MIT OpenCourseWare http://ocw.mit.edu

3.23 Electrical, Optical, and Magnetic Properties of MaterialsFall 2007

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

http://ocw.mit.edu/termshttp://ocw.mit.edu

3.23 Fall 2007 – Lecture 9

BAND STRUCTURE

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

Last time

1. Hamiltonian in a periodic potential, translation operators

2 BlBloch’s ththeorem (f(from common eiigensttattes of H, TTR))2. h’ f H

3. n, k quantum numbers

4. Born-von Karman boundary conditions

5. Explicity proof of Bloch’s theorem

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

1

•

Study

• ChapChap. 3 Singleton3 Singleton

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

Bloch Theorem (in two equiv forms)

rr r r r r rΨ =r ik r exp u( ) ( ) ( )nk nk

rr r r r

r rΨ + = exp ( ik R )( ) ( ) Ψ ( )rr Rnk nk

2

Hamiltonian in the Bloch

representation

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

Explicit solution for the Bloch orbitals

⎛ ⎞h2 −− +⎜ ⎟( )q G′

2

E Cq G− −′ ′∑VG G′ ′Cq G− ′′ = 0⎜ ⎟2m⎝ ⎠ G′′

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

3

⎛ ⎞

Explicit solution for the Bloch orbitals

⎛ ⎞⎜ ⎟

( )q G′ − +′ ′∑V ′h2 − 2

E Cq G G− −′ G Cq G− ′′ = 0⎜ ⎟ ′′2m⎝ ⎠ G ⎝ ⎠

h2 2⎛ ⎞( )− 2 V V V⎜ ⎟ ⎜ ⎟

q G G G2 3G V 42m − − − − G

h2 ⎛ ⎞⎜ ⎟2 C C⎛ ⎞V q( )− V 2 2q GG V V q G− −G G 2 3− G⎜ ⎟2m − − G ⎜ ⎟ ⎜ ⎟⎜ ⎟ C C 2 ⎜ ⎟q G ⎜ q G ⎟− − ⎜ ⎟h 2 ⎜ ⎟ ⎜ ⎟ ) q q = EV V ((q) V V C C2G G − −G 2G⎜ ⎟⎜ ⎟2 ⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎟2m⎜ ⎟ C C

2 ⎜ ⎟q G ⎜ q G ⎟⎜ ⎟+ +

⎟h 2V V V q + ⎜ ⎟( )G V C C⎜⎜ ⎟3 2 G 2m −G + +2 2⎝ ⎠G G q G G⎝ ⎠ q⎜ ⎟ ⎜ ⎟V V V V h

2

( )q + 2G 2⎜ ⎟4 3 2 GG G G 2m⎝ ⎠

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

Free, free, set me free

h2 2⎛ ⎞− V V V⎜ ⎟( )q G G G2 3 V− G2 − − − G 42m⎜ ⎟ h2 2⎜ ⎟

⎜ ⎟ 2 2G GV V G C Cq G q G− −V q( )− G − −2 3V− G ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ 2 ⎜ ⎟ ⎜ ⎟

2m C Cq G q G⎜ ⎟ − − ⎜ ⎟h 2 ⎜ ⎟C C= E ⎜ ⎟V V2 ( )q V G V q qG G − −2G⎜ ⎟⎜ ⎟ ⎜ ⎟2m⎜ ⎟ C C

2 ⎜ ⎟q G ⎜ q G ⎟⎜ ⎟+ +

⎜ ⎟⎜ ⎟ ⎜⎜ ⎟⎟ V V V G G + G) V 2 2GV V V q G C C 3 2 G h ((q + G)2 V− C Cq G+ +q⎜ ⎟⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎜ ⎟2m⎜ ⎟2⎜ ⎟V V V V q + 2G⎜ ⎟4 3G G 2G G 2

h

m( )2

⎝ ⎠

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

4

Free electron dispersions, 1-d

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

Band Structures: Free Electron Gas, Silicon

5

Figure by MIT OpenCourseWare.

Band Structures: Free Electron Gas, Silicon

Band Structure of Diamond

6

Figure by MIT OpenCourseWare.

Courtesy L. J. Terminello and F. J. Himpsel. Used with permission.

Band Edge h2 2⎛ ⎞( )− 2 V V V ⎜ ⎟

⎜ ⎟2mq G G G2 3G V−4G− − −

h2 2⎜ ⎟ C C⎜ ⎟

C CG G ⎛ ⎞⎜ ⎟V GG V V ) V V 2 3V ⎛ ⎞⎛ ⎞q 2 22 2G ⎛ ⎞ q G− −VG G ((q − ) − − G V − G2m ⎜ ⎟ ⎜ ⎟⎜ ⎟ C Cq G q G− −2 ⎜ ⎟ ⎜ ⎟⎜ ⎟h 2 C C⎜ ⎟V ⎜ ⎟ = EV V ( )q V q q2G G G 22m − − G ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ C C2 q G+ +⎜ q G⎜ ⎟ ⎟h 2 ⎜ ⎟ ⎜⎜ ⎟V V V q + G V C C ⎟ 3 2G G G ( ) q G q G2m −G + +2 2⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎜ ⎟

⎜ ⎟V V V V h2

( )q + 2G 2 ⎜ ⎟4 3 2GG G G 2m⎝ ⎠

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

Band Edge

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

7

Ψnk (r) is not a momentum eigenstate

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

Group velocity

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

8

Effective mass

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

Fermi energy

-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_

Copper

9

( )2

( )r f rn k, ,n kn k,

rρ = Ψ∑ r rrr

Figure by MIT OpenCourseWare.

The Fermi surface

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

The Fermi surface

http://www phys ufl edu/fermisurface/http://www.phys.ufl.edu/fermisurface/

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

10

Images from the Fermi Surface Database. Used with permission.Please see: http://www.phys.ufl.edu/fermisurface/jpg/K.jpg,http://www.phys.ufl.edu/fermisurface/jpg/Cu.jpg.

http://www.phys.ufl.edu/fermisurface/jpg/K.jpghttp://www.phys.ufl.edu/fermisurface/jpg/Cu.jpghttp://www.phys.ufl.edu/fermisurface/