IMPRS band structure pdf.ppt - Max Planck · PDF file1.4 Sommerfeld model in 3D free electrons...
Transcript of IMPRS band structure pdf.ppt - Max Planck · PDF file1.4 Sommerfeld model in 3D free electrons...
IMPRS Block Course, 22.3. – 26.3.2013
Band structure of solids - „a physicist point of view“
Martin WolfFritz-Haber-Institut of the Max-Planck-Society, Berlin, Germany
1. Electrons in solids: the free electron gas
2 Electronic band structure of periodic crystals2. Electronic band structure of periodic crystals
3. Examples & angle-resolved photoemission
4. Surface states
2Solid state physics - concepts
Structure of solid state materials
Introduction
Structure of solid state materials
- model system: single crystals with periodic boundary conditions - but defects play important role doping in semiconductors, optical materials - alloys and amorphous solids technological relevance (steel, glass, ...)
Theoretical description of solids
Schrödinger equation for electrons and nuclei
electron density of Si
- Schrödinger equation for electrons and nuclei
+ Si+ SiBUT: how for ~1023 atoms / cm3 ! ? Born-Oppenheimer approximation
- Maxwell equations for electromagnetic fields
- Statistical physics and thermodynamics
+ Si Density functional theory (DFT): Hohnberg+Kohn: „All physical observablesare uniquely described by (electron) density (r)”
→ distribution functions for electrons (Fermions) and phonons (Bosons)
Solids are many body systemsq y y ( ) y ( ) ab initio calculation of solid state properties
Solids are many body systems
- electron-phonon interaction - exchange and correlation effects
useful concept: single particle picture
EFermi
occupied- useful concept: single particle picture- however simple models (e.g. Drude model)
describe often already essential physics
occupiedstates
41.1 Electrons in solids: Sommerfeld model free electrons in solids
Effective one-electron potential: (rigid lattice)
V - Born-Oppenheimer approximation- Single electron approximation
Sommerfeld model:particle in a box“„particle in a box
71.4 Sommerfeld model in 3D free electrons in solids
Sommerfeld model in 3DSommerfeld model in 3D
for periodic boundary conditions !
91.6 Fermi gas at T = 0 free electrons in solids
Fermi distribution: Density of states: Fermi sphere:
f (E T 0) EmED
23
22
22
1
Fermi distribution: Density of states: Fermi sphere:
22
2 FF km
E
f (E, T=0)
112.1 Electronic band structure of solids band structure
Effective one-electron potential:
B O h i i ti- Born-Oppenheimer approximation- Single electron approximation- Periodic potential
- Diffraction of electron waves deviation from free electron
b d t t (b d )
p
band structure (band gaps,…)
12Reminder: Lattice planes and Miller Indexes Diffraction
Lattice planes: 3 lattice vectors 321ian which do not fall ona straight line define a lattice plane, which is characterizedby 3 integers, the Miller indices hkl:
Lattice planes:33an 3 lattice vectors 3,2,1, iiian which do not fall on
1a2a
3a
11an
22an
Zlkhn
mln
mkn
mh ,,,1,1,1
321
The reciprocal lattice vector:
321 gggG lkhhkl Ghkl
is perpendicular to the lattice plane (hkl) and the distance b t dj t l ibetween adjacent planes is:
Gdhkl
2
Ghkl
dhkl
13Reminder: Diffraction and Bragg law Diffraction
2Gk hkl
hkl dk 2sin2 0 GThe Laue condition is equivalent to:
This yields the Bragg law for diffraction sin2 hkldhkl
Bragg reflection
k
Ghkl
Bragg reflectionat crystal planes hkl
k
k0dhkl
2k0 Ghkl
Bragg: Laue:
Diffraction Reminder: The reciprocal lattice
Zlkhlkh GZ3 RF d
Zmm ,2RG
Zlkhlkh ,,,321 gggG
ijji 2agthe condition requires:
Znan ii
ii 1
,RFor and
The resulting basis vectors of the reciprocal lattice are:
lkjlkj ,,aagand therefore
)(2,
)(2,
)(2
321
213
321
132
321
321 aaa
aagaaa
aagaaa
aag
Real space latticeExample: 2D - lattice Reciprocal lattice
units [m] units [m-1]
15Reminder: Wigner-Seitz cell Structure
1/21/2
Goal: primitive unit cell (with minimum volume)
Example: construction in 2 dimensions
bcc fcc
C4
C3
C4
bcc fccC3
C3
C2C2
The Wigner Seitz cell comprises all points closer to a certain lattice point
The Wigner-Seitz cell reflects point group symmetry of the lattice
The Wigner Seitz cell comprises all points closer to a certain lattice point than to any other point of the Bravais lattice
16Reminder: Brillouin zones Structure
bcc lattice fcc lattice hexagonal
The first Brillouin zone is defined as the Wigner Seitz cell of the reciprocal lattice
bcc lattice fcc lattice hexagonallattice
dirrez a
a 4
dirrez a
a 4
dirrez a
a34
dir
rez cc 4
Note: The reciprocal lattice of the bcc lattice is the fcc lattice and vice versa.
172.1 Electronic band structure of solids band structure
Effective one-electron potential:
B O h i i tiE(k) = ħk2/2m
- Born-Oppenheimer approximation- Single electron approximation- Periodic potential
- Bragg reflexion at Brillouin zoneboundary standing waves
p
- Opening of band gaps
202.3 Bloch Theorem band structure
U t ti S h ödi tiUse stationary Schrödinger equation:
for a periodic potential in a crystal: with lattice vector
This implies (non-degenerate wave functions):
Solution Bloch theorem:
The wave function in a periodic potential is given by aThe wave function in a periodic potential is given by aa periodic function uk(r) modulated by a plane wave:
212.4 Bloch Theorem band structure
Translational symmetryTranslational symmetry
Bloch-wave:
Symmetry properties:
222.5 Consequences of the Bloch Theorem band structure
Th Bl h th i ld f th f ti d Ei t t
)Gk()k( nn
The Bloch theorem yields for the wave functions and Eigenstates:
)Gk(E)k(E nn
where the index n counts the Eigenstates En of electrons in the periodic potential.B d t t b d d t th 1 B ill iBand structure can be reduced to the 1. Brillouin zone
Repeated zonescheme:
Extended zonescheme:
Reduced zonescheme:
G
232.6 Consequences of the Bloch Theorem band structure
ExtendedExtended zone sheme
-G
Reduced zone sheme o e s e e
252.9 Tight binding approximation band structure
“perturbation”HA i = Ei i
p
kk HkE
)(
i h 1
kk
kE
)(
with m = n ± 1
m = n ± 1
302.14 Tight binding approximation: Examples band structure
Tight binding Bloch wave:
Band structure of Aluminium using 1 s, 3 p and 5 d basis set (j = 1 . . . 9) for tight binding calc.
from: D. A. Papaconstantopoulos, Handbook of the band structure of elemental solids, Plenum Press (New York) 1986.
333.2 Photoemission spectroscopy band structure
verticaltransitiontransition
Determination of band structure:
Assume free electron final state:
374.1 Surface states surface states
B k t l ti l t t f l d t l ti f• Broken translational symmetry at surface can lead to solutions ofthe Schrödinger equation which are localized at the surface.
• Surface state exists in band gaps of the bulk projected band structured f tl di l ll l t b lk b d dand frequently disperse nearly parallel to bulk band edges
• Surface states do not disperse normal to the surface
6 Elid
F n=1
6
4
EvacCu(111)solid vacuum
E
E -
EF
n=02
E
sp-inverted Band gap
Surface t t
projected band gap
Evac
0
2
EFstate
0.0 0.5 1.0-2
k (Å )||-1 M
zsurface
384.1 Surface states surface states
B k t l ti l t t f l d t l ti f• Broken translational symmetry at surface can lead to solutions ofthe Schrödinger equation which are localized at the surface.
• Surface state exists in band gaps of the bulk projected band structured f tl di l ll l t b lk b d dand frequently disperse nearly parallel to bulk band edges
• Surface states do not disperse normal to the surface
Shockley state: 6 EShockley state:- contribution of s/p-orbitals dominate- delocalized parallel to surface- pronounced dispersion E
F n=14
EvacCu(111)
p p
E - n=0
0
2
EFTamm state- contribution of d-orbitals dominate (or pz)- more localized character
0.0 0.5 1.0-2
k (Å )||-1 M
more localized character- small dispersion
Surface resonance:- Resonant with bulk bands (& strong mixing)Resonant with bulk bands (& strong mixing)- Penetration into the bulk (weak localization at surface)
394.2 Surface states surface states
B k t l ti l t t f l d t l ti f• Broken translational symmetry at surface can lead to solutions ofthe Schrödinger equation which are localized at the surface.
• Example: two-band model for the Cu(111) surface
~e-z