6

Free electron bands for fcc structureG – center of the BZ

X – [100] intercept; G - X path D

K – [110] intercept; G - K path S

L – [111] intercept; G - L path L

Band structure of Al (fcc)

3

Free Electron Bands

Ene

rgy

k

†

Ek = Ek k + G( )

1D : E =h2

2mk +

n2pa

Ê

Ë Á

ˆ

¯ ˜

2

; n = 0, ±1, ± 2 ...n=1

n=-2

n=-1

n=0

Brillouin zones - 2DBZ construction•reciprocal lattice

•bisect vectors to the nearest neighbors

•area defined by bisecting linesrepresents 1BZ

Zonas de brillouin en red cuadrada

Zonas de brillouin (red cúbica simple)

1 2 3

Zonas de brillouin (red bcc)

1 2 3

Zonas de brillouin (red fcc)

1 2 3

Zonas de Brillouin y Superficie de Fermi

* Red cuadrada* Aprox. e libres

(densidad arbitraria)

Ejemplo:

Esquema de zona extendida

Segunda zona de BrillouinEsquema de zona repetida:

Tercera zona de BrillouinEsquema de zona repetida:

Cuarta zona de BrillouinEsquema de zona repetida:

Superficie de Fermi en esquema de zona reducida

1 2 3 4

The Fermi surface ofalkali metals

kF = (3p2n)1/3,n = 2/a3

à kF = (3/4p)1/3(2p/a)GN=(2p/a)[(1/2)2+(1/2)2]1/2

\ kF = 0.877 GN

First BZ (of a bcc lattice) andthe Fermi sphere

The Fermi surface of copper (similar for Ag and Au)

Band structure

(Cu is amonovalent metalwith fcc lattice)

Fermi surface

(a cross-section)

Periodic zone scheme

The Fermi surface of Al

(Al is a trivalent metalwith fcc lattice)

Empty lattice result

Actual Fermi surface

11

Distorting the Fermi Surface

2nd zone 3rd zone 3rd zone in periodic zone scheme

Distortion due to weak periodic potential

2nd zone 3rd zone

3D Fermi Surface

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

Na Ca Al

1 valence e- 2 valence e- 3 valence e-

Periodic Table of the Fermi Surfaces of Elemental Solids http://www.phys.ufl.edu/fermisurface

Ferromagnets:

Alternate Structures :

Tat-Sang Choy, Jeffery Naset , Selman Hershfield, and Christopher StantonPhysics Department, University of Florida

Seagate TechnologyJian Chen

Source of tight binding parameters (except for fcc Co ferromagnet): D.A. Papaconstantopoulos, Handbook of the band structure of elemental solids, Plenum 1986.This work is supported by NSF, AFOSR, Research Corporation, and a Sun Microsystems Academic Equipment Grant.

(15 March, 2000)

Co_fcc Co_fcc