

−∂∂

−∂∂

µεµσ

( )tkxioeEE ω−=

textof (1.38) 2 ik ωµσµεω +=

( ) ( ) E EitE

EitE

EkEikxE 22

2

222

2

2

and , , ωωω −==∂∂

−=∂∂

−==∂∂

Dispersion Equation: k=complex number

( ) iikik get we, from Then, . 222 ωµσµεωβαβα +=+=+=

±

+=

1121

2

ωεσ

µεωβα

(keeping the same sign)

Let’s consider two limiting cases.

Case 1. Good Dielectric:

A wave solution for electric field (E) is

0 and 0 == βσ

fvvik λλπωεµωαβα =====+= where2

( )[ ]txkiEE o ω−= exp

Case 2. Good Conductor: Then,

Thus

( ) 1 >>ωεσ

same as the case obtained for insulator: no attenuation

2σµω

βα ≈≈

( )[ ] ( )[ ]txieEtxkiEE xoo ωαω β −=−= − expexp

wave amplitude decays exponentially with x: strong attenuation

EM Wave Equation: Conductor


Define “skin depth” as the distance where amplitude reduces to 1/e of surface value.

( )σµπσµωβδδβ

fe

121 exp 1 ====− −

Note 1) The above analysis is only valid for frequencies below

metals.most for Hz102or 18≈<<<< επσ

εσω f (X-ray)

Note 2) EM wave is absorbed (or damped) in a metal due to conductivity.

Note 3) We have treated as a constant, which is not true. Let’s look at the frequency dependence of electrical conductivity.

σ

Skin Depth


Applying the Newton’ 2nd Law of Motion to free electron gas,

Assuming, at a fixed location (x=constant, e.g. x=0), the electron gas is driven at the same frequency with the applied field,

AC Dependence of Conductivity

textof (1.48) τvm

Eqdtvd

mF −==

ACee

imqN

EvqN

EJ

Eim

qv σ

τωτ

στω

τ≡

−===

−=

)1( and

)1(

2

tio

tio evveEE ωω −− == and

Then, the solution of the Eq. (1.48) becomes


DCσ=

Case 1: : Most Common

No frequency dependence = constant conductivity

Electron motion depends more on collision than EM wave frequency

To the electron, the EM wave looks like DC.

Collision frequency is much higher than EM wave frequency

DCAC σστω ≈<< 1

ωτ >> 1 sec 10 10 1512 −−≈τ( ~ )

( ) textof (1.51) 1

1τωεω

σεµω

ii

k DC

−+=

The complete dispersion equation is obtained by subsituting the previous equation into Eq. (1.38) of text.

General Dispersion Equation


Plasma Frequency: Natural Oscillation

Case 2: : Pure Imaginary

and J are 90 degree out of phase with E .

No Power Absorption Electron gas is transparent to EM wave.

Then, Eq. (1.51) becomes

( ) τωστωσστω DCDCAC ii =−=>> 1

σ

τεσ

εω

ω

ωεµω

ωεεµω

τωσ

ωµεµω

=≡

−=

−=+=

mqN

mqNi

ik

ep

peDC

22

2

2

22

2222

where

11

!number! real

frequency plasma

2 =

≡

k

pω


Case 2-1: : k is real !!

Then, the EM wave in a conductor such as metal or semiconductor propagates as in a

good dielectric. Transmission or Metal is transparent to light !!

and 1 pωωτω >>>

( )

materials magnetic-nonfor offactor by the

and , , where

),(

Kcv

kkK

c

K

ckv

eEtxEE

oor

tkxio

<⇒

>=≅==

== −

ωωµ

ω

ω


Incident EM wave is partly reflected and partly transmitted, but no absorption.

Energy in the incident wave = Energy in the transmitted wave + Energy in the reflected wave


Case 2-2: : k is pure imaginary number !!

Then, the EM wave in a conductor such as metal or semiconductor decays

exponentially, BUT No Absorption !!

1 ùùô p<<<

( )[ ] ( )[ ] ( )tieEtxiiEtxkiEtxEE xooo ωωβω β −=−=−== − expexpexp),(


Incident EM wave is totally reflected if conductor is thick, or partly reflected and partly transmitted if conductor is thin.


Cyclotron Resonance

- Electrons driven in a DC magnetic field (Bo) by an AC electric field.

- Used to measure “effective mass” of an electron in a solid

rvm

BvqF o

2*==

r

velectron

Bo

co f

mBq

rv

vrT

==

===

*2

2211

Frequency

π

ππ

*mBq o

c =ω

Cyclotron Resonance

(source: Fig. 1-7 of text)

chapter 1. the electron as a particleyu.ac.kr/~hyulee/ssp_lecture/chap1.pdf · the principle of...

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