Understanding Conductivity in Solids: Metals, Semiconductors, and Insulators - Prof. Paul , Study notes of Quantum Physics

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Lecture 14: Quantum Consequences:

Effects on our everyday lives

The big picture of science,technology,

and the universe

Overview^ Overview

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Why do some solids conduct – others do not – others are intermediate

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Metals, Insulators and Semiconductors – understood with quantum mechanics!

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Understood in terms of Energy Bands and the Exclusion Principle

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Metallic conduction - a purely quantum phenomenon

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Solid-state semiconductor devices

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The electronic states in semiconductors - a purely quantum phenomenon

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Transistors, lasers,...

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The laser

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Bosons and Fermions

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Stimulated emission of photons – a purely quantum phenomenon

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Superconductivity

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Electrical conduction with zero resistance! – a purely quantum phenomenon

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All the electrons in a metal cooperate to form a single quantum state

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Supplement:

FYI

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Quantum mechanics in the universe

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The smallest particles to neutron stars!

Electron in a crystal – periodic array of atoms

Bands and Band gaps occur for all types^ Bands and Band gaps occur for all types

of waves in periodic systemsof waves in periodic systems

Bands of “allowed” energiesfor electrons

Bands Gap – range of energy wherethere are no “allowed states”

Light propagating through a periodic set of layers withdifferent index of refraction – an interference filter

Photon

Transmitted?

Reflected?

Layers with alternating index n

1

and n

2

Phys 214 – Lecture 14, Slide 7

Light propagating through a periodic layers^ Light propagating through a periodic layers

Demonstration of Bands and Band gapsDemonstration of Bands and Band gaps

Light propagating through a periodic set of layers withdifferent index of refraction – an interference filter

Photon

Transmitted?

Reflected?

Layers with alternating index n

1

and n

2

Demonstration with a commercial interference filter

Interference filters do not absorb light (do not get hot) and are used in many applications •

Used for color separation, e.g., color TV, color photography,...

Examples: Red, Green, Blue ----- Yellow, Cyan, Magenta

What is magenta?

Answer:

Magenta =

Red + Blue (no green)

A magenta filter must have a stop band in the green, but pass red and blue

Electron in a crystal – periodic array of atoms

Bands and Band Gaps occur for electrons in^ Bands and Band Gaps occur for electrons in

crystals because electrons act like waves!crystals because electrons act like waves!

Not expected for particle in classical mechanicsNot expected for particle in classical mechanics

Can be understood only in quantum mechanicsCan be understood only in quantum mechanics

Bands of “allowed” energiesfor electrons

Bands Gap – range of energy wherethere are no “allowed states”

ψ

2 ψ

5

Examples

of electron

waves

Band gaps are

“stop bands” for

electrons with

spacing between atoms

Semi-^ Semi

-classical Picture of Conduction

classical Picture of Conduction

n =

free electrons/volume

time between

scattering events

nev

J

drift

eE m

F m

a

v

drift

ty

conductivi

m

ne

E

E

m

ne

J

2

2

J =

current density = I/A

F = force = -eEa =

acceleration = F/m

e

Wire withcross section A

E

Metal:

scattering time gets

shorter with increasing T

Temperature, T

Resistivity

τ

σ

ρ

2

ne

m

Why do some solids conduct^ Why do some solids conduct

current and others don’current and others don

’t?

t?

How do we understand why someHow do we understand why some

materials are metals, somematerials are metals, some

semiconductors, some insulators?semiconductors, some insulators?

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The answer is provided by quantum mechanicsin terms of:

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1. Energy Bands for electrons in solids, which can beunderstood in terms of the wave nature of theelectrons (review from last lecture)

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2. The Pauli exclusion principle, which applies toelectrons (and all fermions)

Insulators, Semiconductors^ Insulators, Semiconductors

and Metalsand Metals

Energy bands and the gaps between them determine the

conductivity and other properties of solids.

E insulators

metals

semi-

conductors

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Insulators

have a valence band which is

full and a large energy gap (few eV)

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Apply an electric field - no states ofhigher energy available for electron

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Metals

have an upper band which is partly

full so that electrons are free to move

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Apply an electric field – empty statesare available for electrons to move

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Semiconductors

are insulators at T = 0.

They have a small energy gap (~1 eV)between valence and conduction bands,so they become conducting at higher T.

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The conductivity can be controlled byapplied voltages (Field Effect Transistor)or by adding foreign atoms (“doping”)

Phys 214 – Lecture 14, Slide 17

Electronic Conduction in Metals^ Electronic Conduction in Metals

--example: Na--

example: Na

Z = 11

1s

2

2s

2

2p

6

3s

1

N states4N statesN states

1s 2s, 2p

3s

2N electrons fillthese states.

8N electrons fillthese states.

Total # atoms = N

Total # electrons = 11N

Fill the Bloch statesaccording to Pauli Principle:

The 3s band is only halffilled (N orbital statesand N electrons)

These electrons are easilypromoted to higher statesin the band. Therefore,Na is a good conductor.

Partially filled band

good conductor

Phys 214 – Lecture 14, Slide 19

Z = 14

1s

2

2s

2

2p

6

3s

2

3p

2

4N states4N statesN states

1s 3s, 3p2s, 2p

2N electrons fillthese states.

8N electrons fillthese states.

Total # atoms = NTotal # electrons = 14N

Fill the Bloch statesaccording to Pauli Principle

It appears that, like Na,Si will also have a halffilled band: The 3s3pband has 4N orbitalstates and 4N electrons.

By this analysis, Si should bea good metal, just like Na.But something specialhappens for Group IVelements.

What about semiconductors like silicon?^ What about semiconductors like silicon?