estado solido

1. 6
Free electron bands for fcc structure
G – 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)

2. 3
Free Electron Bands
Energy
k
†
Ek = Ek k + G( )
1D : E =
h2
2m
k +
n2p
a
Ê
Ë
Á
ˆ
¯
˜
2
; n = 0, ±1, ± 2 ...
n=1
n=-2
n=-1
n=0
Brillouin zones - 2D
BZ construction
•reciprocal lattice
•bisect vectors to the nearest neighbors
•area defined by bisecting lines
represents 1BZ

3. Zonas de brillouin en red cuadrada

4. Zonas de brillouin (red cúbica simple)
1 2 3

5. Zonas de brillouin (red bcc)
1 2 3

6. Zonas de brillouin (red fcc)
1 2 3

7. Zonas de Brillouin y Superficie de Fermi
* Red cuadrada
* Aprox. e libres
(densidad arbitraria)
Ejemplo:
Esquema de zona extendida

8. Segunda zona de BrillouinEsquema de zona repetida:

9. Tercera zona de BrillouinEsquema de zona repetida:

10. Cuarta zona de BrillouinEsquema de zona repetida:

11. Superficie de Fermi en esquema de zona reducida
1 2 3 4

12. The Fermi surface of
alkali 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) and
the Fermi sphere

13. The Fermi surface of copper (similar for Ag and Au)
Band structure
(Cu is a
monovalent metal
with fcc lattice)
Fermi surface
(a cross-section)
Periodic zone scheme

14. The Fermi surface of Al
(Al is a trivalent metal
with fcc lattice)
Empty lattice result
Actual Fermi surface

15. 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-

16. 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 Stanton
Physics Department, University of Florida
Seagate Technology
Jian 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