From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Directionof Propagation

y

x

E y

Bz

x

Velocity = c

z

Fig. 3.1: The classical view of light as an electromagnetic wave. Anelectromagnetic wave is a travelling wave which has time varyingelectric and magnetic fields which are perpendicular to each otherand to the direction of propagation.

Chap. 3. Elementary Quantum Physics3.1 Photons

- Light: e.m "waves" - interference, diffraction, refraction, reflection

with

The instantaneous intensity (energy flow per unit area per second)

→

*Light: a stream of discrete energy packets (photons: "particles" of zero rest-mass), each carrying energy and momentum .

- Young's interference experiment:

Path difference = for constructive interference

= for destructive interference

- Bragg's law: x-ray beam from "a single crystal" (diffracted patterns of "spots") or "a polycrystalline material, powered crystal" (diffracted patterns of bright "rings" - no unique orientation of crystal axes)

Existence of a diffracted beam:

X-rays with singlewavelength

Powdered crystal orpolycrystalline material

Scattered X-rays

Photographic film

X-rays with allwavelengths

Single crystal

Photographic film

Scattered X-rays

(b)(a)

X-rays

θ

dsinθ dsinθ

12

d

d

Atomic planes

θ

Crystal

A

B

Detector1

2

(c)

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Fig. 3.3: Diffraction patterns obtained by passing X-rays through crystalscan only be explained by using ideas based on the interference of waves.(a) Diffraction of X-rays from a single crystal gives a diffraction patternof bright spots on a photographic film. (b) Diffraction of X-rays from apowdered crystalline material or a polycrystalline material gives adiffraction pattern of bright rings on a photographic film. (c) X-raydiffraction involves constructive interference of waves being "reflected"by various atomic planes in the crystal.

From P rincip les o f E lectron ic M ateria ls and D ev ices, S econd E dition , S .O . K asap (© M cG raw -H ill, 2002)http:/ /M a te ria ls.U sask .C a

KEm

0

-Φ3

Cs

υ03

-Φ2

υ02

K

-Φ1

υ01

W

υ

slope = h

Fig. 3 .6: T he effect of varying the frequency of light and thecathode m aterial in the photoelectric experim ent. T he lines for thedifferent m aterials have the sam e slope of h but different in tercepts.

3.1.1. The Photoelectric Effect

For an incident light with onto a metal surface, the electrons will be emitted (the current I is generated).

where Vo = the negative anode voltage at which the current I extinguishes. where h = Plank's constant.- The work function: →

Fig. 3.9: Scattering of an x-ray photon by a "free" electron in aconductor.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

c φ

θ

Recoiling electron

ElectronX-ray photon

Scattered photon

υ, λ

y

x c

υ', λ'

3.1.2. Compton Scattering

- X-ray scattering by an electron: →′′ KE of the elelctron = ′ momentum of the photon

From Principles of E lectronic Materials and Devices, Second Edition , S .O . Kasap (© McGraw-Hill, 2002)http://Materia ls.U sask.Ca

Collimator

Source ofmonochromaticX-rays

X-ray beamUnscattered x-rays

X-ray spectrometer

θ

Path of the spectrometer

λ'

λ0λ0

(a) A schematic diagram of the Compton experiment.

λ0

Intensity ofX-rays

λ

θ = 0°Primary beam

Intensity ofX-rays

λ

θ = 90°

λ'λ0

Intensity ofx-rays

λ

θ = 135°

λ0 λ'

(b) Results from the Compton experiment

Fig. 3.10. The Compton experiment and its results.

The scattered x-rays are detected at various angles with respect to the original direction ( ), their wavelength ′ is measured.

Photon energy: Photon momentum: where

Iλ

λ ( µm)1 2 3 4 50

3000 K

2500 K

Classical theory

Planck's radiation law

Hot body

Escaping black bodyradiation

Small hole acts as a black body

From Principles of Electronic Materials and Devices, Second Edition , S.O. Kasap (© McGraw-H ill, 2002)http://Materia ls.U sask.Ca

Fig. 3.11. Schematic illustration of black body radiation and itscharacteristics. Spectral irradiance vs wavelength at two temperatures(3000K is about the temperature of the incandescent tungstenfilament in a light bulb).

Spe

ctra

l irra

dian

ce

3.1.3, Blackbody Radiation

- Rayleigh-Jeans law (classical): ∝ and ∝

where the spectral irradiance = the emitted radiation intensity (power per unit area) per unit wavelength, so that = the intensity in a small range of wavelength - UV catastrophe in the short wavelength range

- Plank's blackbody radiation formula: → "Classical" → "Quantum mechanically": light quanta = photon

3.2. The Electron as a Wave

3.2.1. De Broglie Relationship

- Electron: a wave of wavelength (wave-like & particle-like)

- Electron diffraction experiments:

where ("real" particle)

3.2.2. Time-Independent Schroedinger Equation

- A travelling em wave: where = the spatial dependence.

- The average intensity

- 1926, Max Born : a probability wave interpretation for "the wave-like behavior of the electron"

: a plane traveling wavefunction for an electric field →

x

ψ(x) ψ(x) not continuous

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

not continuousdψdx

x

ψ(x)

ψ(x) not single valued

x

ψ(x)

Fig. 3.14: Unacceptable forms of ψ(x)

- The wave property of the electron described by :

1) = the probability of finding the electron per unit vol. at (x,y,z) at time t. or = the probability in a small vol. dxdydz. 2) has physical meaning, not itself. 3) : single-valued (See Fig. 3.14)

4) : continuous (See Fig. 3.14)

- Total wavefunction where , the angular frequency.

- Time-independent Schroedinger equation for = the spatial dependence

;

in 3-dim. space.

3.3. Infinite Potential Well: A Confined Electron

- For a certain region, , an electron is confined.

→

Using the b.c. of ,

The energy of the electron: →

x = 0 x = a0E1

E3

E2

E4

n = 1

n = 2

n = 3

n = 4

Energy levels in the well

ψ1

ψ2

ψ3

ψ4ψ(x) ∝ sin(nπx/a) Probability density ∝ |ψ(x)|2

0 a a0

0 ax

V(x)

0

∞

V = 0

Electron

V = ∞ V = ∞

x

Fig. 3.15: Electron in a one-dimensional infinite PE well. The energyof the electron is quantized. Possible wavefunctions and theprobability distributions for the electron are shown.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

Ener

gy o

f el

ectro

n

Normalization condition (A) :

The normalized wavefunction

- Quantized energy levels:

→ →

(free electron case: continuous)

3.4. Heisenberg's Uncertainty Principle

- For an electron trapped in a 1-dim infinite PE well in the region , The uncertainty in position = a, the uncertainty in momentum = For n = 1 (ground state), →

- Heisenberg's uncertainty principle: ≥ and ≥

I II III

A

B

C

D

E

Start here from rest

(a)V(x)

Vo

x = 0 x = a

E < VoψΙ(x)

ψΙΙ(x)ψΙΙΙ(x)

Incident

Reflected Transmitted

A1

A2

x(b)Fig. 3.16

(a) The roller coaster released from A can at most make to C, but not to E. Its PE at A isless than the PE at D. When the car is the bottom its energy is totally KE. CD is the energybarrier which prevents the car making to E. In quantum theory, on the other hand, there isa chance that the car could tunnel (leak) through the potential energy barrier between Cand E and emerge on the other side of the hill at E .

(b) The wavefunction of the electron incident on a potential energy barrier (Vo). Theincident and reflected waves interfere to give ψI(x). There is no reflected wave in regionIII. In region II the wavefunction decays with x because E < Vo.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

3.5. Tunneling Phenomena: Quantum Leak: Finite PE Well

- Three regions: I, II, and III (boundary conditions, C2 = 0, normalization)

where

- Transmission coefficient T : the relative probability that the electron will tunnel from I to III.

→

where

- For a "wide" or "high" barrier, using ≫ ≈

≈ where

x

V(x)

Metal

ψ(x) Second MetalVacuum

Vo

(b)x

V(x)

Metal

ψ(x)Vacuum

Vo

E < Vo

(a)

Materialsurface

Probe ScanItunnel

x

Itunnel

Image of surface (schematic sketch)

(c)Fig. 3.17: (a) The wavefunction decays exponentially as we move away from thesurface because the PE outside the metal is Vo and the energy of the electron, E < Vo..(b) If we bring a second metal close to the first metal, then the wavefunction canpenetrate into the second metal. The electron can tunnel from the first metal to thesecond. (c) The principle of the Scanning Tunneling Microscope. The tunneling currentdepends on exp(-αa) where a is the distance of the probe from the surface of thematerial and α is a constant.From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

3.6. Potential Box (3-Dim. Quantum Numbers)

with V= 0 in 0<x<a, 0<y<b, and 0<z<c.

Let , then

Using b.c,

with = the quantum numbers.- The eigenfunctions of the electron:

- The energy eigenvalues:

→

For a=b=c,

3.7. Hydrogen Atom

3.7.1. Electron Wavefunctions

- Potential: ; the wavefunction:

- Principle quantum number: n = 1,2,3,.... Orbital angular momentum quantum number: l = 0,1,2,3......(n-1) < n Magnetic quantum number: ml = -l, -(l-1),.....0,.....(l-1), l- Labeling of various n l possibilities : n=1 (K), L, M, N ; l=0 (s), l=1 (p), l=2 (d), l=3 (f),...

- The probability that the electron is in the spherical shell of thickness :

3.7.2. Quantized Electron Energy

- The electron energy:

where

×

Energy

0

1s1

2s 2p2

3s 3p3 3d

4s 4p4 4d 4f

5s 5p5 5d 5f

l = 0 l = 1 l = 2 l = 3

-13.6eV

n l

Photon

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materia ls.U sask.Ca

Fig. 3.27: An illustration of the allowed photon emission processes.Photon emission involves ∆l = ±1.

3.7.3. Orbital Angular Momentum and Space Quantization

- Orbital angular momentum: where where ≤ states- Selection rules for EM radiation: ± ±

Fig. 3.29: (a) The orbitting electron is equivalent to a current loopwhich behaves like a bar of magnet.(b) The spinning electron ican be imagined to be equivalent to acurrent loop as shown. This current loop behaves like a bar ofmagnet just as in orbital case.

S

Equivalent current

Spin direction

S

N

=

Magnetic momentµspin

N

SB

B

iA ===

ω-e

µorbital

(a)

(b)

From Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap (© McGraw-Hill, 2002)http://Materials.Usask.Ca

3.7.4. Electron Spin and Intrinsic Angular Momentum S

- Spin: ±

- Magnetic dipole moment of the electron:

Since ,

L

S

J

L

S

J

z

Jz=mjh

B

(a) (b)

Fig. 3.31 (a) The angular momentum vectors L and S precessaround their resultant total angular momentum vector J.(b) The total angular momentum vector is space quantized.Vector J precesses about the z-axis along which its componentmust be mjh.

3.7.5. Total Angular Momentum J

- Total angular momentum: J = L + S

[Reading Assignment]

3.8. The He Atom and The Periodic Table

3.9. Stimulated Emission and Lasers

3.10. Time-Dependent Schroedinger Equation

[Homework]

Prob. #3.5