Principles of Semiconductor Devices-L3

8/8/2019 Principles of Semiconductor Devices-L3

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Alam ECE-606 S09 1

www.nanohub.org

NCN

EE-606: Solid State DevicesLecture 3: Elements of Quantum Mechanics

Muhammad Ashraful [email protected]



Outline

Alam ECE-606 S09 2

1) Why do we need quantum physics

2) Quantum concepts

3) Formulation of quantum mechanics

4) Conclusions

Reference: Vol. 6, Ch. 1 (pages 23-32)



Do I really need Quantum Mechanics ?

Alam ECE-606 S09 3

OriginalProblem

PeriodicStructure

Electrons in periodic

potential: Problemwe want to solve

If it were large objects, like a skier skiing past a set of obstacles,

Newton’s mechanics would work fine, but in a micro-world ……



Carrier Density

Alam ECE-606 S09 4

Carrier number = Number of states x filling factor

Chapters 2-3 Chapter 4

Total number of occupants= Number of apartments

X The fraction occupied



Outline

Alam ECE-606 S09 5


2) Quantum concepts

3) Formulation of quantum mechanics

4) Conclusions



Four Quantum Concepts ..

Alam ECE-606 S09 6

• Blackbody Radiation

• Photoelectric Effect

• Bohr Atom

• Wave Particle Duality


http://slidepdf.com/reader/full/principles-of-semiconductor-devices-l3 7/19Alam ECE-606 S087

(1) black-body radiation

300K

1000K

2000K



Rayleigh-Jeans Formula

4( , ) / B

u T k T λ λ ∝

log( ) 4 log( ) log( )u T λ = − +

Wein’s Formula /

5e

T

u

β λ

λ

−

∝

( u )

2000K

( ) 5 /

1 1,

1

T u T

e

β λ λ

λ

⎡ ⎤∝ ⎢ ⎥

−⎣ ⎦

Plank’s fitting formula



Interpretation of Plank’s Formula

Alam ECE-606 S09 9

(( ), ,) uu T df

T d

f λ λ

=

EM emission occurs in discrete quanta of

E hf = n=1,2,……..N

Energyof mode

nos. ofmodes

OccupationProbability

2

/

1~e 1hf kT

hf f ⎛ ⎞× × ⎜ ⎟−⎝ ⎠

5 /

1 1~

1T

d c

e df f β λ

λ λ

λ

⎡ ⎤=⎢ ⎥−⎣ ⎦



Recent Example: COBE Data

Alam ECE-606 S09 10

Show that the cosmic background temperature is

approximately 3K. Can you “see” this radiation?

J.C. Mather, Astrophysics J., 1990.





Origin of Quantization

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, 2 2

1 1m n E const

m n

⎛ ⎞= × −⎜ ⎟

⎝ ⎠

frequency



(3) Bohr Atom ..

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Assume that angular momentum is quantized:

0 / n

n m r υ =



(3) Bohr Atom (continued) …

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, 2 2

1 1

m n E const

m n

⎛ ⎞= × −

⎜ ⎟⎝ ⎠



(4) Wave-Particle Duality

Alam ECE-606 S09 15

2 4 2 20 E m c p c= +

pchf =

) / 2(because

)(because / /

λ π

λ λ

==

===

k k

f chchf p

Photons act both as wave and particle, what about electrons ?

m0=0 (photon rest mass)



Outline

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2) Quantum concepts

3) Formulation of Schrodinger Equation

4) Conclusions



Schrodinger Equation for electrons

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2 2

0 0( / 2 ) E m c V p m− = +

2 2

0

( / 2 )V k mω = + hf =

2 4 2 2 2 2 2 2 4

0 0 01 / 2 ... E m c p c m c p c m c⎡ ⎤= + ≈ + +⎣ ⎦



Schrodinger Equation (continued)

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))(exp(),( kxt i At x −−=Ψ ω

2 20( / 2 )k m V ω = +

2 2 2 / and / d dt i d dx k ω Ψ = − Ψ Ψ = − Ψ

2 2

2

02

d d i V

dt m dx

⎛ ⎞Ψ Ψ= − + Ψ⎜ ⎟

⎝ ⎠

Assume,



Conclusions

Alam ECE-606 S09 19

1. Given chemical composition and atomic arrangements,we can compute electron density by using quantum mechanics.

2. We discussed the origin of quantum mechanics – experiments were inconsistent with the classical theory.

3. We saw how Schrodinger equation can ariseas a consequence of quantization and relativity,but this is not a derivation .

4. We will solve some toy problems in the next classto get a feeling of how to use quantum mechanics.

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Principles of Semiconductor Devices-L3