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3.23 Electrical, Optical, and Magnetic Properties of Materials 2007

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Fall

3.23 Fall 2007 – Lecture 12

SEMICONDUCTORS

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Last time

1. Periodic potential: atomic + pertubation

2 BlBloch sums of l f localili zedd orbitbitalls (atomiic, or LCAO)2. h ( t LCAO)

3. Tight-binding formulation (in the case only one orbital has significant overlap)

4. From flat atomic “bands” to dispersive cosines

5. Bandwidths

6. Tigght-bindingg vs. emppirical ppseudoppotential ((i.e. a perturbation of the free electron gas)

7. Band structure (DETAILED) of a semiconductor

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

1

2

Ferroelectric perovskites

Image removed due to copyright restrictions. Please see: Fig. 3 inKing-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricityin Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Ferroelectric perovskites

Image removed due to copyright restrictions. Please see: Fig. 4 inKing-Smith, R. D., and David Vanderbilt. "First-principles Investigation of Ferroelectricityin Perovskite Compounds." Physical Review B 49 (March 1994): 5828-5844.

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Silicon Lead

Copper

Silver

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

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Image removed due to copyright restrictions.

http://www.bandstructure.jp/Table/BAND/band_png/pb4800b.ps.png

Image removed due to copyright restrictions.

http://www.bandstructure.jp/Table/BAND/band_png/ag39275a.ps.png

Please see any band gap diagram of lead, such as

Please see and band gap diagram of silver, such as

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

-0.939

-0.539

X W L

Γ12

Γ25'

Γ1

Γ Κ

∆1

Ζ3

Ζ2

Α1

Σ1

Σ2 K2

K3

K1K1

K4Σ4 Σ1Σ3

Σ1

Α1Α3

Α3

Α1

Ζ1Ζ3Ζ4

Ζ1

∆2

∆5∆2'

∆1

5 3

333 2'

2'

1 1

1

1'2

Q_

Q+

Q+

Q+

Q_

Q_

Figure by MIT OpenCourseWare.

Images removed due to copyright restrictions.Please see Fig. 2.24 in Yu, Peter Y., and Cardona, Manuel.Fundamentals of Semiconductors: Physics and Materials Properties.New York, NY: Springer, 2001.

Platinum Gold

3.23

Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Band structure of graphene

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

4

Courtesy Hongki Min. Used with permission.

Used with permission.

Band structure of graphene

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Carbon nanotubes

Image from Wikimedia Commons, http://commons.wikimedia.org

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

5

Images removed due to copyright restrictions. Please see: Fig. 2.4 and 2.6 inMinot, Ethan. "Tuning the Band Structure of Carbon Nanotubes." PhD dissertation, Cornell University, 2004.

•

Zone folding: Band structure of nanotubes

(8,0) semiconducting (5,5) metallic

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

The independent-electron gas

• HamiltonianHamiltonian

• Eigenvalues and eigenfunctions

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

6

KB

�

�

B

A

KB

Figure by MIT OpenCourseWare.

•

•

The independent-electron gas

• BvKBvK boundary conditionsboundary conditions

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

The independent-electron gas

• Counting the statesCounting the states

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

7

Image removed due to copyright restrictions.Please see any diagram of free electron band gaps, such as http://leung.uwaterloo.ca/CHEM/750/Lectures%202007/SSNT-5-Electronic%20Structure%20II_files/image008.jpg.

2 � /L x

2 � /L y

ky

kx

•

•

The independent-electron gas

• Particle densityParticle density

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

The independent-electron gas

• Energy densityEnergy density

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

8

∫

Density of states (for any solid)

r r 1 r r 1 g k( ) = −2∫ 3 (ε ε ( )) dε δ kn n8π

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Band structure of graphene

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

9

Courtesy Hongki Min. Used with permission.

Massive vs massless bands Dimensions d=1 d=2 d=3

Massless (E≈k) const E EE22Massless (E≈k) const E

Massive (E≈k2) 1/sqrt(E) const sqrt(E)

1 1g dS 8π ∇εn ( )kn ( )ε = 2∫ 3 r

• S goes as kd-1 , where d is the dimensionality , where d is the dimensionalityS goes as k

• 1 r for a band that has kl dispersions goes as k-(l-1),

( )∇ε k

• the integral goes as kd-l

• energy is proportional to kl, the integral goes as ε(d-l)/l

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Statistics of classical and quantum particles

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

10

Probability and Partition Function

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

Chemical potential

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

11

Fermi-Dirac distribution

3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)

12

Images from Wikimedia Commons, http://commons.wikimedia.org.